idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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10,401 | How many lags to use in the Ljung-Box test of a time series? | Let me suggest you our R package hwwntest. It has implemented Wavelet-based white noise tests that do not require any tuning parameters and have good statistical size and power.
Additionally, I have recently found "Thoughts on the Ljung-Box test" which is excellent discussion on the topic from Rob Hyndman.
Update: Cons... | How many lags to use in the Ljung-Box test of a time series? | Let me suggest you our R package hwwntest. It has implemented Wavelet-based white noise tests that do not require any tuning parameters and have good statistical size and power.
Additionally, I have r | How many lags to use in the Ljung-Box test of a time series?
Let me suggest you our R package hwwntest. It has implemented Wavelet-based white noise tests that do not require any tuning parameters and have good statistical size and power.
Additionally, I have recently found "Thoughts on the Ljung-Box test" which is exc... | How many lags to use in the Ljung-Box test of a time series?
Let me suggest you our R package hwwntest. It has implemented Wavelet-based white noise tests that do not require any tuning parameters and have good statistical size and power.
Additionally, I have r |
10,402 | Why break down the denominator in Bayes' Theorem? | The short answer to your question is, "most of the time we don't know what P(cheese) is, and it is often (relatively) difficult to calculate."
The longer answer why Bayes' Rule/Theorem is normally stated in the way that you wrote is because in Bayesian problems we have - sitting in our lap - a prior distribution (the ... | Why break down the denominator in Bayes' Theorem? | The short answer to your question is, "most of the time we don't know what P(cheese) is, and it is often (relatively) difficult to calculate."
The longer answer why Bayes' Rule/Theorem is normally st | Why break down the denominator in Bayes' Theorem?
The short answer to your question is, "most of the time we don't know what P(cheese) is, and it is often (relatively) difficult to calculate."
The longer answer why Bayes' Rule/Theorem is normally stated in the way that you wrote is because in Bayesian problems we have... | Why break down the denominator in Bayes' Theorem?
The short answer to your question is, "most of the time we don't know what P(cheese) is, and it is often (relatively) difficult to calculate."
The longer answer why Bayes' Rule/Theorem is normally st |
10,403 | Why break down the denominator in Bayes' Theorem? | One reason for using the total probability rule is that we often deal with the component probabilities in that expression and it's straightforward to find the marginal probability by simply plugging in the values. For an illustration of this, see the following example on Wikipedia:
Bayes' Theorem > Example 1: Drug Te... | Why break down the denominator in Bayes' Theorem? | One reason for using the total probability rule is that we often deal with the component probabilities in that expression and it's straightforward to find the marginal probability by simply plugging i | Why break down the denominator in Bayes' Theorem?
One reason for using the total probability rule is that we often deal with the component probabilities in that expression and it's straightforward to find the marginal probability by simply plugging in the values. For an illustration of this, see the following example ... | Why break down the denominator in Bayes' Theorem?
One reason for using the total probability rule is that we often deal with the component probabilities in that expression and it's straightforward to find the marginal probability by simply plugging i |
10,404 | Why break down the denominator in Bayes' Theorem? | Previous replies are detailed enough, but an intuitive way of looking why $P (A) $ (ie dinominator in the Bayes theorem) is broken into two cases.
It is hard to comment about what is the $P(A)$ without any knowledge whether the email is ham or spam. You are correct that "cheese" appears in spam as well as in ham, but i... | Why break down the denominator in Bayes' Theorem? | Previous replies are detailed enough, but an intuitive way of looking why $P (A) $ (ie dinominator in the Bayes theorem) is broken into two cases.
It is hard to comment about what is the $P(A)$ withou | Why break down the denominator in Bayes' Theorem?
Previous replies are detailed enough, but an intuitive way of looking why $P (A) $ (ie dinominator in the Bayes theorem) is broken into two cases.
It is hard to comment about what is the $P(A)$ without any knowledge whether the email is ham or spam. You are correct that... | Why break down the denominator in Bayes' Theorem?
Previous replies are detailed enough, but an intuitive way of looking why $P (A) $ (ie dinominator in the Bayes theorem) is broken into two cases.
It is hard to comment about what is the $P(A)$ withou |
10,405 | How LDA, a classification technique, also serves as dimensionality reduction technique like PCA | As I've noted in the comment to your question, discriminant analysis is a composite procedure with two distinct stages - dimensionality reduction (supervised) and classification stage. At dimensionality reduction we extract discriminant functions which replace the original explanatory variables. Then we classify (typic... | How LDA, a classification technique, also serves as dimensionality reduction technique like PCA | As I've noted in the comment to your question, discriminant analysis is a composite procedure with two distinct stages - dimensionality reduction (supervised) and classification stage. At dimensionali | How LDA, a classification technique, also serves as dimensionality reduction technique like PCA
As I've noted in the comment to your question, discriminant analysis is a composite procedure with two distinct stages - dimensionality reduction (supervised) and classification stage. At dimensionality reduction we extract ... | How LDA, a classification technique, also serves as dimensionality reduction technique like PCA
As I've noted in the comment to your question, discriminant analysis is a composite procedure with two distinct stages - dimensionality reduction (supervised) and classification stage. At dimensionali |
10,406 | Do we still need to do feature selection while using Regularization algorithms? | Feature selection sometimes improves the performance of regularized models, but in my experience it generally makes generalization performance worse. The reason for this is that the more choices we make regarding our model (including the values of the parameters, the choice of features, the setting of hyper-parameters... | Do we still need to do feature selection while using Regularization algorithms? | Feature selection sometimes improves the performance of regularized models, but in my experience it generally makes generalization performance worse. The reason for this is that the more choices we m | Do we still need to do feature selection while using Regularization algorithms?
Feature selection sometimes improves the performance of regularized models, but in my experience it generally makes generalization performance worse. The reason for this is that the more choices we make regarding our model (including the v... | Do we still need to do feature selection while using Regularization algorithms?
Feature selection sometimes improves the performance of regularized models, but in my experience it generally makes generalization performance worse. The reason for this is that the more choices we m |
10,407 | Do we still need to do feature selection while using Regularization algorithms? | A lot of people do think that regularization is enough to take care of extraneous variables and no variable selection is needed if you appropriately regularize, do partial pooling, create hierarchical models, etc. when the goal is predictive accuracy. For example, if a parameter estimate for a particular variable $j$ i... | Do we still need to do feature selection while using Regularization algorithms? | A lot of people do think that regularization is enough to take care of extraneous variables and no variable selection is needed if you appropriately regularize, do partial pooling, create hierarchical | Do we still need to do feature selection while using Regularization algorithms?
A lot of people do think that regularization is enough to take care of extraneous variables and no variable selection is needed if you appropriately regularize, do partial pooling, create hierarchical models, etc. when the goal is predictiv... | Do we still need to do feature selection while using Regularization algorithms?
A lot of people do think that regularization is enough to take care of extraneous variables and no variable selection is needed if you appropriately regularize, do partial pooling, create hierarchical |
10,408 | Do we still need to do feature selection while using Regularization algorithms? | I don't think overfitting is the reason that we need feature selection in the first place. In fact, overfitting is something that happens if we don't give our model enough data, and feature selection further reduces the amount of data that we pass our algorithm.
I would instead say that feature selection is needed as a... | Do we still need to do feature selection while using Regularization algorithms? | I don't think overfitting is the reason that we need feature selection in the first place. In fact, overfitting is something that happens if we don't give our model enough data, and feature selection | Do we still need to do feature selection while using Regularization algorithms?
I don't think overfitting is the reason that we need feature selection in the first place. In fact, overfitting is something that happens if we don't give our model enough data, and feature selection further reduces the amount of data that ... | Do we still need to do feature selection while using Regularization algorithms?
I don't think overfitting is the reason that we need feature selection in the first place. In fact, overfitting is something that happens if we don't give our model enough data, and feature selection |
10,409 | Do we still need to do feature selection while using Regularization algorithms? | In the case of lasso, preprocessing the data to remove nuisance features is actually pretty common. For a recent paper discussing ways to do this, please see Xiang et al's Screening Tests for Lasso Problems. The common motivation mentioned in the papers I've seen is to reduce the computational burden of computing the s... | Do we still need to do feature selection while using Regularization algorithms? | In the case of lasso, preprocessing the data to remove nuisance features is actually pretty common. For a recent paper discussing ways to do this, please see Xiang et al's Screening Tests for Lasso Pr | Do we still need to do feature selection while using Regularization algorithms?
In the case of lasso, preprocessing the data to remove nuisance features is actually pretty common. For a recent paper discussing ways to do this, please see Xiang et al's Screening Tests for Lasso Problems. The common motivation mentioned ... | Do we still need to do feature selection while using Regularization algorithms?
In the case of lasso, preprocessing the data to remove nuisance features is actually pretty common. For a recent paper discussing ways to do this, please see Xiang et al's Screening Tests for Lasso Pr |
10,410 | Do we still need to do feature selection while using Regularization algorithms? | I think if you do not have sufficient number of data points to robustly optimize the parameters you can do feature selection to remove some variables. But I would not suggest doing too much of it since you can lose the signal you want to model.
Plus there might be certain features you do not want in your models based ... | Do we still need to do feature selection while using Regularization algorithms? | I think if you do not have sufficient number of data points to robustly optimize the parameters you can do feature selection to remove some variables. But I would not suggest doing too much of it sinc | Do we still need to do feature selection while using Regularization algorithms?
I think if you do not have sufficient number of data points to robustly optimize the parameters you can do feature selection to remove some variables. But I would not suggest doing too much of it since you can lose the signal you want to mo... | Do we still need to do feature selection while using Regularization algorithms?
I think if you do not have sufficient number of data points to robustly optimize the parameters you can do feature selection to remove some variables. But I would not suggest doing too much of it sinc |
10,411 | Why does logistic regression produce well-calibrated models? | Yes.
The predicted probability vector $p$ from logistic regression satisfies the matrix equation
$$ X^t(p - y) = 0$$
Where $X$ is the design matrix and $y$ is the response vector. This can be viewed as a collection of linear equations, one arising from each column of the design matrix $X$.
Specializing to the intercep... | Why does logistic regression produce well-calibrated models? | Yes.
The predicted probability vector $p$ from logistic regression satisfies the matrix equation
$$ X^t(p - y) = 0$$
Where $X$ is the design matrix and $y$ is the response vector. This can be viewed | Why does logistic regression produce well-calibrated models?
Yes.
The predicted probability vector $p$ from logistic regression satisfies the matrix equation
$$ X^t(p - y) = 0$$
Where $X$ is the design matrix and $y$ is the response vector. This can be viewed as a collection of linear equations, one arising from each ... | Why does logistic regression produce well-calibrated models?
Yes.
The predicted probability vector $p$ from logistic regression satisfies the matrix equation
$$ X^t(p - y) = 0$$
Where $X$ is the design matrix and $y$ is the response vector. This can be viewed |
10,412 | Why does logistic regression produce well-calibrated models? | I think I can provide you an easy-to-understand explanation as follows:
We know that its loss function can be expressed as the following function:
$$
J(\theta) =
-\frac{1}{m}\sum_{i=1}^m
\left[ y^{(i)}\log\left(h_\theta \left(x^{(i)}\right)\right) +
(1 -y^{(i)})\log\left(1-h_\theta \left(x^{(i)}\right)\right)\right... | Why does logistic regression produce well-calibrated models? | I think I can provide you an easy-to-understand explanation as follows:
We know that its loss function can be expressed as the following function:
$$
J(\theta) =
-\frac{1}{m}\sum_{i=1}^m
\left[ y^ | Why does logistic regression produce well-calibrated models?
I think I can provide you an easy-to-understand explanation as follows:
We know that its loss function can be expressed as the following function:
$$
J(\theta) =
-\frac{1}{m}\sum_{i=1}^m
\left[ y^{(i)}\log\left(h_\theta \left(x^{(i)}\right)\right) +
(1 -y... | Why does logistic regression produce well-calibrated models?
I think I can provide you an easy-to-understand explanation as follows:
We know that its loss function can be expressed as the following function:
$$
J(\theta) =
-\frac{1}{m}\sum_{i=1}^m
\left[ y^ |
10,413 | Why does logistic regression produce well-calibrated models? | Logistic regression models are typically trained by minimizing the negative log likelihood. Maximum-likelihood models are known to asymptotically identify the solution minimizing the Kullback-Leibler divergence $D_{KL}(P, Q)$ between the empirical data distribution $P$ and the distribution $Q$ described by the model, s... | Why does logistic regression produce well-calibrated models? | Logistic regression models are typically trained by minimizing the negative log likelihood. Maximum-likelihood models are known to asymptotically identify the solution minimizing the Kullback-Leibler | Why does logistic regression produce well-calibrated models?
Logistic regression models are typically trained by minimizing the negative log likelihood. Maximum-likelihood models are known to asymptotically identify the solution minimizing the Kullback-Leibler divergence $D_{KL}(P, Q)$ between the empirical data distri... | Why does logistic regression produce well-calibrated models?
Logistic regression models are typically trained by minimizing the negative log likelihood. Maximum-likelihood models are known to asymptotically identify the solution minimizing the Kullback-Leibler |
10,414 | Why is a "negative binomial" random variable called that? | It's a reference to the fact that a certain binomial coefficient that appears in the formula for that distribution can be written more simply with negative numbers.
When you conduct a series of experiment with success probability $p$, the likelihood that you will see $r$ failures after exactly $k$ trials is
${k+r−1}\ch... | Why is a "negative binomial" random variable called that? | It's a reference to the fact that a certain binomial coefficient that appears in the formula for that distribution can be written more simply with negative numbers.
When you conduct a series of experi | Why is a "negative binomial" random variable called that?
It's a reference to the fact that a certain binomial coefficient that appears in the formula for that distribution can be written more simply with negative numbers.
When you conduct a series of experiment with success probability $p$, the likelihood that you wil... | Why is a "negative binomial" random variable called that?
It's a reference to the fact that a certain binomial coefficient that appears in the formula for that distribution can be written more simply with negative numbers.
When you conduct a series of experi |
10,415 | Why is a "negative binomial" random variable called that? | I think the most likely origin is that if you take any moment formula from $\mathrm{Bin}(n,q)$ and you replace the parameters $n$ and $q$ in that formula with negative values $-\alpha$ and $-\theta$, then the result will be the equivalent moment formula for a Negative Binomial distribution with parameters $\alpha$ and ... | Why is a "negative binomial" random variable called that? | I think the most likely origin is that if you take any moment formula from $\mathrm{Bin}(n,q)$ and you replace the parameters $n$ and $q$ in that formula with negative values $-\alpha$ and $-\theta$, | Why is a "negative binomial" random variable called that?
I think the most likely origin is that if you take any moment formula from $\mathrm{Bin}(n,q)$ and you replace the parameters $n$ and $q$ in that formula with negative values $-\alpha$ and $-\theta$, then the result will be the equivalent moment formula for a Ne... | Why is a "negative binomial" random variable called that?
I think the most likely origin is that if you take any moment formula from $\mathrm{Bin}(n,q)$ and you replace the parameters $n$ and $q$ in that formula with negative values $-\alpha$ and $-\theta$, |
10,416 | Why is a "negative binomial" random variable called that? | Below are some alternative explanations.
They relate to the currently accepted answer. It boils down to a negative index somewhere that contrasts to the positive index for the 'regular' binomial distribution. But, in these historic cases, there is a different focus in relation to the place where the 'negative' index oc... | Why is a "negative binomial" random variable called that? | Below are some alternative explanations.
They relate to the currently accepted answer. It boils down to a negative index somewhere that contrasts to the positive index for the 'regular' binomial distr | Why is a "negative binomial" random variable called that?
Below are some alternative explanations.
They relate to the currently accepted answer. It boils down to a negative index somewhere that contrasts to the positive index for the 'regular' binomial distribution. But, in these historic cases, there is a different fo... | Why is a "negative binomial" random variable called that?
Below are some alternative explanations.
They relate to the currently accepted answer. It boils down to a negative index somewhere that contrasts to the positive index for the 'regular' binomial distr |
10,417 | How do I calculate a confidence interval for the mean of a log-normal data set? | There are several ways for calculating confidence intervals for the mean of a lognormal distribution. I am going to present two methods: Bootstrap and Profile likelihood. I will also present a discussion on the Jeffreys prior.
Bootstrap
For the MLE
In this case, the MLE of $(\mu,\sigma)$ for a sample $(x_1,...,x_n)$ ar... | How do I calculate a confidence interval for the mean of a log-normal data set? | There are several ways for calculating confidence intervals for the mean of a lognormal distribution. I am going to present two methods: Bootstrap and Profile likelihood. I will also present a discuss | How do I calculate a confidence interval for the mean of a log-normal data set?
There are several ways for calculating confidence intervals for the mean of a lognormal distribution. I am going to present two methods: Bootstrap and Profile likelihood. I will also present a discussion on the Jeffreys prior.
Bootstrap
For... | How do I calculate a confidence interval for the mean of a log-normal data set?
There are several ways for calculating confidence intervals for the mean of a lognormal distribution. I am going to present two methods: Bootstrap and Profile likelihood. I will also present a discuss |
10,418 | How do I calculate a confidence interval for the mean of a log-normal data set? | You might try the Bayesian approach with Jeffreys' prior. It should yield credibility intervals with a correct frequentist-matching property: the confidence level of the credibility interval is close to its credibility level.
# required package
library(bayesm)
# simulated data
mu <- 0
sdv <- 1
y <- exp(rnorm(100... | How do I calculate a confidence interval for the mean of a log-normal data set? | You might try the Bayesian approach with Jeffreys' prior. It should yield credibility intervals with a correct frequentist-matching property: the confidence level of the credibility interval is close | How do I calculate a confidence interval for the mean of a log-normal data set?
You might try the Bayesian approach with Jeffreys' prior. It should yield credibility intervals with a correct frequentist-matching property: the confidence level of the credibility interval is close to its credibility level.
# required pa... | How do I calculate a confidence interval for the mean of a log-normal data set?
You might try the Bayesian approach with Jeffreys' prior. It should yield credibility intervals with a correct frequentist-matching property: the confidence level of the credibility interval is close |
10,419 | How do I calculate a confidence interval for the mean of a log-normal data set? | Another approximate confidence interval for the mean $\delta = \exp\left(\mu+\sigma^2/2\right)$ of the i.i.d. random variables in a random sample $\left(X_1, \ldots, X_n\right)$ of size $n$ from a $\mathcal{LN}\left(\mu,\sigma^2\right)$ distribution can be established by considering the asymptotic distribution of the M... | How do I calculate a confidence interval for the mean of a log-normal data set? | Another approximate confidence interval for the mean $\delta = \exp\left(\mu+\sigma^2/2\right)$ of the i.i.d. random variables in a random sample $\left(X_1, \ldots, X_n\right)$ of size $n$ from a $\m | How do I calculate a confidence interval for the mean of a log-normal data set?
Another approximate confidence interval for the mean $\delta = \exp\left(\mu+\sigma^2/2\right)$ of the i.i.d. random variables in a random sample $\left(X_1, \ldots, X_n\right)$ of size $n$ from a $\mathcal{LN}\left(\mu,\sigma^2\right)$ dis... | How do I calculate a confidence interval for the mean of a log-normal data set?
Another approximate confidence interval for the mean $\delta = \exp\left(\mu+\sigma^2/2\right)$ of the i.i.d. random variables in a random sample $\left(X_1, \ldots, X_n\right)$ of size $n$ from a $\m |
10,420 | How do I calculate a confidence interval for the mean of a log-normal data set? | However, I'm a bit suspicious of this method, simply because it doesn't work for the mean itself: 10mean(log10(X))≠mean(X)
You're right -- that's the formula for the geometric mean, not the arithmetic mean. The arithmetic mean is a parameter from the normal distribution, and is often not very meaningful for lognormal ... | How do I calculate a confidence interval for the mean of a log-normal data set? | However, I'm a bit suspicious of this method, simply because it doesn't work for the mean itself: 10mean(log10(X))≠mean(X)
You're right -- that's the formula for the geometric mean, not the arithmeti | How do I calculate a confidence interval for the mean of a log-normal data set?
However, I'm a bit suspicious of this method, simply because it doesn't work for the mean itself: 10mean(log10(X))≠mean(X)
You're right -- that's the formula for the geometric mean, not the arithmetic mean. The arithmetic mean is a paramet... | How do I calculate a confidence interval for the mean of a log-normal data set?
However, I'm a bit suspicious of this method, simply because it doesn't work for the mean itself: 10mean(log10(X))≠mean(X)
You're right -- that's the formula for the geometric mean, not the arithmeti |
10,421 | How do I calculate a confidence interval for the mean of a log-normal data set? | As @dnidz said, you probably want to be computing the geometric mean and its CI, not the arithmetic mean and its CI, when data are sampled from a lognormal distribution. Why?
First, let's think about normal distributions. For an ideal normal population (distribution), the arithmetic mean and the median are identical. F... | How do I calculate a confidence interval for the mean of a log-normal data set? | As @dnidz said, you probably want to be computing the geometric mean and its CI, not the arithmetic mean and its CI, when data are sampled from a lognormal distribution. Why?
First, let's think about | How do I calculate a confidence interval for the mean of a log-normal data set?
As @dnidz said, you probably want to be computing the geometric mean and its CI, not the arithmetic mean and its CI, when data are sampled from a lognormal distribution. Why?
First, let's think about normal distributions. For an ideal norma... | How do I calculate a confidence interval for the mean of a log-normal data set?
As @dnidz said, you probably want to be computing the geometric mean and its CI, not the arithmetic mean and its CI, when data are sampled from a lognormal distribution. Why?
First, let's think about |
10,422 | How do I calculate a confidence interval for the mean of a log-normal data set? | Ulf Olsson (2005)$^{[1]}$ presents several possibilities of calculating confidence intervals for a lognormal mean. First, let's clarify the notation. Let $X$ be a random variable following a log-normal distribution with mean $\operatorname{E}(X)=\theta$. Let $Y = \log(X)$ be the log-transformed variable which is normal... | How do I calculate a confidence interval for the mean of a log-normal data set? | Ulf Olsson (2005)$^{[1]}$ presents several possibilities of calculating confidence intervals for a lognormal mean. First, let's clarify the notation. Let $X$ be a random variable following a log-norma | How do I calculate a confidence interval for the mean of a log-normal data set?
Ulf Olsson (2005)$^{[1]}$ presents several possibilities of calculating confidence intervals for a lognormal mean. First, let's clarify the notation. Let $X$ be a random variable following a log-normal distribution with mean $\operatorname{... | How do I calculate a confidence interval for the mean of a log-normal data set?
Ulf Olsson (2005)$^{[1]}$ presents several possibilities of calculating confidence intervals for a lognormal mean. First, let's clarify the notation. Let $X$ be a random variable following a log-norma |
10,423 | How do I calculate a confidence interval for the mean of a log-normal data set? | On R you can try EnvStats::elnorm(data_vector, ci = TRUE)$interval$limits
Otherwise, on the below answer you can see how to get either prediction on confidence intervals with different prob values.
https://stats.stackexchange.com/a/109236/162190
Both ways give the same limits of CI. | How do I calculate a confidence interval for the mean of a log-normal data set? | On R you can try EnvStats::elnorm(data_vector, ci = TRUE)$interval$limits
Otherwise, on the below answer you can see how to get either prediction on confidence intervals with different prob values.
ht | How do I calculate a confidence interval for the mean of a log-normal data set?
On R you can try EnvStats::elnorm(data_vector, ci = TRUE)$interval$limits
Otherwise, on the below answer you can see how to get either prediction on confidence intervals with different prob values.
https://stats.stackexchange.com/a/109236/1... | How do I calculate a confidence interval for the mean of a log-normal data set?
On R you can try EnvStats::elnorm(data_vector, ci = TRUE)$interval$limits
Otherwise, on the below answer you can see how to get either prediction on confidence intervals with different prob values.
ht |
10,424 | Dealing with correlated regressors | Principal components make a lot of sense... mathematically. However, I'd be wary of simply using some mathematical trick in this case and hoping that I don't need to think about my problem.
I'd recommend thinking a little about what kind of predictors I have, what the independent variable is, why my predictors are corr... | Dealing with correlated regressors | Principal components make a lot of sense... mathematically. However, I'd be wary of simply using some mathematical trick in this case and hoping that I don't need to think about my problem.
I'd recomm | Dealing with correlated regressors
Principal components make a lot of sense... mathematically. However, I'd be wary of simply using some mathematical trick in this case and hoping that I don't need to think about my problem.
I'd recommend thinking a little about what kind of predictors I have, what the independent vari... | Dealing with correlated regressors
Principal components make a lot of sense... mathematically. However, I'd be wary of simply using some mathematical trick in this case and hoping that I don't need to think about my problem.
I'd recomm |
10,425 | Dealing with correlated regressors | You can use principal components or ridge regression to deal with this problem. On the other hand, if you have two variables that are correlated highly enough to cause problems with parameter estimation, then you could almost certainly drop either one of the two without losing much in terms of prediction--because the ... | Dealing with correlated regressors | You can use principal components or ridge regression to deal with this problem. On the other hand, if you have two variables that are correlated highly enough to cause problems with parameter estimat | Dealing with correlated regressors
You can use principal components or ridge regression to deal with this problem. On the other hand, if you have two variables that are correlated highly enough to cause problems with parameter estimation, then you could almost certainly drop either one of the two without losing much i... | Dealing with correlated regressors
You can use principal components or ridge regression to deal with this problem. On the other hand, if you have two variables that are correlated highly enough to cause problems with parameter estimat |
10,426 | Dealing with correlated regressors | Here is another thought that is inspired by Stephan's answer:
If some of your correlated regressors are meaningfully related (e.g., they are different measures of intelligence i.e., verbal, math etc) then you can create a single variable that measures the same variable using one of the following techniques:
Sum the re... | Dealing with correlated regressors | Here is another thought that is inspired by Stephan's answer:
If some of your correlated regressors are meaningfully related (e.g., they are different measures of intelligence i.e., verbal, math etc) | Dealing with correlated regressors
Here is another thought that is inspired by Stephan's answer:
If some of your correlated regressors are meaningfully related (e.g., they are different measures of intelligence i.e., verbal, math etc) then you can create a single variable that measures the same variable using one of th... | Dealing with correlated regressors
Here is another thought that is inspired by Stephan's answer:
If some of your correlated regressors are meaningfully related (e.g., they are different measures of intelligence i.e., verbal, math etc) |
10,427 | Dealing with correlated regressors | I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly correlated with some measure of size, and things can be improved by using intensive variables, i.e. dividing everything... | Dealing with correlated regressors | I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly | Dealing with correlated regressors
I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly correlated with some measure of size, and things can be improved by using intensive ... | Dealing with correlated regressors
I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly |
10,428 | Dealing with correlated regressors | I'm no expert on this, but my first thought would be to run a principal component analysis on the predictor variables, then use the resulting principal components to predict your dependent variable. | Dealing with correlated regressors | I'm no expert on this, but my first thought would be to run a principal component analysis on the predictor variables, then use the resulting principal components to predict your dependent variable. | Dealing with correlated regressors
I'm no expert on this, but my first thought would be to run a principal component analysis on the predictor variables, then use the resulting principal components to predict your dependent variable. | Dealing with correlated regressors
I'm no expert on this, but my first thought would be to run a principal component analysis on the predictor variables, then use the resulting principal components to predict your dependent variable. |
10,429 | Dealing with correlated regressors | One of the ways to reduce the effects of correlation is to standardize the regressors. In standardizing, all the regressors are subtracted by their respective means and divided by their respective standard deviations. Specifically, if $X$ is the regression matrix:
$$x_{ij}^{standardized}=\frac {x_{ij}-\overline{x_{.j}}... | Dealing with correlated regressors | One of the ways to reduce the effects of correlation is to standardize the regressors. In standardizing, all the regressors are subtracted by their respective means and divided by their respective sta | Dealing with correlated regressors
One of the ways to reduce the effects of correlation is to standardize the regressors. In standardizing, all the regressors are subtracted by their respective means and divided by their respective standard deviations. Specifically, if $X$ is the regression matrix:
$$x_{ij}^{standardiz... | Dealing with correlated regressors
One of the ways to reduce the effects of correlation is to standardize the regressors. In standardizing, all the regressors are subtracted by their respective means and divided by their respective sta |
10,430 | Is accuracy an improper scoring rule in a binary classification setting? | TL;DR
Accuracy is an improper scoring rule. Don't use it.
The slightly longer version
Actually, accuracy is not even a scoring rule. So asking whether it is (strictly) proper is a category error. The most we can say is that under additional assumptions, accuracy is consistent with a scoring rule that is improper, disco... | Is accuracy an improper scoring rule in a binary classification setting? | TL;DR
Accuracy is an improper scoring rule. Don't use it.
The slightly longer version
Actually, accuracy is not even a scoring rule. So asking whether it is (strictly) proper is a category error. The | Is accuracy an improper scoring rule in a binary classification setting?
TL;DR
Accuracy is an improper scoring rule. Don't use it.
The slightly longer version
Actually, accuracy is not even a scoring rule. So asking whether it is (strictly) proper is a category error. The most we can say is that under additional assump... | Is accuracy an improper scoring rule in a binary classification setting?
TL;DR
Accuracy is an improper scoring rule. Don't use it.
The slightly longer version
Actually, accuracy is not even a scoring rule. So asking whether it is (strictly) proper is a category error. The |
10,431 | Bayesian batting average prior | Notice that:
\begin{equation}
\frac{\alpha\cdot\beta}{(\alpha+\beta)^2}=(\frac{\alpha}{\alpha+\beta})\cdot(1-\frac{\alpha}{\alpha+\beta})
\end{equation}
This means the variance can therefore be expressed in terms of the mean as
\begin{equation}
\sigma^2=\frac{\mu\cdot(1-\mu)}{\alpha+\beta+1} \\
\end{equation}
If you wa... | Bayesian batting average prior | Notice that:
\begin{equation}
\frac{\alpha\cdot\beta}{(\alpha+\beta)^2}=(\frac{\alpha}{\alpha+\beta})\cdot(1-\frac{\alpha}{\alpha+\beta})
\end{equation}
This means the variance can therefore be expres | Bayesian batting average prior
Notice that:
\begin{equation}
\frac{\alpha\cdot\beta}{(\alpha+\beta)^2}=(\frac{\alpha}{\alpha+\beta})\cdot(1-\frac{\alpha}{\alpha+\beta})
\end{equation}
This means the variance can therefore be expressed in terms of the mean as
\begin{equation}
\sigma^2=\frac{\mu\cdot(1-\mu)}{\alpha+\beta... | Bayesian batting average prior
Notice that:
\begin{equation}
\frac{\alpha\cdot\beta}{(\alpha+\beta)^2}=(\frac{\alpha}{\alpha+\beta})\cdot(1-\frac{\alpha}{\alpha+\beta})
\end{equation}
This means the variance can therefore be expres |
10,432 | Bayesian batting average prior | I wanted to add this as a comment on the excellent answer but it ran long and will look better with answer formatting.
Something to keep in mind is that not all $(\mu, \sigma^2)$ are possible. It's clear $\mu \in [0,1]$, but not as clear are the limitations for $\sigma^2$.
Using the same reasoning as David, we can expr... | Bayesian batting average prior | I wanted to add this as a comment on the excellent answer but it ran long and will look better with answer formatting.
Something to keep in mind is that not all $(\mu, \sigma^2)$ are possible. It's cl | Bayesian batting average prior
I wanted to add this as a comment on the excellent answer but it ran long and will look better with answer formatting.
Something to keep in mind is that not all $(\mu, \sigma^2)$ are possible. It's clear $\mu \in [0,1]$, but not as clear are the limitations for $\sigma^2$.
Using the same ... | Bayesian batting average prior
I wanted to add this as a comment on the excellent answer but it ran long and will look better with answer formatting.
Something to keep in mind is that not all $(\mu, \sigma^2)$ are possible. It's cl |
10,433 | Reversing PCA back to the original variables [duplicate] | Yes. Basically, what you did was to do:
$$\mathrm{PC}=\mathbf{V}X,$$
where $\mathrm{PC}$ are the principal components, $X$ is your matrix with the data (centered, and with data points in columns) and $\mathbf{V}$ is the matrix with the loadings (the matrix with the eigenvectors of the sample covariance matrix of $X$). ... | Reversing PCA back to the original variables [duplicate] | Yes. Basically, what you did was to do:
$$\mathrm{PC}=\mathbf{V}X,$$
where $\mathrm{PC}$ are the principal components, $X$ is your matrix with the data (centered, and with data points in columns) and | Reversing PCA back to the original variables [duplicate]
Yes. Basically, what you did was to do:
$$\mathrm{PC}=\mathbf{V}X,$$
where $\mathrm{PC}$ are the principal components, $X$ is your matrix with the data (centered, and with data points in columns) and $\mathbf{V}$ is the matrix with the loadings (the matrix with t... | Reversing PCA back to the original variables [duplicate]
Yes. Basically, what you did was to do:
$$\mathrm{PC}=\mathbf{V}X,$$
where $\mathrm{PC}$ are the principal components, $X$ is your matrix with the data (centered, and with data points in columns) and |
10,434 | Reversing PCA back to the original variables [duplicate] | I have a doubt about the above answers. Since after dimension reduction, we only know 2 principal components, and the rest principal components are abandoned. The projection matrix V is not a square matrix (not completely orthonormal, it is a semi-orthogonal matrix). Suppose n is the number of samples and m is the numb... | Reversing PCA back to the original variables [duplicate] | I have a doubt about the above answers. Since after dimension reduction, we only know 2 principal components, and the rest principal components are abandoned. The projection matrix V is not a square m | Reversing PCA back to the original variables [duplicate]
I have a doubt about the above answers. Since after dimension reduction, we only know 2 principal components, and the rest principal components are abandoned. The projection matrix V is not a square matrix (not completely orthonormal, it is a semi-orthogonal matr... | Reversing PCA back to the original variables [duplicate]
I have a doubt about the above answers. Since after dimension reduction, we only know 2 principal components, and the rest principal components are abandoned. The projection matrix V is not a square m |
10,435 | Converting standard error to standard deviation? | Standard error refers to the standard deviation of the sampling distribution of a statistic. Whether or not that formula is appropriate depends on what statistic we are talking about.
The standard deviation of the sample mean is $\sigma/\sqrt{n}$ where $\sigma$ is the (population) standard deviation of the data and $n... | Converting standard error to standard deviation? | Standard error refers to the standard deviation of the sampling distribution of a statistic. Whether or not that formula is appropriate depends on what statistic we are talking about.
The standard de | Converting standard error to standard deviation?
Standard error refers to the standard deviation of the sampling distribution of a statistic. Whether or not that formula is appropriate depends on what statistic we are talking about.
The standard deviation of the sample mean is $\sigma/\sqrt{n}$ where $\sigma$ is the (... | Converting standard error to standard deviation?
Standard error refers to the standard deviation of the sampling distribution of a statistic. Whether or not that formula is appropriate depends on what statistic we are talking about.
The standard de |
10,436 | How does entropy depend on location and scale? | Since the probability element of $X$ is $f(x)\mathrm{d}x,$ the change of variable $y = x\sigma + \mu$ is equivalent to $x = (y-\mu)/\sigma,$ whence from
$$f(x)\mathrm{d}x = f\left(\frac{y-\mu}{\sigma}\right)\mathrm{d}\left(\frac{y-\mu}{\color{red}\sigma}\right) = \frac{1}{\color{red}\sigma} f\left(\frac{y-\mu}{\sigma}\... | How does entropy depend on location and scale? | Since the probability element of $X$ is $f(x)\mathrm{d}x,$ the change of variable $y = x\sigma + \mu$ is equivalent to $x = (y-\mu)/\sigma,$ whence from
$$f(x)\mathrm{d}x = f\left(\frac{y-\mu}{\sigma} | How does entropy depend on location and scale?
Since the probability element of $X$ is $f(x)\mathrm{d}x,$ the change of variable $y = x\sigma + \mu$ is equivalent to $x = (y-\mu)/\sigma,$ whence from
$$f(x)\mathrm{d}x = f\left(\frac{y-\mu}{\sigma}\right)\mathrm{d}\left(\frac{y-\mu}{\color{red}\sigma}\right) = \frac{1}{... | How does entropy depend on location and scale?
Since the probability element of $X$ is $f(x)\mathrm{d}x,$ the change of variable $y = x\sigma + \mu$ is equivalent to $x = (y-\mu)/\sigma,$ whence from
$$f(x)\mathrm{d}x = f\left(\frac{y-\mu}{\sigma} |
10,437 | Why does a decision tree have low bias & high variance? | A bit late to the party but i feel that this question could use answer with concrete examples.
I will write summary of this excellent article: bias-variance-trade-off, which helped me understand the topic.
The prediction error for any machine learning algorithm can be broken down into three parts:
Bias Error
Varianc... | Why does a decision tree have low bias & high variance? | A bit late to the party but i feel that this question could use answer with concrete examples.
I will write summary of this excellent article: bias-variance-trade-off, which helped me understand the | Why does a decision tree have low bias & high variance?
A bit late to the party but i feel that this question could use answer with concrete examples.
I will write summary of this excellent article: bias-variance-trade-off, which helped me understand the topic.
The prediction error for any machine learning algorithm c... | Why does a decision tree have low bias & high variance?
A bit late to the party but i feel that this question could use answer with concrete examples.
I will write summary of this excellent article: bias-variance-trade-off, which helped me understand the |
10,438 | Why does a decision tree have low bias & high variance? | If the number of levels is too high i.e a complicated decision tree, the model tends to overfit.
Intuitively, it can be understood in this way. When there are too many decision nodes to go through before arriving at the result i.e number of nodes to traverse before reaching the leaf nodes is high, the conditions that y... | Why does a decision tree have low bias & high variance? | If the number of levels is too high i.e a complicated decision tree, the model tends to overfit.
Intuitively, it can be understood in this way. When there are too many decision nodes to go through bef | Why does a decision tree have low bias & high variance?
If the number of levels is too high i.e a complicated decision tree, the model tends to overfit.
Intuitively, it can be understood in this way. When there are too many decision nodes to go through before arriving at the result i.e number of nodes to traverse befor... | Why does a decision tree have low bias & high variance?
If the number of levels is too high i.e a complicated decision tree, the model tends to overfit.
Intuitively, it can be understood in this way. When there are too many decision nodes to go through bef |
10,439 | Why does a decision tree have low bias & high variance? | Why does a decision tree have low bias & high variance? Does it depend
on whether the tree is shallow or deep? Or can we say this
irrespective of the depth/levels of the tree? Why is bias low & variance high? Please explain intuitively and mathematically.
Bias vs Variance
More Bias = error from the model being mor... | Why does a decision tree have low bias & high variance? | Why does a decision tree have low bias & high variance? Does it depend
on whether the tree is shallow or deep? Or can we say this
irrespective of the depth/levels of the tree? Why is bias low & va | Why does a decision tree have low bias & high variance?
Why does a decision tree have low bias & high variance? Does it depend
on whether the tree is shallow or deep? Or can we say this
irrespective of the depth/levels of the tree? Why is bias low & variance high? Please explain intuitively and mathematically.
Bia... | Why does a decision tree have low bias & high variance?
Why does a decision tree have low bias & high variance? Does it depend
on whether the tree is shallow or deep? Or can we say this
irrespective of the depth/levels of the tree? Why is bias low & va |
10,440 | Why does a decision tree have low bias & high variance? | A complicated decision tree (e.g. deep) has low bias and high variance. The bias-variance tradeoff does depend on the depth of the tree.
Decision tree is sensitive to where it splits and how it splits. Therefore, even small changes in input variable values might result in very different tree structure. | Why does a decision tree have low bias & high variance? | A complicated decision tree (e.g. deep) has low bias and high variance. The bias-variance tradeoff does depend on the depth of the tree.
Decision tree is sensitive to where it splits and how it splits | Why does a decision tree have low bias & high variance?
A complicated decision tree (e.g. deep) has low bias and high variance. The bias-variance tradeoff does depend on the depth of the tree.
Decision tree is sensitive to where it splits and how it splits. Therefore, even small changes in input variable values might r... | Why does a decision tree have low bias & high variance?
A complicated decision tree (e.g. deep) has low bias and high variance. The bias-variance tradeoff does depend on the depth of the tree.
Decision tree is sensitive to where it splits and how it splits |
10,441 | Why does a decision tree have low bias & high variance? | Context: decision tree has low bias & high variance
Q. Does it depend on whether the tree is shallow or deep?
A. If the tree is shallow then we're not checking a lot of conditions/constrains ie the logic is simple or less complex, hence it automatically reduces over-fitting. This introduces more bias compared to deepe... | Why does a decision tree have low bias & high variance? | Context: decision tree has low bias & high variance
Q. Does it depend on whether the tree is shallow or deep?
A. If the tree is shallow then we're not checking a lot of conditions/constrains ie the l | Why does a decision tree have low bias & high variance?
Context: decision tree has low bias & high variance
Q. Does it depend on whether the tree is shallow or deep?
A. If the tree is shallow then we're not checking a lot of conditions/constrains ie the logic is simple or less complex, hence it automatically reduces o... | Why does a decision tree have low bias & high variance?
Context: decision tree has low bias & high variance
Q. Does it depend on whether the tree is shallow or deep?
A. If the tree is shallow then we're not checking a lot of conditions/constrains ie the l |
10,442 | Expected value and variance of log(a) | If we consider "approximation" in a fairly general sense we can get somewhere.
We have to assume not that we have an actual normal distribution but something that's approximately normal except the density cannot be nonzero in a neighborhood of 0.
So let's say that $a$ is "approximately normal" (and concentrated near th... | Expected value and variance of log(a) | If we consider "approximation" in a fairly general sense we can get somewhere.
We have to assume not that we have an actual normal distribution but something that's approximately normal except the den | Expected value and variance of log(a)
If we consider "approximation" in a fairly general sense we can get somewhere.
We have to assume not that we have an actual normal distribution but something that's approximately normal except the density cannot be nonzero in a neighborhood of 0.
So let's say that $a$ is "approxima... | Expected value and variance of log(a)
If we consider "approximation" in a fairly general sense we can get somewhere.
We have to assume not that we have an actual normal distribution but something that's approximately normal except the den |
10,443 | Keras, how does SGD learning rate decay work? | The documentation that you're referring to includes a reference to the Python source (just click on the [Source] link in the appropriate place), that can be used to answer your questions. Here's the most relevant line, showing how decay modifies the learning rate:
lr = self.lr * (1. / (1. + self.decay * self.iterations... | Keras, how does SGD learning rate decay work? | The documentation that you're referring to includes a reference to the Python source (just click on the [Source] link in the appropriate place), that can be used to answer your questions. Here's the m | Keras, how does SGD learning rate decay work?
The documentation that you're referring to includes a reference to the Python source (just click on the [Source] link in the appropriate place), that can be used to answer your questions. Here's the most relevant line, showing how decay modifies the learning rate:
lr = self... | Keras, how does SGD learning rate decay work?
The documentation that you're referring to includes a reference to the Python source (just click on the [Source] link in the appropriate place), that can be used to answer your questions. Here's the m |
10,444 | Why is the quasi-Poisson in GLM not treated as a special case of negative binomial? | The quasi-Poisson is not a full maximum likelihood (ML) model but a quasi-ML model. You just use the estimating function (or score function) from the Poisson model to estimate the coefficients, and then employ a certain variance function to obtain suitable standard errors (or rather a full covariance matrix) to perform... | Why is the quasi-Poisson in GLM not treated as a special case of negative binomial? | The quasi-Poisson is not a full maximum likelihood (ML) model but a quasi-ML model. You just use the estimating function (or score function) from the Poisson model to estimate the coefficients, and th | Why is the quasi-Poisson in GLM not treated as a special case of negative binomial?
The quasi-Poisson is not a full maximum likelihood (ML) model but a quasi-ML model. You just use the estimating function (or score function) from the Poisson model to estimate the coefficients, and then employ a certain variance functio... | Why is the quasi-Poisson in GLM not treated as a special case of negative binomial?
The quasi-Poisson is not a full maximum likelihood (ML) model but a quasi-ML model. You just use the estimating function (or score function) from the Poisson model to estimate the coefficients, and th |
10,445 | The order of variables in ANOVA matters, doesn't it? | This question evidently came from a study with an unbalanced two-way design, analyzed in R with the aov() function; this page provides a more recent and detailed example of this issue.
The general answer to this question, as to so many, is: "It depends." Here it depends on whether the design is balanced and, if not, wh... | The order of variables in ANOVA matters, doesn't it? | This question evidently came from a study with an unbalanced two-way design, analyzed in R with the aov() function; this page provides a more recent and detailed example of this issue.
The general ans | The order of variables in ANOVA matters, doesn't it?
This question evidently came from a study with an unbalanced two-way design, analyzed in R with the aov() function; this page provides a more recent and detailed example of this issue.
The general answer to this question, as to so many, is: "It depends." Here it depe... | The order of variables in ANOVA matters, doesn't it?
This question evidently came from a study with an unbalanced two-way design, analyzed in R with the aov() function; this page provides a more recent and detailed example of this issue.
The general ans |
10,446 | The order of variables in ANOVA matters, doesn't it? | The term hierarchical model refers to the structure between the factors. For example, a multi-center study is hierarchical: You have the patients nested within the hospitals treating them. Each hospital treats patients with placebo and verum, but recieving each of them in either hospital A or B is slightly different du... | The order of variables in ANOVA matters, doesn't it? | The term hierarchical model refers to the structure between the factors. For example, a multi-center study is hierarchical: You have the patients nested within the hospitals treating them. Each hospit | The order of variables in ANOVA matters, doesn't it?
The term hierarchical model refers to the structure between the factors. For example, a multi-center study is hierarchical: You have the patients nested within the hospitals treating them. Each hospital treats patients with placebo and verum, but recieving each of th... | The order of variables in ANOVA matters, doesn't it?
The term hierarchical model refers to the structure between the factors. For example, a multi-center study is hierarchical: You have the patients nested within the hospitals treating them. Each hospit |
10,447 | Constructing confidence intervals based on profile likelihood | In general, the confidence interval based on the standard error strongly depends on the assumption of normality for the estimator. The "profile likelihood confidence interval" provides an alternative.
I am pretty sure you can find documentation for this. For instance, here and references therein.
Here is a brief overvi... | Constructing confidence intervals based on profile likelihood | In general, the confidence interval based on the standard error strongly depends on the assumption of normality for the estimator. The "profile likelihood confidence interval" provides an alternative. | Constructing confidence intervals based on profile likelihood
In general, the confidence interval based on the standard error strongly depends on the assumption of normality for the estimator. The "profile likelihood confidence interval" provides an alternative.
I am pretty sure you can find documentation for this. For... | Constructing confidence intervals based on profile likelihood
In general, the confidence interval based on the standard error strongly depends on the assumption of normality for the estimator. The "profile likelihood confidence interval" provides an alternative. |
10,448 | What are the assumptions of ridge regression and how to test them? | What is an assumption of a statistical procedure?
I am not a statistician and so this might be wrong, but I think the word "assumption" is often used quite informally and can refer to various things. To me, an "assumption" is, strictly speaking, something that only a theoretical result (theorem) can have.
When people t... | What are the assumptions of ridge regression and how to test them? | What is an assumption of a statistical procedure?
I am not a statistician and so this might be wrong, but I think the word "assumption" is often used quite informally and can refer to various things. | What are the assumptions of ridge regression and how to test them?
What is an assumption of a statistical procedure?
I am not a statistician and so this might be wrong, but I think the word "assumption" is often used quite informally and can refer to various things. To me, an "assumption" is, strictly speaking, somethi... | What are the assumptions of ridge regression and how to test them?
What is an assumption of a statistical procedure?
I am not a statistician and so this might be wrong, but I think the word "assumption" is often used quite informally and can refer to various things. |
10,449 | What are the assumptions of ridge regression and how to test them? | I would like to provide some input from the statistics perspective. If Y~N(Xb, sigma2*In), then the mean square error of b^ is
MSE(b^)=E(b^-b).T*(b^-b)=E(|b^-b|^2)=sigma2*trace(inv(X.T*X))
D(|b^-b|^2)=2*sigma4*trace((X.T*X)^(-2))
b^=inv(X.T*X)*X.T*Y
If X.TX is approximately zero,then inv(X.TX) will be very big. So ... | What are the assumptions of ridge regression and how to test them? | I would like to provide some input from the statistics perspective. If Y~N(Xb, sigma2*In), then the mean square error of b^ is
MSE(b^)=E(b^-b).T*(b^-b)=E(|b^-b|^2)=sigma2*trace(inv(X.T*X))
D(|b^-b|^ | What are the assumptions of ridge regression and how to test them?
I would like to provide some input from the statistics perspective. If Y~N(Xb, sigma2*In), then the mean square error of b^ is
MSE(b^)=E(b^-b).T*(b^-b)=E(|b^-b|^2)=sigma2*trace(inv(X.T*X))
D(|b^-b|^2)=2*sigma4*trace((X.T*X)^(-2))
b^=inv(X.T*X)*X.T*Y
... | What are the assumptions of ridge regression and how to test them?
I would like to provide some input from the statistics perspective. If Y~N(Xb, sigma2*In), then the mean square error of b^ is
MSE(b^)=E(b^-b).T*(b^-b)=E(|b^-b|^2)=sigma2*trace(inv(X.T*X))
D(|b^-b|^ |
10,450 | With categorical data, can there be clusters without the variables being related? | Consider the clear-cluster case with uncorrelated scale variables - such as the top-right picture in the question. And categorize its data.
We subdivided the scale range of both variables X and Y into 3 bins which now onward we treat as categorical labels. Moreover, we'll declare them nominal, not ordinal, because the... | With categorical data, can there be clusters without the variables being related? | Consider the clear-cluster case with uncorrelated scale variables - such as the top-right picture in the question. And categorize its data.
We subdivided the scale range of both variables X and Y int | With categorical data, can there be clusters without the variables being related?
Consider the clear-cluster case with uncorrelated scale variables - such as the top-right picture in the question. And categorize its data.
We subdivided the scale range of both variables X and Y into 3 bins which now onward we treat as ... | With categorical data, can there be clusters without the variables being related?
Consider the clear-cluster case with uncorrelated scale variables - such as the top-right picture in the question. And categorize its data.
We subdivided the scale range of both variables X and Y int |
10,451 | With categorical data, can there be clusters without the variables being related? | As I'm sure you know, correlation is a measure of the linear relationship between two variables, not how close the points are to each other. This explains the top four figures.
Of course, you could also create similar graphs for discrete, real-valued data as well.
The problem with more abstract distributions, such as $... | With categorical data, can there be clusters without the variables being related? | As I'm sure you know, correlation is a measure of the linear relationship between two variables, not how close the points are to each other. This explains the top four figures.
Of course, you could al | With categorical data, can there be clusters without the variables being related?
As I'm sure you know, correlation is a measure of the linear relationship between two variables, not how close the points are to each other. This explains the top four figures.
Of course, you could also create similar graphs for discrete,... | With categorical data, can there be clusters without the variables being related?
As I'm sure you know, correlation is a measure of the linear relationship between two variables, not how close the points are to each other. This explains the top four figures.
Of course, you could al |
10,452 | With categorical data, can there be clusters without the variables being related? | Consider the Hamming distance -- the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. From this definition it seems obvious that we can produce data for which we have clusters based on the Hamming distance but no correlations between the v... | With categorical data, can there be clusters without the variables being related? | Consider the Hamming distance -- the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. From this definition it seems obv | With categorical data, can there be clusters without the variables being related?
Consider the Hamming distance -- the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. From this definition it seems obvious that we can produce data for whic... | With categorical data, can there be clusters without the variables being related?
Consider the Hamming distance -- the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. From this definition it seems obv |
10,453 | With categorical data, can there be clusters without the variables being related? | @ttnphns' point about pairwise vs multivariate association is well taken. Related to that is the old saw about the importance of demonstrating association with simple metrics before leaping into a multivariate framework. In other words, if simple pairwise measures of association show no relationship then it becomes inc... | With categorical data, can there be clusters without the variables being related? | @ttnphns' point about pairwise vs multivariate association is well taken. Related to that is the old saw about the importance of demonstrating association with simple metrics before leaping into a mul | With categorical data, can there be clusters without the variables being related?
@ttnphns' point about pairwise vs multivariate association is well taken. Related to that is the old saw about the importance of demonstrating association with simple metrics before leaping into a multivariate framework. In other words, i... | With categorical data, can there be clusters without the variables being related?
@ttnphns' point about pairwise vs multivariate association is well taken. Related to that is the old saw about the importance of demonstrating association with simple metrics before leaping into a mul |
10,454 | What's the difference between a Markov Random Field and a Conditional Random Field? | Ok, I found the answer myself:
Conditinal Random Fields (CRFs) are a special case of Markov Random Fields (MRFs).
1.5.4 Conditional Random Field
A Conditional Random Field (CRF) is a form of MRF that defines a posterior for
variables x given data z, as with the hidden MRF above. Unlike the hidden MRF,
however, the... | What's the difference between a Markov Random Field and a Conditional Random Field? | Ok, I found the answer myself:
Conditinal Random Fields (CRFs) are a special case of Markov Random Fields (MRFs).
1.5.4 Conditional Random Field
A Conditional Random Field (CRF) is a form of MRF that | What's the difference between a Markov Random Field and a Conditional Random Field?
Ok, I found the answer myself:
Conditinal Random Fields (CRFs) are a special case of Markov Random Fields (MRFs).
1.5.4 Conditional Random Field
A Conditional Random Field (CRF) is a form of MRF that defines a posterior for
variables... | What's the difference between a Markov Random Field and a Conditional Random Field?
Ok, I found the answer myself:
Conditinal Random Fields (CRFs) are a special case of Markov Random Fields (MRFs).
1.5.4 Conditional Random Field
A Conditional Random Field (CRF) is a form of MRF that |
10,455 | What's the difference between a Markov Random Field and a Conditional Random Field? | Let's contrast conditional inference under MRFs with modeling using a CRF, settling on definitions along the way, and then address the original question.
MRF
A Markov Random Field (MRF) with respect to a graph $G$ is
a set of random variables (or random "elements" if you like) corresponding to the nodes in $G$ (thus, ... | What's the difference between a Markov Random Field and a Conditional Random Field? | Let's contrast conditional inference under MRFs with modeling using a CRF, settling on definitions along the way, and then address the original question.
MRF
A Markov Random Field (MRF) with respect t | What's the difference between a Markov Random Field and a Conditional Random Field?
Let's contrast conditional inference under MRFs with modeling using a CRF, settling on definitions along the way, and then address the original question.
MRF
A Markov Random Field (MRF) with respect to a graph $G$ is
a set of random va... | What's the difference between a Markov Random Field and a Conditional Random Field?
Let's contrast conditional inference under MRFs with modeling using a CRF, settling on definitions along the way, and then address the original question.
MRF
A Markov Random Field (MRF) with respect t |
10,456 | What's the difference between a Markov Random Field and a Conditional Random Field? | MRF vs Bayes nets: Unpreciesly (but normally) speaking, there are two types of graphical models: undirected graphical models and directed graphical models(one more type, for instance Tanner graph). The former is also known as Markov Random Fields/Markov network and the later Bayes nets/Bayesian network. (Sometimes the ... | What's the difference between a Markov Random Field and a Conditional Random Field? | MRF vs Bayes nets: Unpreciesly (but normally) speaking, there are two types of graphical models: undirected graphical models and directed graphical models(one more type, for instance Tanner graph). Th | What's the difference between a Markov Random Field and a Conditional Random Field?
MRF vs Bayes nets: Unpreciesly (but normally) speaking, there are two types of graphical models: undirected graphical models and directed graphical models(one more type, for instance Tanner graph). The former is also known as Markov Ran... | What's the difference between a Markov Random Field and a Conditional Random Field?
MRF vs Bayes nets: Unpreciesly (but normally) speaking, there are two types of graphical models: undirected graphical models and directed graphical models(one more type, for instance Tanner graph). Th |
10,457 | Is there a statistical application that requires strong consistency? | If you need a reference for the answer in my comment above, here is one, from Andrew Gelman's blog:
Which reminds me of Lucien Le Cam’s reply when I asked him once whether he could think of any examples where the distinction between the strong law of large numbers (convergence with probability 1) and the weak law (con... | Is there a statistical application that requires strong consistency? | If you need a reference for the answer in my comment above, here is one, from Andrew Gelman's blog:
Which reminds me of Lucien Le Cam’s reply when I asked him once whether he could think of any examp | Is there a statistical application that requires strong consistency?
If you need a reference for the answer in my comment above, here is one, from Andrew Gelman's blog:
Which reminds me of Lucien Le Cam’s reply when I asked him once whether he could think of any examples where the distinction between the strong law of... | Is there a statistical application that requires strong consistency?
If you need a reference for the answer in my comment above, here is one, from Andrew Gelman's blog:
Which reminds me of Lucien Le Cam’s reply when I asked him once whether he could think of any examp |
10,458 | Is there a nonparametric equivalent of Tukey HSD? | I did a little google research because I found the question quite interesting, these tests have been mentioned:
Nemenyi-Damico-Wolfe-Dunn test (link, there is an r-package for doing the test)
Dwass-Steel-Chritchlow-Fligner (link, Conover WJ, Practical Nonparametric Statistics (3rd edition). Wiley 1999.
Conover-Inman t... | Is there a nonparametric equivalent of Tukey HSD? | I did a little google research because I found the question quite interesting, these tests have been mentioned:
Nemenyi-Damico-Wolfe-Dunn test (link, there is an r-package for doing the test)
Dwass-S | Is there a nonparametric equivalent of Tukey HSD?
I did a little google research because I found the question quite interesting, these tests have been mentioned:
Nemenyi-Damico-Wolfe-Dunn test (link, there is an r-package for doing the test)
Dwass-Steel-Chritchlow-Fligner (link, Conover WJ, Practical Nonparametric Sta... | Is there a nonparametric equivalent of Tukey HSD?
I did a little google research because I found the question quite interesting, these tests have been mentioned:
Nemenyi-Damico-Wolfe-Dunn test (link, there is an r-package for doing the test)
Dwass-S |
10,459 | Is there a nonparametric equivalent of Tukey HSD? | There is kruskalmc function in pgirmess package in R. Description of the test:
Multiple comparison test between treatments or treatments versus
control after Kruskal-Wallis test. | Is there a nonparametric equivalent of Tukey HSD? | There is kruskalmc function in pgirmess package in R. Description of the test:
Multiple comparison test between treatments or treatments versus
control after Kruskal-Wallis test. | Is there a nonparametric equivalent of Tukey HSD?
There is kruskalmc function in pgirmess package in R. Description of the test:
Multiple comparison test between treatments or treatments versus
control after Kruskal-Wallis test. | Is there a nonparametric equivalent of Tukey HSD?
There is kruskalmc function in pgirmess package in R. Description of the test:
Multiple comparison test between treatments or treatments versus
control after Kruskal-Wallis test. |
10,460 | Is there a nonparametric equivalent of Tukey HSD? | If you want to test for an effect using many Wilcoxon statistics, You can go about by calculating the range of your statistics, and then simulating the distribution of the range under the "all effect are null" hypothesis. I do not think you will find tables for the distribution of the range of sample from a Wilcoxon di... | Is there a nonparametric equivalent of Tukey HSD? | If you want to test for an effect using many Wilcoxon statistics, You can go about by calculating the range of your statistics, and then simulating the distribution of the range under the "all effect | Is there a nonparametric equivalent of Tukey HSD?
If you want to test for an effect using many Wilcoxon statistics, You can go about by calculating the range of your statistics, and then simulating the distribution of the range under the "all effect are null" hypothesis. I do not think you will find tables for the dist... | Is there a nonparametric equivalent of Tukey HSD?
If you want to test for an effect using many Wilcoxon statistics, You can go about by calculating the range of your statistics, and then simulating the distribution of the range under the "all effect |
10,461 | Is there a nonparametric equivalent of Tukey HSD? | JMP does Steel-Dwass comparisons. Use 'Fit Y by X' then on the 'Oneway Analysis of ...' menu choose 'Nonparametric' -> 'Nonparametric Multiple Comparisons' -> 'Steel-Dwass All Pairs' | Is there a nonparametric equivalent of Tukey HSD? | JMP does Steel-Dwass comparisons. Use 'Fit Y by X' then on the 'Oneway Analysis of ...' menu choose 'Nonparametric' -> 'Nonparametric Multiple Comparisons' -> 'Steel-Dwass All Pairs' | Is there a nonparametric equivalent of Tukey HSD?
JMP does Steel-Dwass comparisons. Use 'Fit Y by X' then on the 'Oneway Analysis of ...' menu choose 'Nonparametric' -> 'Nonparametric Multiple Comparisons' -> 'Steel-Dwass All Pairs' | Is there a nonparametric equivalent of Tukey HSD?
JMP does Steel-Dwass comparisons. Use 'Fit Y by X' then on the 'Oneway Analysis of ...' menu choose 'Nonparametric' -> 'Nonparametric Multiple Comparisons' -> 'Steel-Dwass All Pairs' |
10,462 | Why the names Type 1, 2 error? | Great question, motivated me to Google it :) Per Wikipedia (with minor formatting edits):
A type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis.
A type II error (or error of the second kind) is the failure to reject a false null hypothesis.
Further down the page it discuss... | Why the names Type 1, 2 error? | Great question, motivated me to Google it :) Per Wikipedia (with minor formatting edits):
A type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis.
A type II | Why the names Type 1, 2 error?
Great question, motivated me to Google it :) Per Wikipedia (with minor formatting edits):
A type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis.
A type II error (or error of the second kind) is the failure to reject a false null hypothesis.
F... | Why the names Type 1, 2 error?
Great question, motivated me to Google it :) Per Wikipedia (with minor formatting edits):
A type I error (or error of the first kind) is the incorrect rejection of a true null hypothesis.
A type II |
10,463 | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD? | Update: Thanks to this discussion, scikit-learn was updated and works correctly now. Its LDA source code can be found here. The original issue was due to a minor bug (see this github discussion) and my answer was actually not pointing at it correctly (apologies for any confusion caused). As all of that does not matter... | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD? | Update: Thanks to this discussion, scikit-learn was updated and works correctly now. Its LDA source code can be found here. The original issue was due to a minor bug (see this github discussion) and | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD?
Update: Thanks to this discussion, scikit-learn was updated and works correctly now. Its LDA source code can be found here. The original issue was due to a minor bug (see this github discussion) and my answer was actually not p... | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD?
Update: Thanks to this discussion, scikit-learn was updated and works correctly now. Its LDA source code can be found here. The original issue was due to a minor bug (see this github discussion) and |
10,464 | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD? | Just to close this question, the discussed issue with the LDA has been fixed in scikit-learn 0.15.2. | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD? | Just to close this question, the discussed issue with the LDA has been fixed in scikit-learn 0.15.2. | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD?
Just to close this question, the discussed issue with the LDA has been fixed in scikit-learn 0.15.2. | Why is Python's scikit-learn LDA not working correctly and how does it compute LDA via SVD?
Just to close this question, the discussed issue with the LDA has been fixed in scikit-learn 0.15.2. |
10,465 | Definition of autocorrelation time (for effective sample size) | First, the appropriate definition of "effective sample size" is IMO linked to a quite specific question. If $X_1, X_2, \ldots$ are identically distributed with mean $\mu$ and variance 1 the empirical mean
$$\hat{\mu} = \frac{1}{n} \sum_{k=1}^n X_k$$
is an unbiased estimator of $\mu$. But what about its variance? For in... | Definition of autocorrelation time (for effective sample size) | First, the appropriate definition of "effective sample size" is IMO linked to a quite specific question. If $X_1, X_2, \ldots$ are identically distributed with mean $\mu$ and variance 1 the empirical | Definition of autocorrelation time (for effective sample size)
First, the appropriate definition of "effective sample size" is IMO linked to a quite specific question. If $X_1, X_2, \ldots$ are identically distributed with mean $\mu$ and variance 1 the empirical mean
$$\hat{\mu} = \frac{1}{n} \sum_{k=1}^n X_k$$
is an u... | Definition of autocorrelation time (for effective sample size)
First, the appropriate definition of "effective sample size" is IMO linked to a quite specific question. If $X_1, X_2, \ldots$ are identically distributed with mean $\mu$ and variance 1 the empirical |
10,466 | Definition of autocorrelation time (for effective sample size) | see
http://arxiv.org/pdf/1403.5536v1.pdf
and
https://cran.r-project.org/web/packages/mcmcse/mcmcse.pdf
for effective sample size. I think the alternative formulation using the ratio of sample variance and asymptotic Markov chain variance via batch mean is more appropriate estimator. | Definition of autocorrelation time (for effective sample size) | see
http://arxiv.org/pdf/1403.5536v1.pdf
and
https://cran.r-project.org/web/packages/mcmcse/mcmcse.pdf
for effective sample size. I think the alternative formulation using the ratio of sample varian | Definition of autocorrelation time (for effective sample size)
see
http://arxiv.org/pdf/1403.5536v1.pdf
and
https://cran.r-project.org/web/packages/mcmcse/mcmcse.pdf
for effective sample size. I think the alternative formulation using the ratio of sample variance and asymptotic Markov chain variance via batch mean is... | Definition of autocorrelation time (for effective sample size)
see
http://arxiv.org/pdf/1403.5536v1.pdf
and
https://cran.r-project.org/web/packages/mcmcse/mcmcse.pdf
for effective sample size. I think the alternative formulation using the ratio of sample varian |
10,467 | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao? | There exist many such coefficients (most are expressed here). Just try to meditate on what are the consequences of the differences in formulas, especially when you compute a matrix of coefficients.
Imagine, for example, that objects 1 and 2 similar, as objects 3 and 4 are. But 1 and 2 have many of the attributes on the... | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao? | There exist many such coefficients (most are expressed here). Just try to meditate on what are the consequences of the differences in formulas, especially when you compute a matrix of coefficients.
Im | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao?
There exist many such coefficients (most are expressed here). Just try to meditate on what are the consequences of the differences in formulas, especially when you compute a matrix of coefficients.
Imagine, for example, that objects 1 and... | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao?
There exist many such coefficients (most are expressed here). Just try to meditate on what are the consequences of the differences in formulas, especially when you compute a matrix of coefficients.
Im |
10,468 | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao? | The usefulness of the Tanimoto coefficient over the traditional accuracy (i.e. Russell-Rao) is evident in image analysis, when comparing a segmentation to a gold-standard. Consider these two images:
In each of these images which are binary 'masks', we have two objects of the same size but placed at slightly different ... | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao? | The usefulness of the Tanimoto coefficient over the traditional accuracy (i.e. Russell-Rao) is evident in image analysis, when comparing a segmentation to a gold-standard. Consider these two images:
| Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao?
The usefulness of the Tanimoto coefficient over the traditional accuracy (i.e. Russell-Rao) is evident in image analysis, when comparing a segmentation to a gold-standard. Consider these two images:
In each of these images which are bina... | Similarity Coefficients for binary data: Why choose Jaccard over Russell and Rao?
The usefulness of the Tanimoto coefficient over the traditional accuracy (i.e. Russell-Rao) is evident in image analysis, when comparing a segmentation to a gold-standard. Consider these two images:
|
10,469 | What are the assumptions of the permutation test? | The literature distinguishes between two types of permutations tests: (1) the randomization test is the permutation test where exchangeability is satisfied by random assignment of experimental units to conditions; (2) the permutation test is the exact same test but applied to a situation where other assumptions (i.e., ... | What are the assumptions of the permutation test? | The literature distinguishes between two types of permutations tests: (1) the randomization test is the permutation test where exchangeability is satisfied by random assignment of experimental units t | What are the assumptions of the permutation test?
The literature distinguishes between two types of permutations tests: (1) the randomization test is the permutation test where exchangeability is satisfied by random assignment of experimental units to conditions; (2) the permutation test is the exact same test but appl... | What are the assumptions of the permutation test?
The literature distinguishes between two types of permutations tests: (1) the randomization test is the permutation test where exchangeability is satisfied by random assignment of experimental units t |
10,470 | Does the Gibbs Sampling algorithm guarantee detailed balance? | You tried to show detailed balance for the Markov chain that is obtained by considering one transition of the Markov chain to be the 'Gibbs sweep' where you sample each component in turn from its conditional distribution. For this chain, detailed balance is not satisfied. The point is rather that each sampling of a par... | Does the Gibbs Sampling algorithm guarantee detailed balance? | You tried to show detailed balance for the Markov chain that is obtained by considering one transition of the Markov chain to be the 'Gibbs sweep' where you sample each component in turn from its cond | Does the Gibbs Sampling algorithm guarantee detailed balance?
You tried to show detailed balance for the Markov chain that is obtained by considering one transition of the Markov chain to be the 'Gibbs sweep' where you sample each component in turn from its conditional distribution. For this chain, detailed balance is ... | Does the Gibbs Sampling algorithm guarantee detailed balance?
You tried to show detailed balance for the Markov chain that is obtained by considering one transition of the Markov chain to be the 'Gibbs sweep' where you sample each component in turn from its cond |
10,471 | What does the y axis in a kernel density plot mean? [duplicate] | You are correct that
the area under the curve of a density function represents the probability of getting an x value between a range of x values
But remember area is not just height: width is also important. So if you have a spike at 0, if the width is very small (say 0.1) then the height can be quite a bit higher th... | What does the y axis in a kernel density plot mean? [duplicate] | You are correct that
the area under the curve of a density function represents the probability of getting an x value between a range of x values
But remember area is not just height: width is also i | What does the y axis in a kernel density plot mean? [duplicate]
You are correct that
the area under the curve of a density function represents the probability of getting an x value between a range of x values
But remember area is not just height: width is also important. So if you have a spike at 0, if the width is v... | What does the y axis in a kernel density plot mean? [duplicate]
You are correct that
the area under the curve of a density function represents the probability of getting an x value between a range of x values
But remember area is not just height: width is also i |
10,472 | Interpreting the difference between lognormal and power law distribution (network degree distribution) | I think it will be helpful to separate the question into two parts:
What is the functional form of your empirical distribution? and
What does that functional form imply about the generating process in your network?
The first question is a statistics question. If you've applied the methods of Clauset et al. for fittin... | Interpreting the difference between lognormal and power law distribution (network degree distributio | I think it will be helpful to separate the question into two parts:
What is the functional form of your empirical distribution? and
What does that functional form imply about the generating process i | Interpreting the difference between lognormal and power law distribution (network degree distribution)
I think it will be helpful to separate the question into two parts:
What is the functional form of your empirical distribution? and
What does that functional form imply about the generating process in your network?
... | Interpreting the difference between lognormal and power law distribution (network degree distributio
I think it will be helpful to separate the question into two parts:
What is the functional form of your empirical distribution? and
What does that functional form imply about the generating process i |
10,473 | Interpreting the difference between lognormal and power law distribution (network degree distribution) | Coming to this site after counting my bubble distributions and using power law for viscosity data.
Skimming through the example data sets in the power law paper by Clauset et al. they have put up some real horrors of data sets, far from the power law data sets to support their argument. Just from common sense I certain... | Interpreting the difference between lognormal and power law distribution (network degree distributio | Coming to this site after counting my bubble distributions and using power law for viscosity data.
Skimming through the example data sets in the power law paper by Clauset et al. they have put up some | Interpreting the difference between lognormal and power law distribution (network degree distribution)
Coming to this site after counting my bubble distributions and using power law for viscosity data.
Skimming through the example data sets in the power law paper by Clauset et al. they have put up some real horrors of ... | Interpreting the difference between lognormal and power law distribution (network degree distributio
Coming to this site after counting my bubble distributions and using power law for viscosity data.
Skimming through the example data sets in the power law paper by Clauset et al. they have put up some |
10,474 | Interpreting the difference between lognormal and power law distribution (network degree distribution) | I'm having a related conversation about this associated with a question I asked elsewhere on CrossValidated. There, I asked whether the gamma distribution was a good distribution to use in a simulation of a social network where the probability of ties is endogenous to some continuous "popularity" characteristic of node... | Interpreting the difference between lognormal and power law distribution (network degree distributio | I'm having a related conversation about this associated with a question I asked elsewhere on CrossValidated. There, I asked whether the gamma distribution was a good distribution to use in a simulatio | Interpreting the difference between lognormal and power law distribution (network degree distribution)
I'm having a related conversation about this associated with a question I asked elsewhere on CrossValidated. There, I asked whether the gamma distribution was a good distribution to use in a simulation of a social net... | Interpreting the difference between lognormal and power law distribution (network degree distributio
I'm having a related conversation about this associated with a question I asked elsewhere on CrossValidated. There, I asked whether the gamma distribution was a good distribution to use in a simulatio |
10,475 | Interpreting the difference between lognormal and power law distribution (network degree distribution) | The above results show that degree distribution can be both power law and lognormal, which may suggest that small world and scale free properties co-exist in the network under studied. To examine whether the network is scale free (with constant scaling parameter) with preferential attachment, experimental design is oft... | Interpreting the difference between lognormal and power law distribution (network degree distributio | The above results show that degree distribution can be both power law and lognormal, which may suggest that small world and scale free properties co-exist in the network under studied. To examine whet | Interpreting the difference between lognormal and power law distribution (network degree distribution)
The above results show that degree distribution can be both power law and lognormal, which may suggest that small world and scale free properties co-exist in the network under studied. To examine whether the network i... | Interpreting the difference between lognormal and power law distribution (network degree distributio
The above results show that degree distribution can be both power law and lognormal, which may suggest that small world and scale free properties co-exist in the network under studied. To examine whet |
10,476 | Interpreting the difference between lognormal and power law distribution (network degree distribution) | Check out this 2019 article: https://www.nature.com/articles/s41467-019-08746-5
Contrary to the claims of much of network science, applying robust statistical tools to nearly 1000 social, biological, technical, transportation, and information networks showed that a log-normal distribution fit the data as well or better... | Interpreting the difference between lognormal and power law distribution (network degree distributio | Check out this 2019 article: https://www.nature.com/articles/s41467-019-08746-5
Contrary to the claims of much of network science, applying robust statistical tools to nearly 1000 social, biological, | Interpreting the difference between lognormal and power law distribution (network degree distribution)
Check out this 2019 article: https://www.nature.com/articles/s41467-019-08746-5
Contrary to the claims of much of network science, applying robust statistical tools to nearly 1000 social, biological, technical, transp... | Interpreting the difference between lognormal and power law distribution (network degree distributio
Check out this 2019 article: https://www.nature.com/articles/s41467-019-08746-5
Contrary to the claims of much of network science, applying robust statistical tools to nearly 1000 social, biological, |
10,477 | Interpreting the difference between lognormal and power law distribution (network degree distribution) | in our work, log normal implies that the underlying system is limit cycle attractive whereas power law implies that it is unstable periodic or chaos if you like. as limit cycle doesn't really exist in nature, this is really a question of degree. of course, chaos too is limited in its representation of systems as being ... | Interpreting the difference between lognormal and power law distribution (network degree distributio | in our work, log normal implies that the underlying system is limit cycle attractive whereas power law implies that it is unstable periodic or chaos if you like. as limit cycle doesn't really exist in | Interpreting the difference between lognormal and power law distribution (network degree distribution)
in our work, log normal implies that the underlying system is limit cycle attractive whereas power law implies that it is unstable periodic or chaos if you like. as limit cycle doesn't really exist in nature, this is ... | Interpreting the difference between lognormal and power law distribution (network degree distributio
in our work, log normal implies that the underlying system is limit cycle attractive whereas power law implies that it is unstable periodic or chaos if you like. as limit cycle doesn't really exist in |
10,478 | What optimization methods work best for LSTMs? | Ironically the best Optimizers for LSTMs are themselves LSTMs:
https://arxiv.org/abs/1606.04474
Learning to learn by gradient descent by gradient descent.
The basic idea is to use a neural network (specifically here a LSTM network) to co-learn and teach the gradients of the original network. It's called meta learning... | What optimization methods work best for LSTMs? | Ironically the best Optimizers for LSTMs are themselves LSTMs:
https://arxiv.org/abs/1606.04474
Learning to learn by gradient descent by gradient descent.
The basic idea is to use a neural network ( | What optimization methods work best for LSTMs?
Ironically the best Optimizers for LSTMs are themselves LSTMs:
https://arxiv.org/abs/1606.04474
Learning to learn by gradient descent by gradient descent.
The basic idea is to use a neural network (specifically here a LSTM network) to co-learn and teach the gradients of ... | What optimization methods work best for LSTMs?
Ironically the best Optimizers for LSTMs are themselves LSTMs:
https://arxiv.org/abs/1606.04474
Learning to learn by gradient descent by gradient descent.
The basic idea is to use a neural network ( |
10,479 | What optimization methods work best for LSTMs? | There is in general no clear evidence as to which optimisation method to use in what scenario. There has been some analysis in the behaviour of these methods under different scenarios however nothing is conclusive. If you want to dive into this stuff then I recommend:
http://papers.nips.cc/paper/5486-identifying-and-at... | What optimization methods work best for LSTMs? | There is in general no clear evidence as to which optimisation method to use in what scenario. There has been some analysis in the behaviour of these methods under different scenarios however nothing | What optimization methods work best for LSTMs?
There is in general no clear evidence as to which optimisation method to use in what scenario. There has been some analysis in the behaviour of these methods under different scenarios however nothing is conclusive. If you want to dive into this stuff then I recommend:
http... | What optimization methods work best for LSTMs?
There is in general no clear evidence as to which optimisation method to use in what scenario. There has been some analysis in the behaviour of these methods under different scenarios however nothing |
10,480 | What's wrong with t-SNE vs PCA for dimensional reduction using R? | You have to understand what TSNE does before you use it.
It starts by building a neighboorhood graph between feature vectors based on distance.
The graph connects a node(feature vector) to its n nearest nodes(in terms of distance in feature space). This n is called the perplexity parameter.
The purpose of building... | What's wrong with t-SNE vs PCA for dimensional reduction using R? | You have to understand what TSNE does before you use it.
It starts by building a neighboorhood graph between feature vectors based on distance.
The graph connects a node(feature vector) to its n ne | What's wrong with t-SNE vs PCA for dimensional reduction using R?
You have to understand what TSNE does before you use it.
It starts by building a neighboorhood graph between feature vectors based on distance.
The graph connects a node(feature vector) to its n nearest nodes(in terms of distance in feature space). Th... | What's wrong with t-SNE vs PCA for dimensional reduction using R?
You have to understand what TSNE does before you use it.
It starts by building a neighboorhood graph between feature vectors based on distance.
The graph connects a node(feature vector) to its n ne |
10,481 | What's wrong with t-SNE vs PCA for dimensional reduction using R? | It is hard to compare these approaches.
PCA is parameter free. Given the data, you just have to look at the principal components.
On the other hand, t-SNE relies on severe parameters : perplexity, early exaggeration, learning rate, number of iterations - though default values usually provide good results.
So you can't... | What's wrong with t-SNE vs PCA for dimensional reduction using R? | It is hard to compare these approaches.
PCA is parameter free. Given the data, you just have to look at the principal components.
On the other hand, t-SNE relies on severe parameters : perplexity, ear | What's wrong with t-SNE vs PCA for dimensional reduction using R?
It is hard to compare these approaches.
PCA is parameter free. Given the data, you just have to look at the principal components.
On the other hand, t-SNE relies on severe parameters : perplexity, early exaggeration, learning rate, number of iterations -... | What's wrong with t-SNE vs PCA for dimensional reduction using R?
It is hard to compare these approaches.
PCA is parameter free. Given the data, you just have to look at the principal components.
On the other hand, t-SNE relies on severe parameters : perplexity, ear |
10,482 | What's wrong with t-SNE vs PCA for dimensional reduction using R? | I ran t-sne on a dataset to replace PCA and (despite the bug that Rum Wei noticed) got better results. In my application case, rough pca worked well while rough t-sne gave me random looking results. It was due to the scaling/centering step included in the pca (by default in most packages) but not used in the t-sne.
My ... | What's wrong with t-SNE vs PCA for dimensional reduction using R? | I ran t-sne on a dataset to replace PCA and (despite the bug that Rum Wei noticed) got better results. In my application case, rough pca worked well while rough t-sne gave me random looking results. I | What's wrong with t-SNE vs PCA for dimensional reduction using R?
I ran t-sne on a dataset to replace PCA and (despite the bug that Rum Wei noticed) got better results. In my application case, rough pca worked well while rough t-sne gave me random looking results. It was due to the scaling/centering step included in th... | What's wrong with t-SNE vs PCA for dimensional reduction using R?
I ran t-sne on a dataset to replace PCA and (despite the bug that Rum Wei noticed) got better results. In my application case, rough pca worked well while rough t-sne gave me random looking results. I |
10,483 | What's wrong with t-SNE vs PCA for dimensional reduction using R? | An important difference between methods like PCA and SVD with tSNE is that tSNE is using a non-linear scale. This often makes for plots that are more visually balanced but be careful interpreting them in the same manner as you would for PCA. This difference likely accounts for the difference between the plots shown abo... | What's wrong with t-SNE vs PCA for dimensional reduction using R? | An important difference between methods like PCA and SVD with tSNE is that tSNE is using a non-linear scale. This often makes for plots that are more visually balanced but be careful interpreting them | What's wrong with t-SNE vs PCA for dimensional reduction using R?
An important difference between methods like PCA and SVD with tSNE is that tSNE is using a non-linear scale. This often makes for plots that are more visually balanced but be careful interpreting them in the same manner as you would for PCA. This differe... | What's wrong with t-SNE vs PCA for dimensional reduction using R?
An important difference between methods like PCA and SVD with tSNE is that tSNE is using a non-linear scale. This often makes for plots that are more visually balanced but be careful interpreting them |
10,484 | Specifying a difference in differences model with multiple time periods | The typical way to estimate a difference in differences model with more than two time periods is your proposed solution b). Keeping your notation you would regress
$$Y_{ist} = \alpha +\gamma_s (\text{Treatment}_s) + \lambda (\text{year dummy}_t) + \delta D_{st} + \epsilon_{ist}$$
where $D_t \equiv \text{Treatment}_s\cd... | Specifying a difference in differences model with multiple time periods | The typical way to estimate a difference in differences model with more than two time periods is your proposed solution b). Keeping your notation you would regress
$$Y_{ist} = \alpha +\gamma_s (\text{ | Specifying a difference in differences model with multiple time periods
The typical way to estimate a difference in differences model with more than two time periods is your proposed solution b). Keeping your notation you would regress
$$Y_{ist} = \alpha +\gamma_s (\text{Treatment}_s) + \lambda (\text{year dummy}_t) + ... | Specifying a difference in differences model with multiple time periods
The typical way to estimate a difference in differences model with more than two time periods is your proposed solution b). Keeping your notation you would regress
$$Y_{ist} = \alpha +\gamma_s (\text{ |
10,485 | Specifying a difference in differences model with multiple time periods | I would like to clarify something (and indirectly address a question in the comments). In particular, it concerns the use of unit-specific linear time trends. As a robustness check, it would appear you are only interacting dummies for treated units (i.e., $\gamma_{1s}$) with a continuous time trend. However, it is actu... | Specifying a difference in differences model with multiple time periods | I would like to clarify something (and indirectly address a question in the comments). In particular, it concerns the use of unit-specific linear time trends. As a robustness check, it would appear yo | Specifying a difference in differences model with multiple time periods
I would like to clarify something (and indirectly address a question in the comments). In particular, it concerns the use of unit-specific linear time trends. As a robustness check, it would appear you are only interacting dummies for treated units... | Specifying a difference in differences model with multiple time periods
I would like to clarify something (and indirectly address a question in the comments). In particular, it concerns the use of unit-specific linear time trends. As a robustness check, it would appear yo |
10,486 | How should mixed effects models be compared and or validated? | The main problem on model selection in mixed models is to define the degrees of freedom (df) of a model, truly. To compute df of a mixed model, one has to define the number of estimated parameters including fixed and random effects. And this is not straightforward. This paper by Jiming Jiang and others (2008) entitled ... | How should mixed effects models be compared and or validated? | The main problem on model selection in mixed models is to define the degrees of freedom (df) of a model, truly. To compute df of a mixed model, one has to define the number of estimated parameters inc | How should mixed effects models be compared and or validated?
The main problem on model selection in mixed models is to define the degrees of freedom (df) of a model, truly. To compute df of a mixed model, one has to define the number of estimated parameters including fixed and random effects. And this is not straightf... | How should mixed effects models be compared and or validated?
The main problem on model selection in mixed models is to define the degrees of freedom (df) of a model, truly. To compute df of a mixed model, one has to define the number of estimated parameters inc |
10,487 | How should mixed effects models be compared and or validated? | One way to compare models (whether mixed or otherwise) is to plot results. Suppose you havae model A and model B; produce the fitted values from each and graph them against each other in a scatter plot. If the values are very similar (using your judgement as to whether they are) choose the simpler model. Another idea i... | How should mixed effects models be compared and or validated? | One way to compare models (whether mixed or otherwise) is to plot results. Suppose you havae model A and model B; produce the fitted values from each and graph them against each other in a scatter plo | How should mixed effects models be compared and or validated?
One way to compare models (whether mixed or otherwise) is to plot results. Suppose you havae model A and model B; produce the fitted values from each and graph them against each other in a scatter plot. If the values are very similar (using your judgement as... | How should mixed effects models be compared and or validated?
One way to compare models (whether mixed or otherwise) is to plot results. Suppose you havae model A and model B; produce the fitted values from each and graph them against each other in a scatter plo |
10,488 | How likely am I to be descended from a particular person born in the year 1300? | Because this question is receiving answers that vary from astronomically small to almost 100%, I would like to offer a simulation to serve as a reference and inspiration for improved solutions.
I call these "flame plots." Each one documents the dispersion of genetic material within a population as it reproduces in dis... | How likely am I to be descended from a particular person born in the year 1300? | Because this question is receiving answers that vary from astronomically small to almost 100%, I would like to offer a simulation to serve as a reference and inspiration for improved solutions.
I call | How likely am I to be descended from a particular person born in the year 1300?
Because this question is receiving answers that vary from astronomically small to almost 100%, I would like to offer a simulation to serve as a reference and inspiration for improved solutions.
I call these "flame plots." Each one document... | How likely am I to be descended from a particular person born in the year 1300?
Because this question is receiving answers that vary from astronomically small to almost 100%, I would like to offer a simulation to serve as a reference and inspiration for improved solutions.
I call |
10,489 | How likely am I to be descended from a particular person born in the year 1300? | What happens when you try counting ancestors?
You have 2 parents, 4 grandparents, 8 great grandparents, ... So if you go back $n$ generations then you have $2^n$ ancestors. Let's assume an average generation length of $25$ years. Then there have been about $28$ generations since 1300, which gives us about 268 million... | How likely am I to be descended from a particular person born in the year 1300? | What happens when you try counting ancestors?
You have 2 parents, 4 grandparents, 8 great grandparents, ... So if you go back $n$ generations then you have $2^n$ ancestors. Let's assume an average ge | How likely am I to be descended from a particular person born in the year 1300?
What happens when you try counting ancestors?
You have 2 parents, 4 grandparents, 8 great grandparents, ... So if you go back $n$ generations then you have $2^n$ ancestors. Let's assume an average generation length of $25$ years. Then the... | How likely am I to be descended from a particular person born in the year 1300?
What happens when you try counting ancestors?
You have 2 parents, 4 grandparents, 8 great grandparents, ... So if you go back $n$ generations then you have $2^n$ ancestors. Let's assume an average ge |
10,490 | How likely am I to be descended from a particular person born in the year 1300? | The further back you go, the more likely that you are related to a person that successfully passed along their genes that lived in that time. Of the 1/4 billion ancestors that you have that lived in 1300, many of them would show up hundreds (if not thousands, millions) of times in your family tree. Genetic drift and t... | How likely am I to be descended from a particular person born in the year 1300? | The further back you go, the more likely that you are related to a person that successfully passed along their genes that lived in that time. Of the 1/4 billion ancestors that you have that lived in 1 | How likely am I to be descended from a particular person born in the year 1300?
The further back you go, the more likely that you are related to a person that successfully passed along their genes that lived in that time. Of the 1/4 billion ancestors that you have that lived in 1300, many of them would show up hundreds... | How likely am I to be descended from a particular person born in the year 1300?
The further back you go, the more likely that you are related to a person that successfully passed along their genes that lived in that time. Of the 1/4 billion ancestors that you have that lived in 1 |
10,491 | How likely am I to be descended from a particular person born in the year 1300? | The probability is=1-z, every descendant in this problem is related to ancestors above. Whatever the initial rate of reproduction is (1-z) is your probability of being descendant from someone in the initial population.Only uncertain probability is what are the chances of being alive in final population.
I agree with Er... | How likely am I to be descended from a particular person born in the year 1300? | The probability is=1-z, every descendant in this problem is related to ancestors above. Whatever the initial rate of reproduction is (1-z) is your probability of being descendant from someone in the i | How likely am I to be descended from a particular person born in the year 1300?
The probability is=1-z, every descendant in this problem is related to ancestors above. Whatever the initial rate of reproduction is (1-z) is your probability of being descendant from someone in the initial population.Only uncertain probabi... | How likely am I to be descended from a particular person born in the year 1300?
The probability is=1-z, every descendant in this problem is related to ancestors above. Whatever the initial rate of reproduction is (1-z) is your probability of being descendant from someone in the i |
10,492 | How likely am I to be descended from a particular person born in the year 1300? | My updated short answer is:
$$ p > {(1-z)} \times {{{1} \over {n_1(1-z)}} \over {2}} = {2 \over n_1} $$
Answer explained:
Given a particular person today, it is certain that they are a descendant of at least 2 people in 1300.
When picking a particular person in 1300, there is (1-z) chance that person never reproduced, ... | How likely am I to be descended from a particular person born in the year 1300? | My updated short answer is:
$$ p > {(1-z)} \times {{{1} \over {n_1(1-z)}} \over {2}} = {2 \over n_1} $$
Answer explained:
Given a particular person today, it is certain that they are a descendant of a | How likely am I to be descended from a particular person born in the year 1300?
My updated short answer is:
$$ p > {(1-z)} \times {{{1} \over {n_1(1-z)}} \over {2}} = {2 \over n_1} $$
Answer explained:
Given a particular person today, it is certain that they are a descendant of at least 2 people in 1300.
When picking a... | How likely am I to be descended from a particular person born in the year 1300?
My updated short answer is:
$$ p > {(1-z)} \times {{{1} \over {n_1(1-z)}} \over {2}} = {2 \over n_1} $$
Answer explained:
Given a particular person today, it is certain that they are a descendant of a |
10,493 | How likely am I to be descended from a particular person born in the year 1300? | This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.
The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_1}$ and at the limit generation will approach $\lim_{k \to \infty } p_k = (1-z)$... | How likely am I to be descended from a particular person born in the year 1300? | This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.
The % of the population which each generation related to will grow over each | How likely am I to be descended from a particular person born in the year 1300?
This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.
The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_... | How likely am I to be descended from a particular person born in the year 1300?
This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.
The % of the population which each generation related to will grow over each |
10,494 | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood? | As it turns out you can use profile maximum likelihood to ensure the necessary properties. You can prove that for given $\hat\theta$, $l(\hat\theta,\Sigma)$ is maximised by
$$\hat\Sigma=\frac{1}{n}\sum_{i=1}^n\hat{\varepsilon}_i\hat{\varepsilon}_i',$$
where
$$\hat{\varepsilon}_i=y_i-f(x_i,\hat\theta)$$
Then it is possi... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l | As it turns out you can use profile maximum likelihood to ensure the necessary properties. You can prove that for given $\hat\theta$, $l(\hat\theta,\Sigma)$ is maximised by
$$\hat\Sigma=\frac{1}{n}\su | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood?
As it turns out you can use profile maximum likelihood to ensure the necessary properties. You can prove that for given $\hat\theta$, $l(\hat\theta,\Sigma)$ is maximised by
$$\hat\Sigma=\frac{1}{n}\sum_{i=1}^n... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l
As it turns out you can use profile maximum likelihood to ensure the necessary properties. You can prove that for given $\hat\theta$, $l(\hat\theta,\Sigma)$ is maximised by
$$\hat\Sigma=\frac{1}{n}\su |
10,495 | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood? | Assuming that in constructing the covariance matrix, you are automatically taking care of the symmetry issue, your log-likelihood will be $-\infty$ when $\Sigma$ is not positive definite because of the $\log {\rm det} \ \Sigma$ term in the model right? To prevent a numerical error if ${\rm det} \ \Sigma < 0$ I would pr... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l | Assuming that in constructing the covariance matrix, you are automatically taking care of the symmetry issue, your log-likelihood will be $-\infty$ when $\Sigma$ is not positive definite because of th | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood?
Assuming that in constructing the covariance matrix, you are automatically taking care of the symmetry issue, your log-likelihood will be $-\infty$ when $\Sigma$ is not positive definite because of the $\log {... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l
Assuming that in constructing the covariance matrix, you are automatically taking care of the symmetry issue, your log-likelihood will be $-\infty$ when $\Sigma$ is not positive definite because of th |
10,496 | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood? | An alternative parameterization for the covariance matrix is in terms of eigenvalues $\lambda_1,...,\lambda_p$ and $p(p-1)/2$ "Givens" angles $\theta_ij$.
That is, we can write
$$\Sigma = G^T \Lambda G$$
where $G$ is orthonormal, and
$$\Lambda = diag(\lambda_1, ..., \lambda_p)$$
with $\lambda_1 \geq ... \geq \lambda_p ... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l | An alternative parameterization for the covariance matrix is in terms of eigenvalues $\lambda_1,...,\lambda_p$ and $p(p-1)/2$ "Givens" angles $\theta_ij$.
That is, we can write
$$\Sigma = G^T \Lambda | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood?
An alternative parameterization for the covariance matrix is in terms of eigenvalues $\lambda_1,...,\lambda_p$ and $p(p-1)/2$ "Givens" angles $\theta_ij$.
That is, we can write
$$\Sigma = G^T \Lambda G$$
where... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l
An alternative parameterization for the covariance matrix is in terms of eigenvalues $\lambda_1,...,\lambda_p$ and $p(p-1)/2$ "Givens" angles $\theta_ij$.
That is, we can write
$$\Sigma = G^T \Lambda |
10,497 | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood? | Along the lines of charles.y.zheng's solution, you may wish to model $\Sigma = \Lambda + C C^{\top}$, where $\Lambda$ is a diagonal matrix, and $C$ is a Cholesky factorization of a rank update to $\Lambda$. You only then need to keep the diagonal of $\Lambda$ positive to keep $\Sigma$ positive definite. That is, you sh... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l | Along the lines of charles.y.zheng's solution, you may wish to model $\Sigma = \Lambda + C C^{\top}$, where $\Lambda$ is a diagonal matrix, and $C$ is a Cholesky factorization of a rank update to $\La | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum likelihood?
Along the lines of charles.y.zheng's solution, you may wish to model $\Sigma = \Lambda + C C^{\top}$, where $\Lambda$ is a diagonal matrix, and $C$ is a Cholesky factorization of a rank update to $\Lambda$. Yo... | How to ensure properties of covariance matrix when fitting multivariate normal model using maximum l
Along the lines of charles.y.zheng's solution, you may wish to model $\Sigma = \Lambda + C C^{\top}$, where $\Lambda$ is a diagonal matrix, and $C$ is a Cholesky factorization of a rank update to $\La |
10,498 | Having a conjugate prior: Deep property or mathematical accident? | It is not by accident. Here you shall find a brief a very nice review on conjugate priors. Concretely, it mentions that if there exist a set of sufficient statistics of fixed dimension for the given likelihood function, then you can construct a conjugate prior to it. Having a set of sufficient statistics means that you... | Having a conjugate prior: Deep property or mathematical accident? | It is not by accident. Here you shall find a brief a very nice review on conjugate priors. Concretely, it mentions that if there exist a set of sufficient statistics of fixed dimension for the given l | Having a conjugate prior: Deep property or mathematical accident?
It is not by accident. Here you shall find a brief a very nice review on conjugate priors. Concretely, it mentions that if there exist a set of sufficient statistics of fixed dimension for the given likelihood function, then you can construct a conjugate... | Having a conjugate prior: Deep property or mathematical accident?
It is not by accident. Here you shall find a brief a very nice review on conjugate priors. Concretely, it mentions that if there exist a set of sufficient statistics of fixed dimension for the given l |
10,499 | Having a conjugate prior: Deep property or mathematical accident? | I am very new to Bayesian statistics, but it seems to me that all of these distributions (and if not all of them then at least those that are useful) share the property that they are described by some limited metric about the observations that define them. I.e., for a normal distribution, you don't need to know every d... | Having a conjugate prior: Deep property or mathematical accident? | I am very new to Bayesian statistics, but it seems to me that all of these distributions (and if not all of them then at least those that are useful) share the property that they are described by some | Having a conjugate prior: Deep property or mathematical accident?
I am very new to Bayesian statistics, but it seems to me that all of these distributions (and if not all of them then at least those that are useful) share the property that they are described by some limited metric about the observations that define the... | Having a conjugate prior: Deep property or mathematical accident?
I am very new to Bayesian statistics, but it seems to me that all of these distributions (and if not all of them then at least those that are useful) share the property that they are described by some |
10,500 | Having a conjugate prior: Deep property or mathematical accident? | What properties are "deep" is a very subjective issue! so the answer depends on your concept of "deep". But, if having conjugate priors is a "deep" property, in some sense, then that sense is mathematical and not statistical. The only reason that (some) statisticians are interested in conjugate priors is that they simp... | Having a conjugate prior: Deep property or mathematical accident? | What properties are "deep" is a very subjective issue! so the answer depends on your concept of "deep". But, if having conjugate priors is a "deep" property, in some sense, then that sense is mathemat | Having a conjugate prior: Deep property or mathematical accident?
What properties are "deep" is a very subjective issue! so the answer depends on your concept of "deep". But, if having conjugate priors is a "deep" property, in some sense, then that sense is mathematical and not statistical. The only reason that (some) ... | Having a conjugate prior: Deep property or mathematical accident?
What properties are "deep" is a very subjective issue! so the answer depends on your concept of "deep". But, if having conjugate priors is a "deep" property, in some sense, then that sense is mathemat |
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