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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54,001 | Continuity correction in proportion test | In order to consider what you asked, without additional material, consider the $2\times 2$ table TAB as follows:
TAB = rbind(c(80,90), c(20,15))
TAB
[,1] [,2]
[1,] 80 90
[2,] 20 15
A chi-squared test of independence with the Yates
correction is as follows:
chisq.test(TAB)
Pearson's Chi-squared te... | Continuity correction in proportion test | In order to consider what you asked, without additional material, consider the $2\times 2$ table TAB as follows:
TAB = rbind(c(80,90), c(20,15))
TAB
[,1] [,2]
[1,] 80 90
[2,] 20 15
A chi | Continuity correction in proportion test
In order to consider what you asked, without additional material, consider the $2\times 2$ table TAB as follows:
TAB = rbind(c(80,90), c(20,15))
TAB
[,1] [,2]
[1,] 80 90
[2,] 20 15
A chi-squared test of independence with the Yates
correction is as follows:
chisq.te... | Continuity correction in proportion test
In order to consider what you asked, without additional material, consider the $2\times 2$ table TAB as follows:
TAB = rbind(c(80,90), c(20,15))
TAB
[,1] [,2]
[1,] 80 90
[2,] 20 15
A chi |
54,002 | Continuity correction in proportion test | There are many opinions on how to best test for 2 x 2 contingency tables like this. Here the prop.test implements a chi-squared test. The issue is that in some cases data from a 2 x 2 contingency table isn't well approximated by the chi-squared distribution under the null. This can lead to p-values that are too small. ... | Continuity correction in proportion test | There are many opinions on how to best test for 2 x 2 contingency tables like this. Here the prop.test implements a chi-squared test. The issue is that in some cases data from a 2 x 2 contingency tabl | Continuity correction in proportion test
There are many opinions on how to best test for 2 x 2 contingency tables like this. Here the prop.test implements a chi-squared test. The issue is that in some cases data from a 2 x 2 contingency table isn't well approximated by the chi-squared distribution under the null. This ... | Continuity correction in proportion test
There are many opinions on how to best test for 2 x 2 contingency tables like this. Here the prop.test implements a chi-squared test. The issue is that in some cases data from a 2 x 2 contingency tabl |
54,003 | Controlling for non-linear variable in non-linear modeling of response | The intent is to understand how much an increase of 1 in one of the features (e.g. $𝑥_1$) would on average impact $y$ while controlling for $x_c$. (Emphasis added.)
First, given nonlinear associations of predictors with outcome, there isn't a unique answer. You have to specify a particular value of $x_1$ from which t... | Controlling for non-linear variable in non-linear modeling of response | The intent is to understand how much an increase of 1 in one of the features (e.g. $𝑥_1$) would on average impact $y$ while controlling for $x_c$. (Emphasis added.)
First, given nonlinear association | Controlling for non-linear variable in non-linear modeling of response
The intent is to understand how much an increase of 1 in one of the features (e.g. $𝑥_1$) would on average impact $y$ while controlling for $x_c$. (Emphasis added.)
First, given nonlinear associations of predictors with outcome, there isn't a uniq... | Controlling for non-linear variable in non-linear modeling of response
The intent is to understand how much an increase of 1 in one of the features (e.g. $𝑥_1$) would on average impact $y$ while controlling for $x_c$. (Emphasis added.)
First, given nonlinear association |
54,004 | Controlling for non-linear variable in non-linear modeling of response | All you've really told us is that the relationships are nonlinear, and you are asking us to tell you how to get the best model. It's not really possible to do that, since "nonlinear" is a huge class of relationships. Nevertheless, here are some things to think about:
Ideally, there would be some aspect of the variab... | Controlling for non-linear variable in non-linear modeling of response | All you've really told us is that the relationships are nonlinear, and you are asking us to tell you how to get the best model. It's not really possible to do that, since "nonlinear" is a huge class | Controlling for non-linear variable in non-linear modeling of response
All you've really told us is that the relationships are nonlinear, and you are asking us to tell you how to get the best model. It's not really possible to do that, since "nonlinear" is a huge class of relationships. Nevertheless, here are some th... | Controlling for non-linear variable in non-linear modeling of response
All you've really told us is that the relationships are nonlinear, and you are asking us to tell you how to get the best model. It's not really possible to do that, since "nonlinear" is a huge class |
54,005 | Controlling for non-linear variable in non-linear modeling of response | You could train a non-linear model using only $x_c$ to predict $y$, i.e. $f(x_c) = \hat{y}$. Then compute the residual $\epsilon = \hat{y} - y$. The residual $\epsilon$ will have been controlled for with regards to $x_c$ as $f$ has captured the relationship between $x_c$ and $y$ as well as it possibly could.
You can no... | Controlling for non-linear variable in non-linear modeling of response | You could train a non-linear model using only $x_c$ to predict $y$, i.e. $f(x_c) = \hat{y}$. Then compute the residual $\epsilon = \hat{y} - y$. The residual $\epsilon$ will have been controlled for w | Controlling for non-linear variable in non-linear modeling of response
You could train a non-linear model using only $x_c$ to predict $y$, i.e. $f(x_c) = \hat{y}$. Then compute the residual $\epsilon = \hat{y} - y$. The residual $\epsilon$ will have been controlled for with regards to $x_c$ as $f$ has captured the rela... | Controlling for non-linear variable in non-linear modeling of response
You could train a non-linear model using only $x_c$ to predict $y$, i.e. $f(x_c) = \hat{y}$. Then compute the residual $\epsilon = \hat{y} - y$. The residual $\epsilon$ will have been controlled for w |
54,006 | Estimating confidence interval of a parameter from the MLE of another parameter | The question, as clarified in comments, is a general one about confidence intervals. It is best framed generally, because the generality strips away irrelevant details to bring out the main idea.
So, imagine a statistical estimation setting in which a sample $\mathbf X = (X_1,X_2,\ldots,X_n)$ is assumed to be governed... | Estimating confidence interval of a parameter from the MLE of another parameter | The question, as clarified in comments, is a general one about confidence intervals. It is best framed generally, because the generality strips away irrelevant details to bring out the main idea.
So, | Estimating confidence interval of a parameter from the MLE of another parameter
The question, as clarified in comments, is a general one about confidence intervals. It is best framed generally, because the generality strips away irrelevant details to bring out the main idea.
So, imagine a statistical estimation settin... | Estimating confidence interval of a parameter from the MLE of another parameter
The question, as clarified in comments, is a general one about confidence intervals. It is best framed generally, because the generality strips away irrelevant details to bring out the main idea.
So, |
54,007 | Are there classifiers with infinite number of classes? | According to your definition infinite number of classes should be possible.
One way to think about a classification rule is as a division of feature space into segments. Hence, if we consider a one-dimensional feature space $x_1 \in \mathbb{R} $ as an input we can imagine a classifier that returns a class of "$k$" for ... | Are there classifiers with infinite number of classes? | According to your definition infinite number of classes should be possible.
One way to think about a classification rule is as a division of feature space into segments. Hence, if we consider a one-di | Are there classifiers with infinite number of classes?
According to your definition infinite number of classes should be possible.
One way to think about a classification rule is as a division of feature space into segments. Hence, if we consider a one-dimensional feature space $x_1 \in \mathbb{R} $ as an input we can ... | Are there classifiers with infinite number of classes?
According to your definition infinite number of classes should be possible.
One way to think about a classification rule is as a division of feature space into segments. Hence, if we consider a one-di |
54,008 | Are there classifiers with infinite number of classes? | Classifying images to say which integer was depicted would be a classification problem with a countably infinite set of classes, but you would also need an infinitely large image as the input data. It would also be a rather bad way of solving the problem to view it as single classification problem, rather than constru... | Are there classifiers with infinite number of classes? | Classifying images to say which integer was depicted would be a classification problem with a countably infinite set of classes, but you would also need an infinitely large image as the input data. I | Are there classifiers with infinite number of classes?
Classifying images to say which integer was depicted would be a classification problem with a countably infinite set of classes, but you would also need an infinitely large image as the input data. It would also be a rather bad way of solving the problem to view i... | Are there classifiers with infinite number of classes?
Classifying images to say which integer was depicted would be a classification problem with a countably infinite set of classes, but you would also need an infinitely large image as the input data. I |
54,009 | Are there classifiers with infinite number of classes? | The Distance dependent Chinese restaurant process may be an example. You can think of it as a clustering method, in which the number of clusters is adaptively chosen based on the data. As you add more data points, the number of clusters can in principle increase without bound. So there are an infinite possible number o... | Are there classifiers with infinite number of classes? | The Distance dependent Chinese restaurant process may be an example. You can think of it as a clustering method, in which the number of clusters is adaptively chosen based on the data. As you add more | Are there classifiers with infinite number of classes?
The Distance dependent Chinese restaurant process may be an example. You can think of it as a clustering method, in which the number of clusters is adaptively chosen based on the data. As you add more data points, the number of clusters can in principle increase wi... | Are there classifiers with infinite number of classes?
The Distance dependent Chinese restaurant process may be an example. You can think of it as a clustering method, in which the number of clusters is adaptively chosen based on the data. As you add more |
54,010 | Visual Demonstration of Residual Confounding | Confounding is about bias. There are at least two ways I can think of in which dichotomization can bias a regression:
Model Bias. Dichotomizing a continuous predictor transforms the conditional mean from a continuous (possibly smooth, depending on the functional form) function to a discontinuous constant function. ... | Visual Demonstration of Residual Confounding | Confounding is about bias. There are at least two ways I can think of in which dichotomization can bias a regression:
Model Bias. Dichotomizing a continuous predictor transforms the conditional mea | Visual Demonstration of Residual Confounding
Confounding is about bias. There are at least two ways I can think of in which dichotomization can bias a regression:
Model Bias. Dichotomizing a continuous predictor transforms the conditional mean from a continuous (possibly smooth, depending on the functional form) fun... | Visual Demonstration of Residual Confounding
Confounding is about bias. There are at least two ways I can think of in which dichotomization can bias a regression:
Model Bias. Dichotomizing a continuous predictor transforms the conditional mea |
54,011 | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logistic regression model in R? | EDIT: I see you mentioned the Fieller method in your original post. Perhaps you were referring to the solution I provided below.
Here is a great paper on the topic. Using a logistic regression with a logit link function you can model the proportion of fish as a function of length, with $\lambda:=$LD50. Based on the a... | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logis | EDIT: I see you mentioned the Fieller method in your original post. Perhaps you were referring to the solution I provided below.
Here is a great paper on the topic. Using a logistic regression with a | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logistic regression model in R?
EDIT: I see you mentioned the Fieller method in your original post. Perhaps you were referring to the solution I provided below.
Here is a great paper on the topic. Using a logistic regression... | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logis
EDIT: I see you mentioned the Fieller method in your original post. Perhaps you were referring to the solution I provided below.
Here is a great paper on the topic. Using a logistic regression with a |
54,012 | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logistic regression model in R? | TL;DR @kjetilbhalvorsen explains how to compute confidence intervals based on the profile likelihood in Confidence interval for difference between two predicted probabilities in R. The entire thread is worth reading as it describes several methods to compute confidence intervals. I summarize the profile likelihood meth... | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logis | TL;DR @kjetilbhalvorsen explains how to compute confidence intervals based on the profile likelihood in Confidence interval for difference between two predicted probabilities in R. The entire thread i | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logistic regression model in R?
TL;DR @kjetilbhalvorsen explains how to compute confidence intervals based on the profile likelihood in Confidence interval for difference between two predicted probabilities in R. The entire t... | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logis
TL;DR @kjetilbhalvorsen explains how to compute confidence intervals based on the profile likelihood in Confidence interval for difference between two predicted probabilities in R. The entire thread i |
54,013 | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logistic regression model in R? | After some research on the profile likelihood function, it seems that R packages exist to estimate the confidence intervals of parameter (Beta) estimates of a logistic regression model with this approach (e.g., ProfileLikelihood), on top of the confint() function that is already available for this. However, for the pre... | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logis | After some research on the profile likelihood function, it seems that R packages exist to estimate the confidence intervals of parameter (Beta) estimates of a logistic regression model with this appro | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logistic regression model in R?
After some research on the profile likelihood function, it seems that R packages exist to estimate the confidence intervals of parameter (Beta) estimates of a logistic regression model with thi... | How to easily obtain the profile likelihood 95% confidence interval for a predicted value in a logis
After some research on the profile likelihood function, it seems that R packages exist to estimate the confidence intervals of parameter (Beta) estimates of a logistic regression model with this appro |
54,014 | A realistic example of a non identifiable model? | Another classic example of non-identification is a dose response model like the Emax model, which assumes that some (continuous) outcome for patients $i=1,\ldots,N$ for a drug that you can give in different doses obeys the following:
$$Y_i \sim N( E_0 + E_\text{max} \frac{\text{dose}_i}{\text{dose}_i + \text{ED}_{50}},... | A realistic example of a non identifiable model? | Another classic example of non-identification is a dose response model like the Emax model, which assumes that some (continuous) outcome for patients $i=1,\ldots,N$ for a drug that you can give in dif | A realistic example of a non identifiable model?
Another classic example of non-identification is a dose response model like the Emax model, which assumes that some (continuous) outcome for patients $i=1,\ldots,N$ for a drug that you can give in different doses obeys the following:
$$Y_i \sim N( E_0 + E_\text{max} \fra... | A realistic example of a non identifiable model?
Another classic example of non-identification is a dose response model like the Emax model, which assumes that some (continuous) outcome for patients $i=1,\ldots,N$ for a drug that you can give in dif |
54,015 | A realistic example of a non identifiable model? | Non-identification is not just a consequence of having insufficient observations, and indeed, some of the most pernicious examples of not being able to identify parameters from data arise even if we had access to the entire population, as the below example demonstrates. The discussion below is fairly informal and tries... | A realistic example of a non identifiable model? | Non-identification is not just a consequence of having insufficient observations, and indeed, some of the most pernicious examples of not being able to identify parameters from data arise even if we h | A realistic example of a non identifiable model?
Non-identification is not just a consequence of having insufficient observations, and indeed, some of the most pernicious examples of not being able to identify parameters from data arise even if we had access to the entire population, as the below example demonstrates. ... | A realistic example of a non identifiable model?
Non-identification is not just a consequence of having insufficient observations, and indeed, some of the most pernicious examples of not being able to identify parameters from data arise even if we h |
54,016 | A realistic example of a non identifiable model? | In quantum mechanics appears many non-identifiable models in the problem of phase estimation. For example, when you try to estimate a phase $\theta \in [0, 2\pi)$ of a qubit, the $X \sim Ber(p(\theta))$, where $p(\theta) = \frac{1}{2}\left[1 + \sin(\theta) \right]$ which is $\pi$-periodic. | A realistic example of a non identifiable model? | In quantum mechanics appears many non-identifiable models in the problem of phase estimation. For example, when you try to estimate a phase $\theta \in [0, 2\pi)$ of a qubit, the $X \sim Ber(p(\theta) | A realistic example of a non identifiable model?
In quantum mechanics appears many non-identifiable models in the problem of phase estimation. For example, when you try to estimate a phase $\theta \in [0, 2\pi)$ of a qubit, the $X \sim Ber(p(\theta))$, where $p(\theta) = \frac{1}{2}\left[1 + \sin(\theta) \right]$ which... | A realistic example of a non identifiable model?
In quantum mechanics appears many non-identifiable models in the problem of phase estimation. For example, when you try to estimate a phase $\theta \in [0, 2\pi)$ of a qubit, the $X \sim Ber(p(\theta) |
54,017 | Expectation of double quadratic form | Although the solution is essentially and already contained in W. Huber's answer, here is a detailed derivation with a non-zero mean:
\begin{align}
\mathbb E(\overbrace{\hat{Y_k}'A\hat{Y_l}}^{\text{real}}\times\overbrace{\hat{Y_k}'A\hat{Y_l}}^{\text{real}})
&= \mathbb E(\hat{Y_k}'A\hat{Y_l}\hat{Y_l}'A'\hat{Y_k})\\
&= \m... | Expectation of double quadratic form | Although the solution is essentially and already contained in W. Huber's answer, here is a detailed derivation with a non-zero mean:
\begin{align}
\mathbb E(\overbrace{\hat{Y_k}'A\hat{Y_l}}^{\text{rea | Expectation of double quadratic form
Although the solution is essentially and already contained in W. Huber's answer, here is a detailed derivation with a non-zero mean:
\begin{align}
\mathbb E(\overbrace{\hat{Y_k}'A\hat{Y_l}}^{\text{real}}\times\overbrace{\hat{Y_k}'A\hat{Y_l}}^{\text{real}})
&= \mathbb E(\hat{Y_k}'A\h... | Expectation of double quadratic form
Although the solution is essentially and already contained in W. Huber's answer, here is a detailed derivation with a non-zero mean:
\begin{align}
\mathbb E(\overbrace{\hat{Y_k}'A\hat{Y_l}}^{\text{rea |
54,018 | How are True Negative and False Negative converted into True Positive and False Positive in ROC curve? | The ROC curve shows the TPR and FPR as you change the threshold. Your question posits that a classifier only has one TPR statistic and one FPR statistic, and that both of these statistics correspond to a threshold at 0.5. This is a common misconception; there is no reason that the threshold must be 0.5. For each choic... | How are True Negative and False Negative converted into True Positive and False Positive in ROC curv | The ROC curve shows the TPR and FPR as you change the threshold. Your question posits that a classifier only has one TPR statistic and one FPR statistic, and that both of these statistics correspond t | How are True Negative and False Negative converted into True Positive and False Positive in ROC curve?
The ROC curve shows the TPR and FPR as you change the threshold. Your question posits that a classifier only has one TPR statistic and one FPR statistic, and that both of these statistics correspond to a threshold at ... | How are True Negative and False Negative converted into True Positive and False Positive in ROC curv
The ROC curve shows the TPR and FPR as you change the threshold. Your question posits that a classifier only has one TPR statistic and one FPR statistic, and that both of these statistics correspond t |
54,019 | How are True Negative and False Negative converted into True Positive and False Positive in ROC curve? | you vary the threshold which is used to determine positives and negatives, ie at threshold 0.5 we get a point (TP, FP) which we plot, then at threshold 0.45 we get another point (TP, FP) which we plot, etc. These points make the ROC curve | How are True Negative and False Negative converted into True Positive and False Positive in ROC curv | you vary the threshold which is used to determine positives and negatives, ie at threshold 0.5 we get a point (TP, FP) which we plot, then at threshold 0.45 we get another point (TP, FP) which we plo | How are True Negative and False Negative converted into True Positive and False Positive in ROC curve?
you vary the threshold which is used to determine positives and negatives, ie at threshold 0.5 we get a point (TP, FP) which we plot, then at threshold 0.45 we get another point (TP, FP) which we plot, etc. These poi... | How are True Negative and False Negative converted into True Positive and False Positive in ROC curv
you vary the threshold which is used to determine positives and negatives, ie at threshold 0.5 we get a point (TP, FP) which we plot, then at threshold 0.45 we get another point (TP, FP) which we plo |
54,020 | What is the point of using a Bayesian prior? | There are many excellent thoughts here. There is a short answer to the question. If you want to gauge relative evidence you can sometimes get away without using any outside information. Relative evidence can be summarized by a likelihood ratio in the likelihoodist school of statistics. For example one may use study... | What is the point of using a Bayesian prior? | There are many excellent thoughts here. There is a short answer to the question. If you want to gauge relative evidence you can sometimes get away without using any outside information. Relative ev | What is the point of using a Bayesian prior?
There are many excellent thoughts here. There is a short answer to the question. If you want to gauge relative evidence you can sometimes get away without using any outside information. Relative evidence can be summarized by a likelihood ratio in the likelihoodist school ... | What is the point of using a Bayesian prior?
There are many excellent thoughts here. There is a short answer to the question. If you want to gauge relative evidence you can sometimes get away without using any outside information. Relative ev |
54,021 | What is the point of using a Bayesian prior? | Questions
why is using a prior useful?
How do you "not use a prior"? Even the frequentist approach has a prior—it's just unstated, but it's there all the same. There's always a prior, whether you like it or not.
As for which prior you should use, that's a different question, which other answers on this site go over... | What is the point of using a Bayesian prior? | Questions
why is using a prior useful?
How do you "not use a prior"? Even the frequentist approach has a prior—it's just unstated, but it's there all the same. There's always a prior, whether you | What is the point of using a Bayesian prior?
Questions
why is using a prior useful?
How do you "not use a prior"? Even the frequentist approach has a prior—it's just unstated, but it's there all the same. There's always a prior, whether you like it or not.
As for which prior you should use, that's a different quest... | What is the point of using a Bayesian prior?
Questions
why is using a prior useful?
How do you "not use a prior"? Even the frequentist approach has a prior—it's just unstated, but it's there all the same. There's always a prior, whether you |
54,022 | What is the point of using a Bayesian prior? | Here's a example of how to use Bayesian priors in a way that even "frequentists" agree is useful.
Let's say you want to estimate the know how well students at 100 different schools are doing in math, so you can identify schools that are doing particularly well or poorly. But you can only assess math knowledge through a... | What is the point of using a Bayesian prior? | Here's a example of how to use Bayesian priors in a way that even "frequentists" agree is useful.
Let's say you want to estimate the know how well students at 100 different schools are doing in math, | What is the point of using a Bayesian prior?
Here's a example of how to use Bayesian priors in a way that even "frequentists" agree is useful.
Let's say you want to estimate the know how well students at 100 different schools are doing in math, so you can identify schools that are doing particularly well or poorly. But... | What is the point of using a Bayesian prior?
Here's a example of how to use Bayesian priors in a way that even "frequentists" agree is useful.
Let's say you want to estimate the know how well students at 100 different schools are doing in math, |
54,023 | What is the point of using a Bayesian prior? | So an important area of work that I am involved in is decision making under uncertainty, particularly where money is placed at risk. You are asking the wrong question.
The first question should be when and where should I use a Bayesian method and when should I not, maybe never, use a Bayesian method. The subsidiary q... | What is the point of using a Bayesian prior? | So an important area of work that I am involved in is decision making under uncertainty, particularly where money is placed at risk. You are asking the wrong question.
The first question should be wh | What is the point of using a Bayesian prior?
So an important area of work that I am involved in is decision making under uncertainty, particularly where money is placed at risk. You are asking the wrong question.
The first question should be when and where should I use a Bayesian method and when should I not, maybe ne... | What is the point of using a Bayesian prior?
So an important area of work that I am involved in is decision making under uncertainty, particularly where money is placed at risk. You are asking the wrong question.
The first question should be wh |
54,024 | What is the point of using a Bayesian prior? | From whatever little I know about Bayesian method, the best use case is when you have limited pieces of evidence which by themselves are not enough to make a frequentist probability with reliability. But the little evidence you have can add to an informative prior probability to take you closer to the truth.
In the par... | What is the point of using a Bayesian prior? | From whatever little I know about Bayesian method, the best use case is when you have limited pieces of evidence which by themselves are not enough to make a frequentist probability with reliability. | What is the point of using a Bayesian prior?
From whatever little I know about Bayesian method, the best use case is when you have limited pieces of evidence which by themselves are not enough to make a frequentist probability with reliability. But the little evidence you have can add to an informative prior probabilit... | What is the point of using a Bayesian prior?
From whatever little I know about Bayesian method, the best use case is when you have limited pieces of evidence which by themselves are not enough to make a frequentist probability with reliability. |
54,025 | What is the point of using a Bayesian prior? | A prior is useful for incorporating historical information, analogous to a fixed effect meta-analysis. Because Bayesians define parameters as random variables and probability as the belief of the experimenter they may feel much freer to incorporate information from other studies or their personal beliefs. In contrast... | What is the point of using a Bayesian prior? | A prior is useful for incorporating historical information, analogous to a fixed effect meta-analysis. Because Bayesians define parameters as random variables and probability as the belief of the exp | What is the point of using a Bayesian prior?
A prior is useful for incorporating historical information, analogous to a fixed effect meta-analysis. Because Bayesians define parameters as random variables and probability as the belief of the experimenter they may feel much freer to incorporate information from other st... | What is the point of using a Bayesian prior?
A prior is useful for incorporating historical information, analogous to a fixed effect meta-analysis. Because Bayesians define parameters as random variables and probability as the belief of the exp |
54,026 | Are Random Forests trained with the whole dataset? | All previous answers are correct.
I would like just to clarify one point of your question:
My question is, how does this make sense? Because in theory, if we passed the whole training set to each individual model, wouldn't all the models be exactly the same?
As you have seen in previous answers, we are doing Bootstra... | Are Random Forests trained with the whole dataset? | All previous answers are correct.
I would like just to clarify one point of your question:
My question is, how does this make sense? Because in theory, if we passed the whole training set to each ind | Are Random Forests trained with the whole dataset?
All previous answers are correct.
I would like just to clarify one point of your question:
My question is, how does this make sense? Because in theory, if we passed the whole training set to each individual model, wouldn't all the models be exactly the same?
As you h... | Are Random Forests trained with the whole dataset?
All previous answers are correct.
I would like just to clarify one point of your question:
My question is, how does this make sense? Because in theory, if we passed the whole training set to each ind |
54,027 | Are Random Forests trained with the whole dataset? | As noticed in the comment, random forest uses bootstrap resamples of the training data. What this means is that for each tree we sample randomly with replacement the max_samples number of observations from the training data. When using bootstrap in statistics you generally want the number of bootstrap observations to b... | Are Random Forests trained with the whole dataset? | As noticed in the comment, random forest uses bootstrap resamples of the training data. What this means is that for each tree we sample randomly with replacement the max_samples number of observations | Are Random Forests trained with the whole dataset?
As noticed in the comment, random forest uses bootstrap resamples of the training data. What this means is that for each tree we sample randomly with replacement the max_samples number of observations from the training data. When using bootstrap in statistics you gener... | Are Random Forests trained with the whole dataset?
As noticed in the comment, random forest uses bootstrap resamples of the training data. What this means is that for each tree we sample randomly with replacement the max_samples number of observations |
54,028 | Are Random Forests trained with the whole dataset? | From the explanation of the BaggingClassifier:
max_samples: The number of samples to draw from X to train each base
estimator (with replacement by default, see bootstrap for more
details).
So, since the samples are drawn with replacement, it's a bootstrap sample with the same size of the whole dataset, but not exactl... | Are Random Forests trained with the whole dataset? | From the explanation of the BaggingClassifier:
max_samples: The number of samples to draw from X to train each base
estimator (with replacement by default, see bootstrap for more
details).
So, since | Are Random Forests trained with the whole dataset?
From the explanation of the BaggingClassifier:
max_samples: The number of samples to draw from X to train each base
estimator (with replacement by default, see bootstrap for more
details).
So, since the samples are drawn with replacement, it's a bootstrap sample with... | Are Random Forests trained with the whole dataset?
From the explanation of the BaggingClassifier:
max_samples: The number of samples to draw from X to train each base
estimator (with replacement by default, see bootstrap for more
details).
So, since |
54,029 | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$? | $X_1 ~|~ X_1+X_2$ and $X_2 ~|~ X_1+X_2$ are not independent. They are perfectly negatively correlated distributions. | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$? | $X_1 ~|~ X_1+X_2$ and $X_2 ~|~ X_1+X_2$ are not independent. They are perfectly negatively correlated distributions. | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$?
$X_1 ~|~ X_1+X_2$ and $X_2 ~|~ X_1+X_2$ are not independent. They are perfectly negatively correlated distributions. | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$?
$X_1 ~|~ X_1+X_2$ and $X_2 ~|~ X_1+X_2$ are not independent. They are perfectly negatively correlated distributions. |
54,030 | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$? | It's possible if one of them is constant - for example if $X_1$ has a Bernoulli distribution and $X_2$ is always equal to zero. | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$? | It's possible if one of them is constant - for example if $X_1$ has a Bernoulli distribution and $X_2$ is always equal to zero. | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$?
It's possible if one of them is constant - for example if $X_1$ has a Bernoulli distribution and $X_2$ is always equal to zero. | Can $X_1$ and $X_2$ be independent conditioning on $X_1+X_2$?
It's possible if one of them is constant - for example if $X_1$ has a Bernoulli distribution and $X_2$ is always equal to zero. |
54,031 | How does one design a custom loss function? What features make a loss function "good"? | To answer the titular question, a key characteristic of a loss function is that the loss is minimized at the target values $y$. In other words, if you're estimating a quantity, the least loss should be assigned to the estimates that are exactly correct.
Using $\mathcal{L}_\text{subtract}$ and divisive losses are not lo... | How does one design a custom loss function? What features make a loss function "good"? | To answer the titular question, a key characteristic of a loss function is that the loss is minimized at the target values $y$. In other words, if you're estimating a quantity, the least loss should b | How does one design a custom loss function? What features make a loss function "good"?
To answer the titular question, a key characteristic of a loss function is that the loss is minimized at the target values $y$. In other words, if you're estimating a quantity, the least loss should be assigned to the estimates that ... | How does one design a custom loss function? What features make a loss function "good"?
To answer the titular question, a key characteristic of a loss function is that the loss is minimized at the target values $y$. In other words, if you're estimating a quantity, the least loss should b |
54,032 | Why isn't the ROC curve naturally plotted in 3D? [duplicate] | First off, you are right that we could in principle plot the ROC curve as a curve in three-dimensional space. One axis would be the threshold, and the other two would be the TPR and the FPR.
As to why this is not done: I don't know who invented the ROC curve and what their thought processes are (that might be interesti... | Why isn't the ROC curve naturally plotted in 3D? [duplicate] | First off, you are right that we could in principle plot the ROC curve as a curve in three-dimensional space. One axis would be the threshold, and the other two would be the TPR and the FPR.
As to why | Why isn't the ROC curve naturally plotted in 3D? [duplicate]
First off, you are right that we could in principle plot the ROC curve as a curve in three-dimensional space. One axis would be the threshold, and the other two would be the TPR and the FPR.
As to why this is not done: I don't know who invented the ROC curve ... | Why isn't the ROC curve naturally plotted in 3D? [duplicate]
First off, you are right that we could in principle plot the ROC curve as a curve in three-dimensional space. One axis would be the threshold, and the other two would be the TPR and the FPR.
As to why |
54,033 | Why isn't the ROC curve naturally plotted in 3D? [duplicate] | Are the points even plotted according some sorted order of these
thresholds?
No, there is no guarantee that ordered thresholds be reflected on the curve.
Also, shouldn't the threshold be critical at creating your final
classifier? The ROC curve should have a point that corresponds to the
best threshold. But I don't k... | Why isn't the ROC curve naturally plotted in 3D? [duplicate] | Are the points even plotted according some sorted order of these
thresholds?
No, there is no guarantee that ordered thresholds be reflected on the curve.
Also, shouldn't the threshold be critical at | Why isn't the ROC curve naturally plotted in 3D? [duplicate]
Are the points even plotted according some sorted order of these
thresholds?
No, there is no guarantee that ordered thresholds be reflected on the curve.
Also, shouldn't the threshold be critical at creating your final
classifier? The ROC curve should have ... | Why isn't the ROC curve naturally plotted in 3D? [duplicate]
Are the points even plotted according some sorted order of these
thresholds?
No, there is no guarantee that ordered thresholds be reflected on the curve.
Also, shouldn't the threshold be critical at |
54,034 | Why isn't the ROC curve naturally plotted in 3D? [duplicate] | The purpose of an ROC curve is to summarize the performance of a classifier. The majority of binary classifiers in common use produce a score which can be trivially mapped to the interval [0, 1] (indeed it is standard to output a score that has been mapped this way). It stands to reason that as you increase sensitivity... | Why isn't the ROC curve naturally plotted in 3D? [duplicate] | The purpose of an ROC curve is to summarize the performance of a classifier. The majority of binary classifiers in common use produce a score which can be trivially mapped to the interval [0, 1] (inde | Why isn't the ROC curve naturally plotted in 3D? [duplicate]
The purpose of an ROC curve is to summarize the performance of a classifier. The majority of binary classifiers in common use produce a score which can be trivially mapped to the interval [0, 1] (indeed it is standard to output a score that has been mapped th... | Why isn't the ROC curve naturally plotted in 3D? [duplicate]
The purpose of an ROC curve is to summarize the performance of a classifier. The majority of binary classifiers in common use produce a score which can be trivially mapped to the interval [0, 1] (inde |
54,035 | How does R find the dispersion parameter in a GLM? | You can look in the code for summary.glm where you'll see:
sum((object$weights * object$residuals^2)[object$weights > 0])/df.r
df.r is the "residual degrees of freedom" (number of observations - number of parameters)
object$residuals (from ?glm):
residuals: the working residuals, that is the residuals in the final
... | How does R find the dispersion parameter in a GLM? | You can look in the code for summary.glm where you'll see:
sum((object$weights * object$residuals^2)[object$weights > 0])/df.r
df.r is the "residual degrees of freedom" (number of observations - num | How does R find the dispersion parameter in a GLM?
You can look in the code for summary.glm where you'll see:
sum((object$weights * object$residuals^2)[object$weights > 0])/df.r
df.r is the "residual degrees of freedom" (number of observations - number of parameters)
object$residuals (from ?glm):
residuals: the wor... | How does R find the dispersion parameter in a GLM?
You can look in the code for summary.glm where you'll see:
sum((object$weights * object$residuals^2)[object$weights > 0])/df.r
df.r is the "residual degrees of freedom" (number of observations - num |
54,036 | Metropolis Hastings algorithm bivariate normals | EDIT: REFER TO THE CORRECT ANSWER ABOVE:
Note that this is a problem of MIXTURE of NORMALS rather than BIVARIATE NORMALS:
The issue here is that the proposal density does not cover the whole region of interest. You were to calibrate it accordingly to cover the ROI as explained in the answer above. Here is another examp... | Metropolis Hastings algorithm bivariate normals | EDIT: REFER TO THE CORRECT ANSWER ABOVE:
Note that this is a problem of MIXTURE of NORMALS rather than BIVARIATE NORMALS:
The issue here is that the proposal density does not cover the whole region of | Metropolis Hastings algorithm bivariate normals
EDIT: REFER TO THE CORRECT ANSWER ABOVE:
Note that this is a problem of MIXTURE of NORMALS rather than BIVARIATE NORMALS:
The issue here is that the proposal density does not cover the whole region of interest. You were to calibrate it accordingly to cover the ROI as expl... | Metropolis Hastings algorithm bivariate normals
EDIT: REFER TO THE CORRECT ANSWER ABOVE:
Note that this is a problem of MIXTURE of NORMALS rather than BIVARIATE NORMALS:
The issue here is that the proposal density does not cover the whole region of |
54,037 | Metropolis Hastings algorithm bivariate normals | Since the independent Metropolis-Hastings algorithm is formally valid, the issue stands in an inadequate calibration of the proposal to reach the entire support of the target (mixture) distribution. I just modified the code by choosing a larger variance matrix
sig=5*matrix(c(4, 1/2*2*2, 1/2*2*2, 4), nrow=2)
ran the ch... | Metropolis Hastings algorithm bivariate normals | Since the independent Metropolis-Hastings algorithm is formally valid, the issue stands in an inadequate calibration of the proposal to reach the entire support of the target (mixture) distribution. I | Metropolis Hastings algorithm bivariate normals
Since the independent Metropolis-Hastings algorithm is formally valid, the issue stands in an inadequate calibration of the proposal to reach the entire support of the target (mixture) distribution. I just modified the code by choosing a larger variance matrix
sig=5*matri... | Metropolis Hastings algorithm bivariate normals
Since the independent Metropolis-Hastings algorithm is formally valid, the issue stands in an inadequate calibration of the proposal to reach the entire support of the target (mixture) distribution. I |
54,038 | Joint probability measure | Joint Distributions and Expectation
In general, the joint distribution of random variables $X$ and $Y$, defined on a common probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in measurable spaces $(\mathcal{X}, \mathcal{B})$ and $(\mathcal{Y}, \mathcal{C})$, respectively, is the probability measure... | Joint probability measure | Joint Distributions and Expectation
In general, the joint distribution of random variables $X$ and $Y$, defined on a common probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in m | Joint probability measure
Joint Distributions and Expectation
In general, the joint distribution of random variables $X$ and $Y$, defined on a common probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in measurable spaces $(\mathcal{X}, \mathcal{B})$ and $(\mathcal{Y}, \mathcal{C})$, respectively, ... | Joint probability measure
Joint Distributions and Expectation
In general, the joint distribution of random variables $X$ and $Y$, defined on a common probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and taking values in m |
54,039 | Scree plot: $m$ vs $m-1$ components/factors | The following simulations show that the correct Cattell's scree-plot rule is "elbow minus one". That is, select the number of factors one less than the number corresponding to the "elbow" location. The scree-plot considered here is the "classic" one - dealing with eigenvalues of the nonreduced correlation or covariance... | Scree plot: $m$ vs $m-1$ components/factors | The following simulations show that the correct Cattell's scree-plot rule is "elbow minus one". That is, select the number of factors one less than the number corresponding to the "elbow" location. Th | Scree plot: $m$ vs $m-1$ components/factors
The following simulations show that the correct Cattell's scree-plot rule is "elbow minus one". That is, select the number of factors one less than the number corresponding to the "elbow" location. The scree-plot considered here is the "classic" one - dealing with eigenvalues... | Scree plot: $m$ vs $m-1$ components/factors
The following simulations show that the correct Cattell's scree-plot rule is "elbow minus one". That is, select the number of factors one less than the number corresponding to the "elbow" location. Th |
54,040 | Scree plot: $m$ vs $m-1$ components/factors | Significant eigenvalues
You could use some sort of simulation to compute the probability for an eigenvalue exceeding a certain limit and base the selection on that. In the R-package psych, there is a function that does this (demonstrated below).
If you assume samples from a Gaussian distributed population without corre... | Scree plot: $m$ vs $m-1$ components/factors | Significant eigenvalues
You could use some sort of simulation to compute the probability for an eigenvalue exceeding a certain limit and base the selection on that. In the R-package psych, there is a | Scree plot: $m$ vs $m-1$ components/factors
Significant eigenvalues
You could use some sort of simulation to compute the probability for an eigenvalue exceeding a certain limit and base the selection on that. In the R-package psych, there is a function that does this (demonstrated below).
If you assume samples from a G... | Scree plot: $m$ vs $m-1$ components/factors
Significant eigenvalues
You could use some sort of simulation to compute the probability for an eigenvalue exceeding a certain limit and base the selection on that. In the R-package psych, there is a |
54,041 | Doubt in the Invariance Property of Consistent Estimators | $X_{n+1}$ converges not to a constant, but to a distribution with variance 1.
Therefore, Slutsky's theorem (with the two "estimators" $\dfrac{1}{2}\overline{X}_n$ and $\dfrac{1}{2}X_{n+1}$ being two random elements) does not apply. For that, you would need to have $\dfrac{1}{2}X_{n+1}$ converging to a constant.
Intuit... | Doubt in the Invariance Property of Consistent Estimators | $X_{n+1}$ converges not to a constant, but to a distribution with variance 1.
Therefore, Slutsky's theorem (with the two "estimators" $\dfrac{1}{2}\overline{X}_n$ and $\dfrac{1}{2}X_{n+1}$ being two r | Doubt in the Invariance Property of Consistent Estimators
$X_{n+1}$ converges not to a constant, but to a distribution with variance 1.
Therefore, Slutsky's theorem (with the two "estimators" $\dfrac{1}{2}\overline{X}_n$ and $\dfrac{1}{2}X_{n+1}$ being two random elements) does not apply. For that, you would need to ha... | Doubt in the Invariance Property of Consistent Estimators
$X_{n+1}$ converges not to a constant, but to a distribution with variance 1.
Therefore, Slutsky's theorem (with the two "estimators" $\dfrac{1}{2}\overline{X}_n$ and $\dfrac{1}{2}X_{n+1}$ being two r |
54,042 | Causal Inference After Feature Selection | Data mining for potential predictors and causal inference don't go together well. Few problems may arise:
Identification problems:
Confoundness: If some unobserved common causes are still not included in the data, the estimators are still biased.
Bad control: Including too many variables is harmful. You can include c... | Causal Inference After Feature Selection | Data mining for potential predictors and causal inference don't go together well. Few problems may arise:
Identification problems:
Confoundness: If some unobserved common causes are still not include | Causal Inference After Feature Selection
Data mining for potential predictors and causal inference don't go together well. Few problems may arise:
Identification problems:
Confoundness: If some unobserved common causes are still not included in the data, the estimators are still biased.
Bad control: Including too man... | Causal Inference After Feature Selection
Data mining for potential predictors and causal inference don't go together well. Few problems may arise:
Identification problems:
Confoundness: If some unobserved common causes are still not include |
54,043 | Why the confidence intervals in a categorical lm() are not calculated at the group level? | The lm function does its inferential work under a fairly standard assumption that the error term has constant variance. That appears to be a poor assumption for your data, but lm does not know that, so it just treats the groups as having equal variance and calculates the error term variance thinking it is the same for ... | Why the confidence intervals in a categorical lm() are not calculated at the group level? | The lm function does its inferential work under a fairly standard assumption that the error term has constant variance. That appears to be a poor assumption for your data, but lm does not know that, s | Why the confidence intervals in a categorical lm() are not calculated at the group level?
The lm function does its inferential work under a fairly standard assumption that the error term has constant variance. That appears to be a poor assumption for your data, but lm does not know that, so it just treats the groups as... | Why the confidence intervals in a categorical lm() are not calculated at the group level?
The lm function does its inferential work under a fairly standard assumption that the error term has constant variance. That appears to be a poor assumption for your data, but lm does not know that, s |
54,044 | confusion about individual notation | The notation using subscript $i$ is the standard way to express the regression model, although you would also implicitly assume (or explicitly state) that this equation holds over a range of values ---e.g., $i=1,...,n$. When we have different observations/"realisations" of a generic form, we use the subscript notation... | confusion about individual notation | The notation using subscript $i$ is the standard way to express the regression model, although you would also implicitly assume (or explicitly state) that this equation holds over a range of values -- | confusion about individual notation
The notation using subscript $i$ is the standard way to express the regression model, although you would also implicitly assume (or explicitly state) that this equation holds over a range of values ---e.g., $i=1,...,n$. When we have different observations/"realisations" of a generic... | confusion about individual notation
The notation using subscript $i$ is the standard way to express the regression model, although you would also implicitly assume (or explicitly state) that this equation holds over a range of values -- |
54,045 | confusion about individual notation | This matrix equation
$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1}\\ y_{n}\end{bmatrix} = \beta \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1}\\ x_{n}\end{bmatrix} + \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_{n-1}\\ \epsilon_{n}\end{bmatrix}$$
can be written as
$$\forall i : y_i = \be... | confusion about individual notation | This matrix equation
$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1}\\ y_{n}\end{bmatrix} = \beta \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1}\\ x_{n}\end{bmatrix} + \begin{bmatrix} \epsi | confusion about individual notation
This matrix equation
$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1}\\ y_{n}\end{bmatrix} = \beta \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1}\\ x_{n}\end{bmatrix} + \begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_{n-1}\\ \epsilon_{n}\end{bmatrix}$$
can b... | confusion about individual notation
This matrix equation
$$\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_{n-1}\\ y_{n}\end{bmatrix} = \beta \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_{n-1}\\ x_{n}\end{bmatrix} + \begin{bmatrix} \epsi |
54,046 | confusion about individual notation | If both $y$ and $y_i$ are univariate, it doesn't matter if you use index $i$ or not. It's the expected value of the dependent variable given the independent variable, i.e. $E[y|x=a]=E[y_i|x_i=a]$. Sometimes $y$ is the vertical concatenation of each sample response, in which $y=x\beta+\epsilon$ is the matrix notation fo... | confusion about individual notation | If both $y$ and $y_i$ are univariate, it doesn't matter if you use index $i$ or not. It's the expected value of the dependent variable given the independent variable, i.e. $E[y|x=a]=E[y_i|x_i=a]$. Som | confusion about individual notation
If both $y$ and $y_i$ are univariate, it doesn't matter if you use index $i$ or not. It's the expected value of the dependent variable given the independent variable, i.e. $E[y|x=a]=E[y_i|x_i=a]$. Sometimes $y$ is the vertical concatenation of each sample response, in which $y=x\beta... | confusion about individual notation
If both $y$ and $y_i$ are univariate, it doesn't matter if you use index $i$ or not. It's the expected value of the dependent variable given the independent variable, i.e. $E[y|x=a]=E[y_i|x_i=a]$. Som |
54,047 | confusion about individual notation | As far as I know, there are three possible notations:
Individual notation: $y_i = \beta_0 + \beta_1x_{1i} + ... + \beta_Kx_{Ki} + \varepsilon_i$ where $i=1,...,N$.
Without expression $i=1,...,N$, this notation is mostly pointless, however it is common to assume this addition and omit this part. Omitting does not mean... | confusion about individual notation | As far as I know, there are three possible notations:
Individual notation: $y_i = \beta_0 + \beta_1x_{1i} + ... + \beta_Kx_{Ki} + \varepsilon_i$ where $i=1,...,N$.
Without expression $i=1,...,N$, th | confusion about individual notation
As far as I know, there are three possible notations:
Individual notation: $y_i = \beta_0 + \beta_1x_{1i} + ... + \beta_Kx_{Ki} + \varepsilon_i$ where $i=1,...,N$.
Without expression $i=1,...,N$, this notation is mostly pointless, however it is common to assume this addition and om... | confusion about individual notation
As far as I know, there are three possible notations:
Individual notation: $y_i = \beta_0 + \beta_1x_{1i} + ... + \beta_Kx_{Ki} + \varepsilon_i$ where $i=1,...,N$.
Without expression $i=1,...,N$, th |
54,048 | confusion about individual notation | I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine on which level the notation is, everything should be clear. Usually, it is clear when we are on the sample levels since ... | confusion about individual notation | I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine | confusion about individual notation
I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine on which level the notation is, everything should be clear. Usually, it is clear whe... | confusion about individual notation
I believe the confusion arises when distinguishing between population and sample levels. There is some inconsistencies with notation (when to use capital/cursive/bold letters), but once you determine |
54,049 | How to calculate the ACF and PACF for time series | Well if you mean how to estimate the ACF and PACF, here is how it's done:
1. ACF:
In practice, a simple procedure is:
Estimate the sample mean:
$$\bar{y} = \frac{\sum_{t=1}^{T} y_t}{T}$$
Calculate the sample autocorrelation:
$$\hat{\rho_j} = \frac{\sum_{t=j+1}^{T}(y_t - \bar{y})(y_{t-j} - \bar{y})}{\sum_{t=1}^{T}(y_t ... | How to calculate the ACF and PACF for time series | Well if you mean how to estimate the ACF and PACF, here is how it's done:
1. ACF:
In practice, a simple procedure is:
Estimate the sample mean:
$$\bar{y} = \frac{\sum_{t=1}^{T} y_t}{T}$$
Calculate th | How to calculate the ACF and PACF for time series
Well if you mean how to estimate the ACF and PACF, here is how it's done:
1. ACF:
In practice, a simple procedure is:
Estimate the sample mean:
$$\bar{y} = \frac{\sum_{t=1}^{T} y_t}{T}$$
Calculate the sample autocorrelation:
$$\hat{\rho_j} = \frac{\sum_{t=j+1}^{T}(y_t ... | How to calculate the ACF and PACF for time series
Well if you mean how to estimate the ACF and PACF, here is how it's done:
1. ACF:
In practice, a simple procedure is:
Estimate the sample mean:
$$\bar{y} = \frac{\sum_{t=1}^{T} y_t}{T}$$
Calculate th |
54,050 | How to calculate the ACF and PACF for time series | By definition, a ACF is $\gamma_j := E[(Y_{t}-\mu)(Y_{t-j}-\mu)]$ (for covariance) and the is $\rho_j := \frac{\gamma_j}{\gamma_0}$ (for correlation). For a closed formula wrote in function of parameters and such, you need to specify the model that you have (if you say what's your model, i can tell you how to get that ... | How to calculate the ACF and PACF for time series | By definition, a ACF is $\gamma_j := E[(Y_{t}-\mu)(Y_{t-j}-\mu)]$ (for covariance) and the is $\rho_j := \frac{\gamma_j}{\gamma_0}$ (for correlation). For a closed formula wrote in function of paramet | How to calculate the ACF and PACF for time series
By definition, a ACF is $\gamma_j := E[(Y_{t}-\mu)(Y_{t-j}-\mu)]$ (for covariance) and the is $\rho_j := \frac{\gamma_j}{\gamma_0}$ (for correlation). For a closed formula wrote in function of parameters and such, you need to specify the model that you have (if you say ... | How to calculate the ACF and PACF for time series
By definition, a ACF is $\gamma_j := E[(Y_{t}-\mu)(Y_{t-j}-\mu)]$ (for covariance) and the is $\rho_j := \frac{\gamma_j}{\gamma_0}$ (for correlation). For a closed formula wrote in function of paramet |
54,051 | Glmer with count data, nested design and repeated measurements | Adding (1|round) as a random effect to prevent pseudoreplication is right, a good article about this subject can be found here.
To ensure your model assumes everything correct you could compare the df’s of your model summaries and check if they take account for your pseudoreplicates and are therefore lower in the model... | Glmer with count data, nested design and repeated measurements | Adding (1|round) as a random effect to prevent pseudoreplication is right, a good article about this subject can be found here.
To ensure your model assumes everything correct you could compare the df | Glmer with count data, nested design and repeated measurements
Adding (1|round) as a random effect to prevent pseudoreplication is right, a good article about this subject can be found here.
To ensure your model assumes everything correct you could compare the df’s of your model summaries and check if they take account... | Glmer with count data, nested design and repeated measurements
Adding (1|round) as a random effect to prevent pseudoreplication is right, a good article about this subject can be found here.
To ensure your model assumes everything correct you could compare the df |
54,052 | Glmer with count data, nested design and repeated measurements | Nice answer from Thomas! I think we may need some more information though before finalizing a modelling approach.
For example, the currently proposed model ignores the fact that the response variable was collected on different days.
One way I would think about this modelling exercise is like this:
We start out with 2 r... | Glmer with count data, nested design and repeated measurements | Nice answer from Thomas! I think we may need some more information though before finalizing a modelling approach.
For example, the currently proposed model ignores the fact that the response variable | Glmer with count data, nested design and repeated measurements
Nice answer from Thomas! I think we may need some more information though before finalizing a modelling approach.
For example, the currently proposed model ignores the fact that the response variable was collected on different days.
One way I would think ab... | Glmer with count data, nested design and repeated measurements
Nice answer from Thomas! I think we may need some more information though before finalizing a modelling approach.
For example, the currently proposed model ignores the fact that the response variable |
54,053 | The interpretation of a random variable | I think it makes sense to think of the world of mathematics and the real world as separated in principle. However, firstly in order to make real use of mathematics, the world of mathematics and the real world have to be connected by interpretation, including addressing/checking whether the connection is appropriate. Se... | The interpretation of a random variable | I think it makes sense to think of the world of mathematics and the real world as separated in principle. However, firstly in order to make real use of mathematics, the world of mathematics and the re | The interpretation of a random variable
I think it makes sense to think of the world of mathematics and the real world as separated in principle. However, firstly in order to make real use of mathematics, the world of mathematics and the real world have to be connected by interpretation, including addressing/checking w... | The interpretation of a random variable
I think it makes sense to think of the world of mathematics and the real world as separated in principle. However, firstly in order to make real use of mathematics, the world of mathematics and the re |
54,054 | The interpretation of a random variable | something random is something you cannot predict.
so if you can somehow solve an equation that figures out which side a coin will land on before it lands on that side, it is not truly random. you can do this for every random number generating process, so that variable is random if you cannot figure out what it will equ... | The interpretation of a random variable | something random is something you cannot predict.
so if you can somehow solve an equation that figures out which side a coin will land on before it lands on that side, it is not truly random. you can | The interpretation of a random variable
something random is something you cannot predict.
so if you can somehow solve an equation that figures out which side a coin will land on before it lands on that side, it is not truly random. you can do this for every random number generating process, so that variable is random i... | The interpretation of a random variable
something random is something you cannot predict.
so if you can somehow solve an equation that figures out which side a coin will land on before it lands on that side, it is not truly random. you can |
54,055 | How Gradient Descent is used for classification with Decision Trees? | Gradient descent is not used for training decision trees. Not every machine learning algorithm uses a general optimization algorithm (e.g. gradient descent) for training, some of them use specialized algorithms for training them. Examples of such algorithms are $k$-NN, naive Bayes, or decision trees, in case of those a... | How Gradient Descent is used for classification with Decision Trees? | Gradient descent is not used for training decision trees. Not every machine learning algorithm uses a general optimization algorithm (e.g. gradient descent) for training, some of them use specialized | How Gradient Descent is used for classification with Decision Trees?
Gradient descent is not used for training decision trees. Not every machine learning algorithm uses a general optimization algorithm (e.g. gradient descent) for training, some of them use specialized algorithms for training them. Examples of such algo... | How Gradient Descent is used for classification with Decision Trees?
Gradient descent is not used for training decision trees. Not every machine learning algorithm uses a general optimization algorithm (e.g. gradient descent) for training, some of them use specialized |
54,056 | Is this the correct way to run an adaptive LASSO? | Yes, but it depends on what you're goal is. It's a little complicated.
What is adaptive Lasso?
Adaptive Lasso was introduced in Zhou (2006). Adaptive Lasso is a modification of Lasso where each coefficient, $\beta_j$, is given its own weight, $w_j$. The coefficients are estimated by minimizing the objective function,
$... | Is this the correct way to run an adaptive LASSO? | Yes, but it depends on what you're goal is. It's a little complicated.
What is adaptive Lasso?
Adaptive Lasso was introduced in Zhou (2006). Adaptive Lasso is a modification of Lasso where each coeffi | Is this the correct way to run an adaptive LASSO?
Yes, but it depends on what you're goal is. It's a little complicated.
What is adaptive Lasso?
Adaptive Lasso was introduced in Zhou (2006). Adaptive Lasso is a modification of Lasso where each coefficient, $\beta_j$, is given its own weight, $w_j$. The coefficients are... | Is this the correct way to run an adaptive LASSO?
Yes, but it depends on what you're goal is. It's a little complicated.
What is adaptive Lasso?
Adaptive Lasso was introduced in Zhou (2006). Adaptive Lasso is a modification of Lasso where each coeffi |
54,057 | Basic preconditioned gradient descent example | Since the hessian is constant $P = H^{-1}$ is optimal (you fully recover Newton's method): $P = H^{-1} = \text{diag}(1/2, 1/20)$. This is equivalent to making the change of variables $x = \frac{1}{\sqrt{2}} x'$ and $y = \frac{1}{\sqrt{20}} y'$ which gives you the desired $\frac{1}{2} (x'^2 + y'^2)$ form.
As for approac... | Basic preconditioned gradient descent example | Since the hessian is constant $P = H^{-1}$ is optimal (you fully recover Newton's method): $P = H^{-1} = \text{diag}(1/2, 1/20)$. This is equivalent to making the change of variables $x = \frac{1}{\sq | Basic preconditioned gradient descent example
Since the hessian is constant $P = H^{-1}$ is optimal (you fully recover Newton's method): $P = H^{-1} = \text{diag}(1/2, 1/20)$. This is equivalent to making the change of variables $x = \frac{1}{\sqrt{2}} x'$ and $y = \frac{1}{\sqrt{20}} y'$ which gives you the desired $\... | Basic preconditioned gradient descent example
Since the hessian is constant $P = H^{-1}$ is optimal (you fully recover Newton's method): $P = H^{-1} = \text{diag}(1/2, 1/20)$. This is equivalent to making the change of variables $x = \frac{1}{\sq |
54,058 | Basic preconditioned gradient descent example | I'm fairly sure you can just use the Hessian, which in this case is $$
2\begin{bmatrix}
1 & 0\\
0 & 10
\end{bmatrix}
$$
This comes from method 3 (using second-partial derivative information), and possibly also method 1.
The reason we're using the exact matrix here instead of an approximation is because this is a simple... | Basic preconditioned gradient descent example | I'm fairly sure you can just use the Hessian, which in this case is $$
2\begin{bmatrix}
1 & 0\\
0 & 10
\end{bmatrix}
$$
This comes from method 3 (using second-partial derivative information), and poss | Basic preconditioned gradient descent example
I'm fairly sure you can just use the Hessian, which in this case is $$
2\begin{bmatrix}
1 & 0\\
0 & 10
\end{bmatrix}
$$
This comes from method 3 (using second-partial derivative information), and possibly also method 1.
The reason we're using the exact matrix here instead o... | Basic preconditioned gradient descent example
I'm fairly sure you can just use the Hessian, which in this case is $$
2\begin{bmatrix}
1 & 0\\
0 & 10
\end{bmatrix}
$$
This comes from method 3 (using second-partial derivative information), and poss |
54,059 | Basic preconditioned gradient descent example | This is the problem that preconditioning tries to solve:
From this blog (also discussed here on stackexchange: Fitting SIR model with 2019-nCoV data doesn't conververge)
Problems:
With gradient descent methods you follow a path down-hill. But, your algorithm is often not as smooth as water and doesn't flow straight d... | Basic preconditioned gradient descent example | This is the problem that preconditioning tries to solve:
From this blog (also discussed here on stackexchange: Fitting SIR model with 2019-nCoV data doesn't conververge)
Problems:
With gradient desc | Basic preconditioned gradient descent example
This is the problem that preconditioning tries to solve:
From this blog (also discussed here on stackexchange: Fitting SIR model with 2019-nCoV data doesn't conververge)
Problems:
With gradient descent methods you follow a path down-hill. But, your algorithm is often not ... | Basic preconditioned gradient descent example
This is the problem that preconditioning tries to solve:
From this blog (also discussed here on stackexchange: Fitting SIR model with 2019-nCoV data doesn't conververge)
Problems:
With gradient desc |
54,060 | Why we learn $\log{\sigma^2}$ in VAE reparameterization trick instead of standard deviation? [duplicate] | It doesn't make any real difference; since $\log \sigma^2= 2\log \sigma$; learning one is as easy as learning the other
It's traditional in statistics to think of $\sigma^2$ as the second parameter of a Normal distribution (rather than $\sigma$).
There's a simple unbiased estimator for $\sigma^2$ but not for $\sigma$
T... | Why we learn $\log{\sigma^2}$ in VAE reparameterization trick instead of standard deviation? [duplic | It doesn't make any real difference; since $\log \sigma^2= 2\log \sigma$; learning one is as easy as learning the other
It's traditional in statistics to think of $\sigma^2$ as the second parameter of | Why we learn $\log{\sigma^2}$ in VAE reparameterization trick instead of standard deviation? [duplicate]
It doesn't make any real difference; since $\log \sigma^2= 2\log \sigma$; learning one is as easy as learning the other
It's traditional in statistics to think of $\sigma^2$ as the second parameter of a Normal distr... | Why we learn $\log{\sigma^2}$ in VAE reparameterization trick instead of standard deviation? [duplic
It doesn't make any real difference; since $\log \sigma^2= 2\log \sigma$; learning one is as easy as learning the other
It's traditional in statistics to think of $\sigma^2$ as the second parameter of |
54,061 | Does the t statistic have uses unrelated to hypothesis testing? | When you ask about "the t-statistic", I think about the concrete quantity $$\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}$$
To actually calculate this quantity, we have to specify $\mu$. This is typically chosen in reference to some given null hypothesis. So to me it seems awkward to try to disentangle "the statistic" from th... | Does the t statistic have uses unrelated to hypothesis testing? | When you ask about "the t-statistic", I think about the concrete quantity $$\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}$$
To actually calculate this quantity, we have to specify $\mu$. This is typically ch | Does the t statistic have uses unrelated to hypothesis testing?
When you ask about "the t-statistic", I think about the concrete quantity $$\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}$$
To actually calculate this quantity, we have to specify $\mu$. This is typically chosen in reference to some given null hypothesis. So to m... | Does the t statistic have uses unrelated to hypothesis testing?
When you ask about "the t-statistic", I think about the concrete quantity $$\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}$$
To actually calculate this quantity, we have to specify $\mu$. This is typically ch |
54,062 | Does the t statistic have uses unrelated to hypothesis testing? | A "hypothesis test" in the strictest sense always results in a binary outcome of either rejecting or failing to reject a null hypothesis. T-statistics are generally turned into p-values, which are then compared against some pre-defined threshold to make that binary determination. It is possible to use the t-statistic i... | Does the t statistic have uses unrelated to hypothesis testing? | A "hypothesis test" in the strictest sense always results in a binary outcome of either rejecting or failing to reject a null hypothesis. T-statistics are generally turned into p-values, which are the | Does the t statistic have uses unrelated to hypothesis testing?
A "hypothesis test" in the strictest sense always results in a binary outcome of either rejecting or failing to reject a null hypothesis. T-statistics are generally turned into p-values, which are then compared against some pre-defined threshold to make th... | Does the t statistic have uses unrelated to hypothesis testing?
A "hypothesis test" in the strictest sense always results in a binary outcome of either rejecting or failing to reject a null hypothesis. T-statistics are generally turned into p-values, which are the |
54,063 | Measurement error in one indep variable in OLS with multiple regression | Suppose you true matrix is $X^*=\begin{bmatrix} x_1^{*} & x_2 \end{bmatrix}$, but you observe $x_1=x_1^*+v$.
Then the OLS coefficient on $x_1$ has the following probability limit:
$$\mathbf{plim} \hat \beta_{x_1|x_2}=\beta \left[1-\frac{\sigma^2_v}{\sigma^2_{x_1^*}\cdot(1-R^2_{x_1^*,x_2})+\sigma^2_v} \right]=\beta \lef... | Measurement error in one indep variable in OLS with multiple regression | Suppose you true matrix is $X^*=\begin{bmatrix} x_1^{*} & x_2 \end{bmatrix}$, but you observe $x_1=x_1^*+v$.
Then the OLS coefficient on $x_1$ has the following probability limit:
$$\mathbf{plim} \hat | Measurement error in one indep variable in OLS with multiple regression
Suppose you true matrix is $X^*=\begin{bmatrix} x_1^{*} & x_2 \end{bmatrix}$, but you observe $x_1=x_1^*+v$.
Then the OLS coefficient on $x_1$ has the following probability limit:
$$\mathbf{plim} \hat \beta_{x_1|x_2}=\beta \left[1-\frac{\sigma^2_v}... | Measurement error in one indep variable in OLS with multiple regression
Suppose you true matrix is $X^*=\begin{bmatrix} x_1^{*} & x_2 \end{bmatrix}$, but you observe $x_1=x_1^*+v$.
Then the OLS coefficient on $x_1$ has the following probability limit:
$$\mathbf{plim} \hat |
54,064 | Measurement error in one indep variable in OLS with multiple regression | The solution to this problem is in Wooldridge's "Introductory Econometrics" (Chapter 9 Section "Measurement Error in an Explanatory Variable", p320 in the 2012 version) and in Wooldridge's "Econometric analysis of cross-section and_panel data" (Section 4.4.2, p73 in the 2002 version). Here is the takeaway.
Consider the... | Measurement error in one indep variable in OLS with multiple regression | The solution to this problem is in Wooldridge's "Introductory Econometrics" (Chapter 9 Section "Measurement Error in an Explanatory Variable", p320 in the 2012 version) and in Wooldridge's "Econometri | Measurement error in one indep variable in OLS with multiple regression
The solution to this problem is in Wooldridge's "Introductory Econometrics" (Chapter 9 Section "Measurement Error in an Explanatory Variable", p320 in the 2012 version) and in Wooldridge's "Econometric analysis of cross-section and_panel data" (Sec... | Measurement error in one indep variable in OLS with multiple regression
The solution to this problem is in Wooldridge's "Introductory Econometrics" (Chapter 9 Section "Measurement Error in an Explanatory Variable", p320 in the 2012 version) and in Wooldridge's "Econometri |
54,065 | Measurement error in one indep variable in OLS with multiple regression | In the situation that you describe, the true model is like this:
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$
now, you can observe $y$ and $x_2$ but you cannot observe $x_1$. However you can observe
$z = x_1 + \epsilon$
moreover we assume that $\rho(\epsilon,u)=0$
So, if we consider the simplification where $\beta_2 =... | Measurement error in one indep variable in OLS with multiple regression | In the situation that you describe, the true model is like this:
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$
now, you can observe $y$ and $x_2$ but you cannot observe $x_1$. However you can observe
| Measurement error in one indep variable in OLS with multiple regression
In the situation that you describe, the true model is like this:
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$
now, you can observe $y$ and $x_2$ but you cannot observe $x_1$. However you can observe
$z = x_1 + \epsilon$
moreover we assume that $\r... | Measurement error in one indep variable in OLS with multiple regression
In the situation that you describe, the true model is like this:
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$
now, you can observe $y$ and $x_2$ but you cannot observe $x_1$. However you can observe
|
54,066 | Linear Mixed Effect Model - random intercept and slope? Identifiability problems | First, I would almost always advise against deleting observations for any reason, but in your case I definitely advise against it. By deleting observations you lose statistical power but more importantly you can introduce bias.
Think for a moment about what it means to fit random slopes. It means that you allow the slo... | Linear Mixed Effect Model - random intercept and slope? Identifiability problems | First, I would almost always advise against deleting observations for any reason, but in your case I definitely advise against it. By deleting observations you lose statistical power but more importan | Linear Mixed Effect Model - random intercept and slope? Identifiability problems
First, I would almost always advise against deleting observations for any reason, but in your case I definitely advise against it. By deleting observations you lose statistical power but more importantly you can introduce bias.
Think for a... | Linear Mixed Effect Model - random intercept and slope? Identifiability problems
First, I would almost always advise against deleting observations for any reason, but in your case I definitely advise against it. By deleting observations you lose statistical power but more importan |
54,067 | Overlap-tile strategy in U-Nets | On the "overlap-tile strategy" specifically:
The blue box in Fig 2 (left) shows the input to the network. Because they're using valid convolutions, the output is the smaller yellow box (right). Sounds like you understand this part already.
They're trying to show that the image that they want to predict on is bigger t... | Overlap-tile strategy in U-Nets | On the "overlap-tile strategy" specifically:
The blue box in Fig 2 (left) shows the input to the network. Because they're using valid convolutions, the output is the smaller yellow box (right). Soun | Overlap-tile strategy in U-Nets
On the "overlap-tile strategy" specifically:
The blue box in Fig 2 (left) shows the input to the network. Because they're using valid convolutions, the output is the smaller yellow box (right). Sounds like you understand this part already.
They're trying to show that the image that the... | Overlap-tile strategy in U-Nets
On the "overlap-tile strategy" specifically:
The blue box in Fig 2 (left) shows the input to the network. Because they're using valid convolutions, the output is the smaller yellow box (right). Soun |
54,068 | Overlap-tile strategy in U-Nets | Yeah their description is a bit confusing. I agree with your interpretation.
This paper has 15k citations, so if that strategy is effective, it's probably pretty commonly used. Otherwise, my only guess could be that other techniques in the paper were more important. I'm not familiar with biomedical computer vision rese... | Overlap-tile strategy in U-Nets | Yeah their description is a bit confusing. I agree with your interpretation.
This paper has 15k citations, so if that strategy is effective, it's probably pretty commonly used. Otherwise, my only gues | Overlap-tile strategy in U-Nets
Yeah their description is a bit confusing. I agree with your interpretation.
This paper has 15k citations, so if that strategy is effective, it's probably pretty commonly used. Otherwise, my only guess could be that other techniques in the paper were more important. I'm not familiar with... | Overlap-tile strategy in U-Nets
Yeah their description is a bit confusing. I agree with your interpretation.
This paper has 15k citations, so if that strategy is effective, it's probably pretty commonly used. Otherwise, my only gues |
54,069 | Overlap-tile strategy in U-Nets | Instead of having padded convolutions to maintain the spatial size of the feature maps, they pad the original image by mirroring the borders and forward the pre-padded image through the network
I agree with @bogovicj 's explanation.
What do they mean by "only us[ing] the valid part of each convolution"?
Based on you... | Overlap-tile strategy in U-Nets | Instead of having padded convolutions to maintain the spatial size of the feature maps, they pad the original image by mirroring the borders and forward the pre-padded image through the network
I agr | Overlap-tile strategy in U-Nets
Instead of having padded convolutions to maintain the spatial size of the feature maps, they pad the original image by mirroring the borders and forward the pre-padded image through the network
I agree with @bogovicj 's explanation.
What do they mean by "only us[ing] the valid part of ... | Overlap-tile strategy in U-Nets
Instead of having padded convolutions to maintain the spatial size of the feature maps, they pad the original image by mirroring the borders and forward the pre-padded image through the network
I agr |
54,070 | Plotting GAMs on Response Scale with Multiple Smooth and Linear Terms | If the model contains z then the effect of x estimated by the model is that given z is in the model. Hence the fitted response is the additive sum of the two effects, and we can't talk generally about the estimated values of the response for a range of values of x without also stating the value of z.
For Gaussian model... | Plotting GAMs on Response Scale with Multiple Smooth and Linear Terms | If the model contains z then the effect of x estimated by the model is that given z is in the model. Hence the fitted response is the additive sum of the two effects, and we can't talk generally about | Plotting GAMs on Response Scale with Multiple Smooth and Linear Terms
If the model contains z then the effect of x estimated by the model is that given z is in the model. Hence the fitted response is the additive sum of the two effects, and we can't talk generally about the estimated values of the response for a range ... | Plotting GAMs on Response Scale with Multiple Smooth and Linear Terms
If the model contains z then the effect of x estimated by the model is that given z is in the model. Hence the fitted response is the additive sum of the two effects, and we can't talk generally about |
54,071 | Why does conditioning on a mediator variable makes the ancestor and the child independent? | If we know 𝐹𝑖𝑟𝑒 we know the value of 𝐴𝑙𝑎𝑟𝑚. Conditioning on
𝑆𝑚𝑜𝑘𝑒 doesn't change that at all, so why did it render 𝐴 and 𝐶
or 𝐹𝑖𝑟𝑒 and 𝐴𝑙𝑎𝑟𝑚 independent?
Yes, if we know the value of fire we know the value of alarm. But this is not what conditioning on smoke means. Conditioning on smoke me... | Why does conditioning on a mediator variable makes the ancestor and the child independent? | If we know 𝐹𝑖𝑟𝑒 we know the value of 𝐴𝑙𝑎𝑟𝑚. Conditioning on
𝑆𝑚𝑜𝑘𝑒 doesn't change that at all, so why did it render 𝐴 and 𝐶
or 𝐹𝑖𝑟𝑒 and 𝐴𝑙𝑎𝑟𝑚 independent?
Yes, if we know the value of fire we know | Why does conditioning on a mediator variable makes the ancestor and the child independent?
If we know 𝐹𝑖𝑟𝑒 we know the value of 𝐴𝑙𝑎𝑟𝑚. Conditioning on
𝑆𝑚𝑜𝑘𝑒 doesn't change that at all, so why did it render 𝐴 and 𝐶
or 𝐹𝑖𝑟𝑒 and 𝐴𝑙𝑎𝑟𝑚 independent?
Yes, if we know the value of fire we know the... | Why does conditioning on a mediator variable makes the ancestor and the child independent?
If we know 𝐹𝑖𝑟𝑒 we know the value of 𝐴𝑙𝑎𝑟𝑚. Conditioning on
𝑆𝑚𝑜𝑘𝑒 doesn't change that at all, so why did it render 𝐴 and 𝐶
or 𝐹𝑖𝑟𝑒 and 𝐴𝑙𝑎𝑟𝑚 independent?
Yes, if we know the value of fire we know |
54,072 | Why does conditioning on a mediator variable makes the ancestor and the child independent? | Formally, you model implies $P(Alarm|Smoke) = P(Alarm|Smoke, Fire)$.
In short, Fire and Alarm are independent conditional on Smoke.
In words, if you know there is smoke, then you can infer there is an alarm. The additional knowledge of fire does not change you assessment, because there is no other path between fire and... | Why does conditioning on a mediator variable makes the ancestor and the child independent? | Formally, you model implies $P(Alarm|Smoke) = P(Alarm|Smoke, Fire)$.
In short, Fire and Alarm are independent conditional on Smoke.
In words, if you know there is smoke, then you can infer there is an | Why does conditioning on a mediator variable makes the ancestor and the child independent?
Formally, you model implies $P(Alarm|Smoke) = P(Alarm|Smoke, Fire)$.
In short, Fire and Alarm are independent conditional on Smoke.
In words, if you know there is smoke, then you can infer there is an alarm. The additional knowle... | Why does conditioning on a mediator variable makes the ancestor and the child independent?
Formally, you model implies $P(Alarm|Smoke) = P(Alarm|Smoke, Fire)$.
In short, Fire and Alarm are independent conditional on Smoke.
In words, if you know there is smoke, then you can infer there is an |
54,073 | Is Coarsened Exact Matching superior to other matching methods in case-control studies? | CEM does not allow you to estimate the ATE. This is because the matched units in each treatment group will not resemble the overall sample. If no treated units are unmatched, you can estimate the average treatment effect on the treated (ATT). If any treated units are discarded, the estimand is an average treatment effe... | Is Coarsened Exact Matching superior to other matching methods in case-control studies? | CEM does not allow you to estimate the ATE. This is because the matched units in each treatment group will not resemble the overall sample. If no treated units are unmatched, you can estimate the aver | Is Coarsened Exact Matching superior to other matching methods in case-control studies?
CEM does not allow you to estimate the ATE. This is because the matched units in each treatment group will not resemble the overall sample. If no treated units are unmatched, you can estimate the average treatment effect on the trea... | Is Coarsened Exact Matching superior to other matching methods in case-control studies?
CEM does not allow you to estimate the ATE. This is because the matched units in each treatment group will not resemble the overall sample. If no treated units are unmatched, you can estimate the aver |
54,074 | How to deal with incorrect labels in classification? | If you have wrong data and no way to get the true labels then there is nothing "correct" that you can do to obtain this information.
You could treat this as an unsupervised problem first (or semi-supervised), by using say clustering with 2 clusters (since you know there are only 2 labels) to get a model to predict labe... | How to deal with incorrect labels in classification? | If you have wrong data and no way to get the true labels then there is nothing "correct" that you can do to obtain this information.
You could treat this as an unsupervised problem first (or semi-supe | How to deal with incorrect labels in classification?
If you have wrong data and no way to get the true labels then there is nothing "correct" that you can do to obtain this information.
You could treat this as an unsupervised problem first (or semi-supervised), by using say clustering with 2 clusters (since you know th... | How to deal with incorrect labels in classification?
If you have wrong data and no way to get the true labels then there is nothing "correct" that you can do to obtain this information.
You could treat this as an unsupervised problem first (or semi-supe |
54,075 | How to deal with incorrect labels in classification? | Under mild assumptions on the noise mechanism and data distribution (e.g. less than $\frac{1}{2}$ of the data is incorrectly labelled), some classifiers can be shown to be consistent in the binary classification setting. A classifier $C_n$, depending on the training data, is said to be consistent if
$$R(C_n) → R(C_{Bay... | How to deal with incorrect labels in classification? | Under mild assumptions on the noise mechanism and data distribution (e.g. less than $\frac{1}{2}$ of the data is incorrectly labelled), some classifiers can be shown to be consistent in the binary cla | How to deal with incorrect labels in classification?
Under mild assumptions on the noise mechanism and data distribution (e.g. less than $\frac{1}{2}$ of the data is incorrectly labelled), some classifiers can be shown to be consistent in the binary classification setting. A classifier $C_n$, depending on the training ... | How to deal with incorrect labels in classification?
Under mild assumptions on the noise mechanism and data distribution (e.g. less than $\frac{1}{2}$ of the data is incorrectly labelled), some classifiers can be shown to be consistent in the binary cla |
54,076 | How to deal with incorrect labels in classification? | In case of wrong data the best practice, in my experience, is to get rid of it. See, unlike conventional programming where you build the algorithm and apply it to the data, in machine learning, the algorithm comes from the data itself, so if you put a wrong data it will disrupt your algorithm and you will get poor perf... | How to deal with incorrect labels in classification? | In case of wrong data the best practice, in my experience, is to get rid of it. See, unlike conventional programming where you build the algorithm and apply it to the data, in machine learning, the al | How to deal with incorrect labels in classification?
In case of wrong data the best practice, in my experience, is to get rid of it. See, unlike conventional programming where you build the algorithm and apply it to the data, in machine learning, the algorithm comes from the data itself, so if you put a wrong data it w... | How to deal with incorrect labels in classification?
In case of wrong data the best practice, in my experience, is to get rid of it. See, unlike conventional programming where you build the algorithm and apply it to the data, in machine learning, the al |
54,077 | How to deal with incorrect labels in classification? | You have a bunch of known As (if I read correctly) and some other cases that may or may not be As. So you want to find the most similar cases from the unknown set. Sounds relatively straightforward. If As are really different, then you'll get a nice break in the similarity function. | How to deal with incorrect labels in classification? | You have a bunch of known As (if I read correctly) and some other cases that may or may not be As. So you want to find the most similar cases from the unknown set. Sounds relatively straightforward. I | How to deal with incorrect labels in classification?
You have a bunch of known As (if I read correctly) and some other cases that may or may not be As. So you want to find the most similar cases from the unknown set. Sounds relatively straightforward. If As are really different, then you'll get a nice break in the simi... | How to deal with incorrect labels in classification?
You have a bunch of known As (if I read correctly) and some other cases that may or may not be As. So you want to find the most similar cases from the unknown set. Sounds relatively straightforward. I |
54,078 | How to deal with incorrect labels in classification? | I'm a little late to this question, but for future readers: Try giving higher sample weights to data with class A. That way your algorithm will have a higher penalty for misclassifying A than for misclassifying B.
If your algorithm doesn't support sample weights you could try oversampling your data from class A.
There ... | How to deal with incorrect labels in classification? | I'm a little late to this question, but for future readers: Try giving higher sample weights to data with class A. That way your algorithm will have a higher penalty for misclassifying A than for misc | How to deal with incorrect labels in classification?
I'm a little late to this question, but for future readers: Try giving higher sample weights to data with class A. That way your algorithm will have a higher penalty for misclassifying A than for misclassifying B.
If your algorithm doesn't support sample weights you ... | How to deal with incorrect labels in classification?
I'm a little late to this question, but for future readers: Try giving higher sample weights to data with class A. That way your algorithm will have a higher penalty for misclassifying A than for misc |
54,079 | How to deal with incorrect labels in classification? | Learning with noisy label is an active research area. Serval methods are mentioned here: https://youtu.be/8mpBHbjG4E4. | How to deal with incorrect labels in classification? | Learning with noisy label is an active research area. Serval methods are mentioned here: https://youtu.be/8mpBHbjG4E4. | How to deal with incorrect labels in classification?
Learning with noisy label is an active research area. Serval methods are mentioned here: https://youtu.be/8mpBHbjG4E4. | How to deal with incorrect labels in classification?
Learning with noisy label is an active research area. Serval methods are mentioned here: https://youtu.be/8mpBHbjG4E4. |
54,080 | Lasso, Ridge and Best Subset estimator for orthogonal cases | From the context, I'm assuming that the $\beta_j's$ are the regular least squares estimates, and the table is showing how they would be transformed under each of the listed methods.
Best Subset:
Because the columns are orthonormal, the least squares coefficients are simply $\hat{B_j} = {x_j^{T}y}$. (Orthogonality impl... | Lasso, Ridge and Best Subset estimator for orthogonal cases | From the context, I'm assuming that the $\beta_j's$ are the regular least squares estimates, and the table is showing how they would be transformed under each of the listed methods.
Best Subset:
Becau | Lasso, Ridge and Best Subset estimator for orthogonal cases
From the context, I'm assuming that the $\beta_j's$ are the regular least squares estimates, and the table is showing how they would be transformed under each of the listed methods.
Best Subset:
Because the columns are orthonormal, the least squares coefficien... | Lasso, Ridge and Best Subset estimator for orthogonal cases
From the context, I'm assuming that the $\beta_j's$ are the regular least squares estimates, and the table is showing how they would be transformed under each of the listed methods.
Best Subset:
Becau |
54,081 | Inverse moment of Multivariate Normal Norm | You have to use the so-called law of the unconscious statistician to write the integral you need to compute. In this case, we are asked to calculate the first and second moments of the reciprocal distribution 1/x.
\begin{align} \mathbb{E}\left[ \frac{1}{ |x|^n } \right] =& \int d x^p \frac{1}{ \sqrt{(2 \pi)^p \d... | Inverse moment of Multivariate Normal Norm | You have to use the so-called law of the unconscious statistician to write the integral you need to compute. In this case, we are asked to calculate the first and second moments of the reciprocal di | Inverse moment of Multivariate Normal Norm
You have to use the so-called law of the unconscious statistician to write the integral you need to compute. In this case, we are asked to calculate the first and second moments of the reciprocal distribution 1/x.
\begin{align} \mathbb{E}\left[ \frac{1}{ |x|^n } \right] =&... | Inverse moment of Multivariate Normal Norm
You have to use the so-called law of the unconscious statistician to write the integral you need to compute. In this case, we are asked to calculate the first and second moments of the reciprocal di |
54,082 | Inverse moment of Multivariate Normal Norm | For the case where $\Sigma = \mathbf{I}\sigma$, we have the following formulas:
\begin{equation}
\mathbb{E}\left( \frac{1}{||x||} \right) = \frac{1}{\sqrt{2}} {}_1F_1 \left(\frac{1}{2}, \frac{P}{2}, -\frac{||\frac{\mu}{\sigma}||^2}{2}\right) \frac{\Gamma\left(\frac{P-1}{2}\right)}{\Gamma\left(\frac{P}{2}\right)}
\end{e... | Inverse moment of Multivariate Normal Norm | For the case where $\Sigma = \mathbf{I}\sigma$, we have the following formulas:
\begin{equation}
\mathbb{E}\left( \frac{1}{||x||} \right) = \frac{1}{\sqrt{2}} {}_1F_1 \left(\frac{1}{2}, \frac{P}{2}, - | Inverse moment of Multivariate Normal Norm
For the case where $\Sigma = \mathbf{I}\sigma$, we have the following formulas:
\begin{equation}
\mathbb{E}\left( \frac{1}{||x||} \right) = \frac{1}{\sqrt{2}} {}_1F_1 \left(\frac{1}{2}, \frac{P}{2}, -\frac{||\frac{\mu}{\sigma}||^2}{2}\right) \frac{\Gamma\left(\frac{P-1}{2}\rig... | Inverse moment of Multivariate Normal Norm
For the case where $\Sigma = \mathbf{I}\sigma$, we have the following formulas:
\begin{equation}
\mathbb{E}\left( \frac{1}{||x||} \right) = \frac{1}{\sqrt{2}} {}_1F_1 \left(\frac{1}{2}, \frac{P}{2}, - |
54,083 | Updating the inverse covariance matrix after deleting the i-th column and row of the covariance matrix | When $i=n,$ write $\mathbb{A}$ in block matrix form
$$\mathbb A = \pmatrix{A & B \\ C & D}$$
where $A$ is the $n-1 \times n-1$ matrix obtained by omitting the last row and column of $\mathbb{A},$ $B = C^\prime$ is the first $n-1$ entries in the last column, and $D = \mathbb{A}_{nn}$ is a nonzero number because $\mathb... | Updating the inverse covariance matrix after deleting the i-th column and row of the covariance matr | When $i=n,$ write $\mathbb{A}$ in block matrix form
$$\mathbb A = \pmatrix{A & B \\ C & D}$$
where $A$ is the $n-1 \times n-1$ matrix obtained by omitting the last row and column of $\mathbb{A},$ $B | Updating the inverse covariance matrix after deleting the i-th column and row of the covariance matrix
When $i=n,$ write $\mathbb{A}$ in block matrix form
$$\mathbb A = \pmatrix{A & B \\ C & D}$$
where $A$ is the $n-1 \times n-1$ matrix obtained by omitting the last row and column of $\mathbb{A},$ $B = C^\prime$ is th... | Updating the inverse covariance matrix after deleting the i-th column and row of the covariance matr
When $i=n,$ write $\mathbb{A}$ in block matrix form
$$\mathbb A = \pmatrix{A & B \\ C & D}$$
where $A$ is the $n-1 \times n-1$ matrix obtained by omitting the last row and column of $\mathbb{A},$ $B |
54,084 | How Does L2 Norm Regularization Work with Negative Weights? | This is our objective function, composed of a loss function and a regularizer.
$$\mathcal O(w,x,y) = \mathcal L(w,x,y)+\mathcal R(w)$$
So $\mathcal R(w)=\|w\|_2^2=\sum{w_i^2}$ in the case of $\ell_2$ regularization.
Let's perform gradient-based minimization, i.e. we will update params based on the negative partial deri... | How Does L2 Norm Regularization Work with Negative Weights? | This is our objective function, composed of a loss function and a regularizer.
$$\mathcal O(w,x,y) = \mathcal L(w,x,y)+\mathcal R(w)$$
So $\mathcal R(w)=\|w\|_2^2=\sum{w_i^2}$ in the case of $\ell_2$ | How Does L2 Norm Regularization Work with Negative Weights?
This is our objective function, composed of a loss function and a regularizer.
$$\mathcal O(w,x,y) = \mathcal L(w,x,y)+\mathcal R(w)$$
So $\mathcal R(w)=\|w\|_2^2=\sum{w_i^2}$ in the case of $\ell_2$ regularization.
Let's perform gradient-based minimization, i... | How Does L2 Norm Regularization Work with Negative Weights?
This is our objective function, composed of a loss function and a regularizer.
$$\mathcal O(w,x,y) = \mathcal L(w,x,y)+\mathcal R(w)$$
So $\mathcal R(w)=\|w\|_2^2=\sum{w_i^2}$ in the case of $\ell_2$ |
54,085 | How Does L2 Norm Regularization Work with Negative Weights? | L2 regularization adds $w_i^2$ term to the loss function. In iterative approaches using gradients, we subtract the gradient of the loss function not the magnitude of the weight itself. And in the loss function, the regularization part's derivative with respect to $w_i$ is going to be ${d\over dw_i}(w_i^2)=2w_i$. Typica... | How Does L2 Norm Regularization Work with Negative Weights? | L2 regularization adds $w_i^2$ term to the loss function. In iterative approaches using gradients, we subtract the gradient of the loss function not the magnitude of the weight itself. And in the loss | How Does L2 Norm Regularization Work with Negative Weights?
L2 regularization adds $w_i^2$ term to the loss function. In iterative approaches using gradients, we subtract the gradient of the loss function not the magnitude of the weight itself. And in the loss function, the regularization part's derivative with respect... | How Does L2 Norm Regularization Work with Negative Weights?
L2 regularization adds $w_i^2$ term to the loss function. In iterative approaches using gradients, we subtract the gradient of the loss function not the magnitude of the weight itself. And in the loss |
54,086 | Closed form of the integral of the difference of two Gaussian CDFs? | The $(b-d)/a$ result is correct when $a \gt 0.$ This post explains why. It generalizes the question broadly in order to reveal the underlying ideas.
Because $\Phi$ is not special in this regard, let's consider any distribution function $F_X$ for a random variable $X.$ Recall that by definition, $F_X(x) = \Pr(X\le x... | Closed form of the integral of the difference of two Gaussian CDFs? | The $(b-d)/a$ result is correct when $a \gt 0.$ This post explains why. It generalizes the question broadly in order to reveal the underlying ideas.
Because $\Phi$ is not special in this regard, le | Closed form of the integral of the difference of two Gaussian CDFs?
The $(b-d)/a$ result is correct when $a \gt 0.$ This post explains why. It generalizes the question broadly in order to reveal the underlying ideas.
Because $\Phi$ is not special in this regard, let's consider any distribution function $F_X$ for a r... | Closed form of the integral of the difference of two Gaussian CDFs?
The $(b-d)/a$ result is correct when $a \gt 0.$ This post explains why. It generalizes the question broadly in order to reveal the underlying ideas.
Because $\Phi$ is not special in this regard, le |
54,087 | Closed form of the integral of the difference of two Gaussian CDFs? | A geometrical intuition
A geometrical intuition to accompany Whuber's answer is the following:
It relates to answers to other questions here and here.
The mean of the variable relates to the area of the gray striped surfaces, which can be computed in two directions
The vertical stripes: as an integral of the quantile... | Closed form of the integral of the difference of two Gaussian CDFs? | A geometrical intuition
A geometrical intuition to accompany Whuber's answer is the following:
It relates to answers to other questions here and here.
The mean of the variable relates to the area of | Closed form of the integral of the difference of two Gaussian CDFs?
A geometrical intuition
A geometrical intuition to accompany Whuber's answer is the following:
It relates to answers to other questions here and here.
The mean of the variable relates to the area of the gray striped surfaces, which can be computed in ... | Closed form of the integral of the difference of two Gaussian CDFs?
A geometrical intuition
A geometrical intuition to accompany Whuber's answer is the following:
It relates to answers to other questions here and here.
The mean of the variable relates to the area of |
54,088 | Does poisson penalized quasi likelihood regression use biased estimators? | The Penalized Quasi Likelihood (PQL) method has been proposed to fit generalized linear mixed-effects models. The way it works is by doing a kind of a Laplace approximation in a quasi-likelihood formulation of the model. This approximation results in a transformation of the original outcome variable. The aim of the tra... | Does poisson penalized quasi likelihood regression use biased estimators? | The Penalized Quasi Likelihood (PQL) method has been proposed to fit generalized linear mixed-effects models. The way it works is by doing a kind of a Laplace approximation in a quasi-likelihood formu | Does poisson penalized quasi likelihood regression use biased estimators?
The Penalized Quasi Likelihood (PQL) method has been proposed to fit generalized linear mixed-effects models. The way it works is by doing a kind of a Laplace approximation in a quasi-likelihood formulation of the model. This approximation result... | Does poisson penalized quasi likelihood regression use biased estimators?
The Penalized Quasi Likelihood (PQL) method has been proposed to fit generalized linear mixed-effects models. The way it works is by doing a kind of a Laplace approximation in a quasi-likelihood formu |
54,089 | R-square and Intrumental Regression | No.
$R^2$ in instrumental variables regression is not useful.
Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ into $\beta^2 Var(x) + Var(\epsilon)$, so the obtained $R^2$ has neither a natural interpretation, nor can it be used for ... | R-square and Intrumental Regression | No.
$R^2$ in instrumental variables regression is not useful.
Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ in | R-square and Intrumental Regression
No.
$R^2$ in instrumental variables regression is not useful.
Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ into $\beta^2 Var(x) + Var(\epsilon)$, so the obtained $R^2$ has neither a natural int... | R-square and Intrumental Regression
No.
$R^2$ in instrumental variables regression is not useful.
Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ in |
54,090 | Why one result is so wide in this logistic multiple regession | You have too many variables for the amount of data you have. A rough rule of thumb is that, in logistic regression, you can have $1$ predictor variable for every $15$ observations in the less commonly occurring category. With $193$ data, you can have at most $97$ instances of yeses or noes. That implies you should u... | Why one result is so wide in this logistic multiple regession | You have too many variables for the amount of data you have. A rough rule of thumb is that, in logistic regression, you can have $1$ predictor variable for every $15$ observations in the less commonl | Why one result is so wide in this logistic multiple regession
You have too many variables for the amount of data you have. A rough rule of thumb is that, in logistic regression, you can have $1$ predictor variable for every $15$ observations in the less commonly occurring category. With $193$ data, you can have at mo... | Why one result is so wide in this logistic multiple regession
You have too many variables for the amount of data you have. A rough rule of thumb is that, in logistic regression, you can have $1$ predictor variable for every $15$ observations in the less commonl |
54,091 | Intuition behind posterior predictive distribution | Let $X$ denotes the observations and $\theta \in \Theta$ the parameter. In a Bayesian approach, both are considered random quantities.
The first step of modeling is to define a statistical model, i.e. the distribution of $X$ given $\theta$, which can be written as $X \mid \theta \sim p(\cdot \mid \theta)$. This is mai... | Intuition behind posterior predictive distribution | Let $X$ denotes the observations and $\theta \in \Theta$ the parameter. In a Bayesian approach, both are considered random quantities.
The first step of modeling is to define a statistical model, i.e | Intuition behind posterior predictive distribution
Let $X$ denotes the observations and $\theta \in \Theta$ the parameter. In a Bayesian approach, both are considered random quantities.
The first step of modeling is to define a statistical model, i.e. the distribution of $X$ given $\theta$, which can be written as $X ... | Intuition behind posterior predictive distribution
Let $X$ denotes the observations and $\theta \in \Theta$ the parameter. In a Bayesian approach, both are considered random quantities.
The first step of modeling is to define a statistical model, i.e |
54,092 | Why is R-squared equal to the sum of standardized coefficients times the correlation? | I offer two solutions. The first (geometric) solution reverses the usual least-squares perspective by starting with its solution--the fitted values--and working backwards to the problem(!), which exposes the basic nature and simplicity of this result. The second (algebraic) solution cranks the standard least-squares ... | Why is R-squared equal to the sum of standardized coefficients times the correlation? | I offer two solutions. The first (geometric) solution reverses the usual least-squares perspective by starting with its solution--the fitted values--and working backwards to the problem(!), which exp | Why is R-squared equal to the sum of standardized coefficients times the correlation?
I offer two solutions. The first (geometric) solution reverses the usual least-squares perspective by starting with its solution--the fitted values--and working backwards to the problem(!), which exposes the basic nature and simplici... | Why is R-squared equal to the sum of standardized coefficients times the correlation?
I offer two solutions. The first (geometric) solution reverses the usual least-squares perspective by starting with its solution--the fitted values--and working backwards to the problem(!), which exp |
54,093 | Multiple metrics in keras - why and when might we want to use it? | loss='mean_squared_error'
We divide these terms into differentiable loss function that's used to train neural network weights, and quality metrics that are used to assess the quality of the training convergence.
In your example, $$L = (Y - Y') ^ 2 / n$$ is the loss function which is minimzed along the training phase.
... | Multiple metrics in keras - why and when might we want to use it? | loss='mean_squared_error'
We divide these terms into differentiable loss function that's used to train neural network weights, and quality metrics that are used to assess the quality of the training | Multiple metrics in keras - why and when might we want to use it?
loss='mean_squared_error'
We divide these terms into differentiable loss function that's used to train neural network weights, and quality metrics that are used to assess the quality of the training convergence.
In your example, $$L = (Y - Y') ^ 2 / n$$... | Multiple metrics in keras - why and when might we want to use it?
loss='mean_squared_error'
We divide these terms into differentiable loss function that's used to train neural network weights, and quality metrics that are used to assess the quality of the training |
54,094 | Multiple metrics in keras - why and when might we want to use it? | For classification problems, sometimes cross-entropy is preferable for the "objective function" (metric), as compared with the MSE (mean square error). MSE is absolutely required if you use ANNs for function approximation problems (vs. classification problems). Would recommend looking at texts (books) like Bishop or ... | Multiple metrics in keras - why and when might we want to use it? | For classification problems, sometimes cross-entropy is preferable for the "objective function" (metric), as compared with the MSE (mean square error). MSE is absolutely required if you use ANNs for | Multiple metrics in keras - why and when might we want to use it?
For classification problems, sometimes cross-entropy is preferable for the "objective function" (metric), as compared with the MSE (mean square error). MSE is absolutely required if you use ANNs for function approximation problems (vs. classification pr... | Multiple metrics in keras - why and when might we want to use it?
For classification problems, sometimes cross-entropy is preferable for the "objective function" (metric), as compared with the MSE (mean square error). MSE is absolutely required if you use ANNs for |
54,095 | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$? | Taking $X_1,X_2 \sim \text{IID Exp}(1)$ we have:
$$R \equiv \ln \bigg( \frac{X_1+X_2}{X_1} \bigg) \sim \text{Exp}(1).$$
There are various ways to demonstrate this. In cases of difficulty, the simplest way is to derive the CDF. For all $r \geqslant 0$ we have:
$$\begin{equation} \begin{aligned}
F_R(r) = \mathbb{P}(R... | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$? | Taking $X_1,X_2 \sim \text{IID Exp}(1)$ we have:
$$R \equiv \ln \bigg( \frac{X_1+X_2}{X_1} \bigg) \sim \text{Exp}(1).$$
There are various ways to demonstrate this. In cases of difficulty, the simpl | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$?
Taking $X_1,X_2 \sim \text{IID Exp}(1)$ we have:
$$R \equiv \ln \bigg( \frac{X_1+X_2}{X_1} \bigg) \sim \text{Exp}(1).$$
There are various ways to demonstrate this. In cases of difficulty, the simplest way is to derive the ... | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$?
Taking $X_1,X_2 \sim \text{IID Exp}(1)$ we have:
$$R \equiv \ln \bigg( \frac{X_1+X_2}{X_1} \bigg) \sim \text{Exp}(1).$$
There are various ways to demonstrate this. In cases of difficulty, the simpl |
54,096 | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$? | Observe that $ln(\frac{X_1+X_2}{X_1})$ = $-ln(\frac{X_1}{X_1+X_2})$. Since $-ln({U})$ is exp(1) when U ~ Uniform(0,1). If we can prove $\frac{X_1}{X_1+X_2}$ is standard uniform then we will have completed the proof.
Observe that
$P(\frac{X_1}{X_1+X_2}\leq t) = P(\frac{X_1+X_2}{X_1}\leq \frac{1}{t}) = P\{X_2\leq X_1(\f... | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$? | Observe that $ln(\frac{X_1+X_2}{X_1})$ = $-ln(\frac{X_1}{X_1+X_2})$. Since $-ln({U})$ is exp(1) when U ~ Uniform(0,1). If we can prove $\frac{X_1}{X_1+X_2}$ is standard uniform then we will have compl | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$?
Observe that $ln(\frac{X_1+X_2}{X_1})$ = $-ln(\frac{X_1}{X_1+X_2})$. Since $-ln({U})$ is exp(1) when U ~ Uniform(0,1). If we can prove $\frac{X_1}{X_1+X_2}$ is standard uniform then we will have completed the proof.
Observe ... | What is the distribution of $\ln(\frac{X_1+X_2}{X_1})$ when $X_1, X_2 \sim \text{IID Exp}(1)$?
Observe that $ln(\frac{X_1+X_2}{X_1})$ = $-ln(\frac{X_1}{X_1+X_2})$. Since $-ln({U})$ is exp(1) when U ~ Uniform(0,1). If we can prove $\frac{X_1}{X_1+X_2}$ is standard uniform then we will have compl |
54,097 | What is the probability distribution used in logistic regression called? | Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:
$$Y_i | \mathbf{x}, \mathbf{w} \sim \text{Bern} \Bigg( \text{Prob} = \frac{\exp(\mathbf{w}^\text{T} \mathbf{x})}{1 + \exp(\mathbf{w}^\text{T} \mathbf{x})} \Bigg).$$
One alternative way of looking at the logistic regression is to regard th... | What is the probability distribution used in logistic regression called? | Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:
$$Y_i | \mathbf{x}, \mathbf{w} \sim \text{Bern} \Bigg( \text{Prob} = \frac{\exp(\mathbf{w}^\text{T} \mathbf{x})}{1 + \ | What is the probability distribution used in logistic regression called?
Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:
$$Y_i | \mathbf{x}, \mathbf{w} \sim \text{Bern} \Bigg( \text{Prob} = \frac{\exp(\mathbf{w}^\text{T} \mathbf{x})}{1 + \exp(\mathbf{w}^\text{T} \mathbf{x})} \Bigg).$$
... | What is the probability distribution used in logistic regression called?
Since $Y_i$ is a binary variable, its distribution is the Bernoulli distribution:
$$Y_i | \mathbf{x}, \mathbf{w} \sim \text{Bern} \Bigg( \text{Prob} = \frac{\exp(\mathbf{w}^\text{T} \mathbf{x})}{1 + \ |
54,098 | What is the probability distribution used in logistic regression called? | In your link, you have the cumulative distribution function for the logistic distribution as $$\frac{1}{1+e^{-\frac{x-\mu}{s}}}$$
while in your question you have $$\dfrac{\exp(w^TX)}{(1+\exp(w^TX))} \text{ which is } \dfrac{1}{1+e^{-w^TX}}$$
and these are essentially the same so long as $w^TX$ has mean $0$ and varianc... | What is the probability distribution used in logistic regression called? | In your link, you have the cumulative distribution function for the logistic distribution as $$\frac{1}{1+e^{-\frac{x-\mu}{s}}}$$
while in your question you have $$\dfrac{\exp(w^TX)}{(1+\exp(w^TX))} \ | What is the probability distribution used in logistic regression called?
In your link, you have the cumulative distribution function for the logistic distribution as $$\frac{1}{1+e^{-\frac{x-\mu}{s}}}$$
while in your question you have $$\dfrac{\exp(w^TX)}{(1+\exp(w^TX))} \text{ which is } \dfrac{1}{1+e^{-w^TX}}$$
and ... | What is the probability distribution used in logistic regression called?
In your link, you have the cumulative distribution function for the logistic distribution as $$\frac{1}{1+e^{-\frac{x-\mu}{s}}}$$
while in your question you have $$\dfrac{\exp(w^TX)}{(1+\exp(w^TX))} \ |
54,099 | What is the relation between "conjugate priors" and the approximate inference? | Priors are not chosen for convenience but for reflecting one's own beliefs or absence thereof about the parameters of the model. There is thus no foundational reason for always choosing conjugate priors, which main justification is computational. Furthermore,
Conjugate priors only exist for exponential family models.... | What is the relation between "conjugate priors" and the approximate inference? | Priors are not chosen for convenience but for reflecting one's own beliefs or absence thereof about the parameters of the model. There is thus no foundational reason for always choosing conjugate prio | What is the relation between "conjugate priors" and the approximate inference?
Priors are not chosen for convenience but for reflecting one's own beliefs or absence thereof about the parameters of the model. There is thus no foundational reason for always choosing conjugate priors, which main justification is computati... | What is the relation between "conjugate priors" and the approximate inference?
Priors are not chosen for convenience but for reflecting one's own beliefs or absence thereof about the parameters of the model. There is thus no foundational reason for always choosing conjugate prio |
54,100 | What is the relation between "conjugate priors" and the approximate inference? | There's no relation whatsoever. If you use conjugate priors, then the solution exists in closed-form, so you don't need to use things like MCMC or approximate inference to obtain the solution. | What is the relation between "conjugate priors" and the approximate inference? | There's no relation whatsoever. If you use conjugate priors, then the solution exists in closed-form, so you don't need to use things like MCMC or approximate inference to obtain the solution. | What is the relation between "conjugate priors" and the approximate inference?
There's no relation whatsoever. If you use conjugate priors, then the solution exists in closed-form, so you don't need to use things like MCMC or approximate inference to obtain the solution. | What is the relation between "conjugate priors" and the approximate inference?
There's no relation whatsoever. If you use conjugate priors, then the solution exists in closed-form, so you don't need to use things like MCMC or approximate inference to obtain the solution. |
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