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 |
|---|---|---|---|---|---|---|
10,101 | Would a Random Forest with multiple outputs be possible/practical? | Multiple output decision trees (and hence, random forests) have been developed and published. Pierre Guertz distributes a package for this (download). See also Segal & Xiao, Multivariate random forests, WIREs Data Mining Knowl Discov 2011 1 80–87, DOI: 10.1002/widm.12
I believe the latest version of Scikit-learn also ... | Would a Random Forest with multiple outputs be possible/practical? | Multiple output decision trees (and hence, random forests) have been developed and published. Pierre Guertz distributes a package for this (download). See also Segal & Xiao, Multivariate random forest | Would a Random Forest with multiple outputs be possible/practical?
Multiple output decision trees (and hence, random forests) have been developed and published. Pierre Guertz distributes a package for this (download). See also Segal & Xiao, Multivariate random forests, WIREs Data Mining Knowl Discov 2011 1 80–87, DOI: ... | Would a Random Forest with multiple outputs be possible/practical?
Multiple output decision trees (and hence, random forests) have been developed and published. Pierre Guertz distributes a package for this (download). See also Segal & Xiao, Multivariate random forest |
10,102 | Would a Random Forest with multiple outputs be possible/practical? | As stated here:
All classifiers in scikit-learn do multiclass classification out-of-the-box.
And that includes Random Forest.
Also the page: http://scikit-learn.org/stable/modules/tree.html#tree-multioutput has a lot of references on that topic. | Would a Random Forest with multiple outputs be possible/practical? | As stated here:
All classifiers in scikit-learn do multiclass classification out-of-the-box.
And that includes Random Forest.
Also the page: http://scikit-learn.org/stable/modules/tree.html#tree-mul | Would a Random Forest with multiple outputs be possible/practical?
As stated here:
All classifiers in scikit-learn do multiclass classification out-of-the-box.
And that includes Random Forest.
Also the page: http://scikit-learn.org/stable/modules/tree.html#tree-multioutput has a lot of references on that topic. | Would a Random Forest with multiple outputs be possible/practical?
As stated here:
All classifiers in scikit-learn do multiclass classification out-of-the-box.
And that includes Random Forest.
Also the page: http://scikit-learn.org/stable/modules/tree.html#tree-mul |
10,103 | Would a Random Forest with multiple outputs be possible/practical? | Version 0.24.2 is taking this use case into account: https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier
As specified in the documentation, the shape of the target you're passing to the fit function is like (n_samples, n_outputs). | Would a Random Forest with multiple outputs be possible/practical? | Version 0.24.2 is taking this use case into account: https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier
As specified | Would a Random Forest with multiple outputs be possible/practical?
Version 0.24.2 is taking this use case into account: https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier
As specified in the documentation, the shape of the target you're... | Would a Random Forest with multiple outputs be possible/practical?
Version 0.24.2 is taking this use case into account: https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html#sklearn.ensemble.RandomForestClassifier
As specified |
10,104 | What diagnostic plots exists for quantile regression? | Quantile regression does not make distributional assumptions, i.e., assumptions about residuals, other than assuming that the response variable is almost continuous. If you are addressing the problem of estimating a single quantile as a function predictors X, far and away the major things that can go wrong are misspec... | What diagnostic plots exists for quantile regression? | Quantile regression does not make distributional assumptions, i.e., assumptions about residuals, other than assuming that the response variable is almost continuous. If you are addressing the problem | What diagnostic plots exists for quantile regression?
Quantile regression does not make distributional assumptions, i.e., assumptions about residuals, other than assuming that the response variable is almost continuous. If you are addressing the problem of estimating a single quantile as a function predictors X, far a... | What diagnostic plots exists for quantile regression?
Quantile regression does not make distributional assumptions, i.e., assumptions about residuals, other than assuming that the response variable is almost continuous. If you are addressing the problem |
10,105 | Multiple hypothesis testing correction with Benjamini-Hochberg, p-values or q-values? | As Robin said, you've got the Benjamini-Hochberg method backwards. With that method, you set a value for Q (upper case Q; the maximum desired FDR) and it then sorts your comparisons into two piles. The goal is that no more than Q% of the comparisons in the "discovery" pile are false, and thus at least 100%-Q% are true.... | Multiple hypothesis testing correction with Benjamini-Hochberg, p-values or q-values? | As Robin said, you've got the Benjamini-Hochberg method backwards. With that method, you set a value for Q (upper case Q; the maximum desired FDR) and it then sorts your comparisons into two piles. Th | Multiple hypothesis testing correction with Benjamini-Hochberg, p-values or q-values?
As Robin said, you've got the Benjamini-Hochberg method backwards. With that method, you set a value for Q (upper case Q; the maximum desired FDR) and it then sorts your comparisons into two piles. The goal is that no more than Q% of ... | Multiple hypothesis testing correction with Benjamini-Hochberg, p-values or q-values?
As Robin said, you've got the Benjamini-Hochberg method backwards. With that method, you set a value for Q (upper case Q; the maximum desired FDR) and it then sorts your comparisons into two piles. Th |
10,106 | Intuition behind logistic regression | The logistic regression model is maximum likelihood using the natural parameter (the log-odds ratio) to contrast the relative changes in the risk of the outcome per unit difference in the predictor. This is assuming, of course, a binomial probability model for the outcome. That means that the consistency and robustness... | Intuition behind logistic regression | The logistic regression model is maximum likelihood using the natural parameter (the log-odds ratio) to contrast the relative changes in the risk of the outcome per unit difference in the predictor. T | Intuition behind logistic regression
The logistic regression model is maximum likelihood using the natural parameter (the log-odds ratio) to contrast the relative changes in the risk of the outcome per unit difference in the predictor. This is assuming, of course, a binomial probability model for the outcome. That mean... | Intuition behind logistic regression
The logistic regression model is maximum likelihood using the natural parameter (the log-odds ratio) to contrast the relative changes in the risk of the outcome per unit difference in the predictor. T |
10,107 | Intuition behind logistic regression | One way to think about logistic regression is as a threshold response model. In these models, you have a binary dependent variable, $Y$, which is influenced by the values of a vector of independent variables $X$. The dependent variable $Y$ can only take on the values 0 and 1, so you can't model the dependence of $Y$ ... | Intuition behind logistic regression | One way to think about logistic regression is as a threshold response model. In these models, you have a binary dependent variable, $Y$, which is influenced by the values of a vector of independent v | Intuition behind logistic regression
One way to think about logistic regression is as a threshold response model. In these models, you have a binary dependent variable, $Y$, which is influenced by the values of a vector of independent variables $X$. The dependent variable $Y$ can only take on the values 0 and 1, so y... | Intuition behind logistic regression
One way to think about logistic regression is as a threshold response model. In these models, you have a binary dependent variable, $Y$, which is influenced by the values of a vector of independent v |
10,108 | Intuition behind logistic regression | Logistic regression is not originally developed by machine learning community but statistics community. There are a lot of probabilistic views behind it. You may search the following terms to get more information: odds, log odds, generalized linear model, binomial link function.
But for machine learning, the goal is sl... | Intuition behind logistic regression | Logistic regression is not originally developed by machine learning community but statistics community. There are a lot of probabilistic views behind it. You may search the following terms to get more | Intuition behind logistic regression
Logistic regression is not originally developed by machine learning community but statistics community. There are a lot of probabilistic views behind it. You may search the following terms to get more information: odds, log odds, generalized linear model, binomial link function.
But... | Intuition behind logistic regression
Logistic regression is not originally developed by machine learning community but statistics community. There are a lot of probabilistic views behind it. You may search the following terms to get more |
10,109 | Link between moment-generating function and characteristic function | As mentioned in the comments, characteristic functions always exist, because they require integration of a function of modulus $1$. However, the moment generating function doesn't need to exist because in particular it requires the existence of moments of any order.
When we know that $E[e^{tX}]$ is integrable for all ... | Link between moment-generating function and characteristic function | As mentioned in the comments, characteristic functions always exist, because they require integration of a function of modulus $1$. However, the moment generating function doesn't need to exist becaus | Link between moment-generating function and characteristic function
As mentioned in the comments, characteristic functions always exist, because they require integration of a function of modulus $1$. However, the moment generating function doesn't need to exist because in particular it requires the existence of moments... | Link between moment-generating function and characteristic function
As mentioned in the comments, characteristic functions always exist, because they require integration of a function of modulus $1$. However, the moment generating function doesn't need to exist becaus |
10,110 | Link between moment-generating function and characteristic function | From Proakis, Digital communication 5th ed., the straightforward relationship is
$$
\varphi(\omega) = M(j\omega)
$$
and
$$
M_x(t) = \varphi_x(-j t)
$$ | Link between moment-generating function and characteristic function | From Proakis, Digital communication 5th ed., the straightforward relationship is
$$
\varphi(\omega) = M(j\omega)
$$
and
$$
M_x(t) = \varphi_x(-j t)
$$ | Link between moment-generating function and characteristic function
From Proakis, Digital communication 5th ed., the straightforward relationship is
$$
\varphi(\omega) = M(j\omega)
$$
and
$$
M_x(t) = \varphi_x(-j t)
$$ | Link between moment-generating function and characteristic function
From Proakis, Digital communication 5th ed., the straightforward relationship is
$$
\varphi(\omega) = M(j\omega)
$$
and
$$
M_x(t) = \varphi_x(-j t)
$$ |
10,111 | Why is my derivation of a closed form lasso solution incorrect? | This is normally done with least angle regression, you can find the paper here.
Sorry about my confusion in the beginning, I am going to make another attempt at this.
So after the expansion of your function $f(\beta)$ you get
$$
f(\beta)=\sum_{i=1}^n y_i^2 -2\sum_{i=1}^n y_i X_i \beta + \sum_{i=1}^n \beta^T X_i^T X_i \... | Why is my derivation of a closed form lasso solution incorrect? | This is normally done with least angle regression, you can find the paper here.
Sorry about my confusion in the beginning, I am going to make another attempt at this.
So after the expansion of your fu | Why is my derivation of a closed form lasso solution incorrect?
This is normally done with least angle regression, you can find the paper here.
Sorry about my confusion in the beginning, I am going to make another attempt at this.
So after the expansion of your function $f(\beta)$ you get
$$
f(\beta)=\sum_{i=1}^n y_i^2... | Why is my derivation of a closed form lasso solution incorrect?
This is normally done with least angle regression, you can find the paper here.
Sorry about my confusion in the beginning, I am going to make another attempt at this.
So after the expansion of your fu |
10,112 | Difference of two i.i.d. lognormal random variables | This is a difficult problem. I thought first about using (some approximation of) the moment generating function of the lognormal distribution. That doesn't work, as I will explain. But first some notation:
Let $\phi$ be the standard normal density and $\Phi$ the corresponding cumulative distribution function. We wil... | Difference of two i.i.d. lognormal random variables | This is a difficult problem. I thought first about using (some approximation of) the moment generating function of the lognormal distribution. That doesn't work, as I will explain. But first some no | Difference of two i.i.d. lognormal random variables
This is a difficult problem. I thought first about using (some approximation of) the moment generating function of the lognormal distribution. That doesn't work, as I will explain. But first some notation:
Let $\phi$ be the standard normal density and $\Phi$ the cor... | Difference of two i.i.d. lognormal random variables
This is a difficult problem. I thought first about using (some approximation of) the moment generating function of the lognormal distribution. That doesn't work, as I will explain. But first some no |
10,113 | Difference of two i.i.d. lognormal random variables | This does not strictly answer your question, but wouldn't it be easier to look at the ratio of the $X$ and $Y$? You then simply arrive at
$$
\begin{align}
\Pr\left(\frac{X}{Y} \leq t\right) &= \Pr\left(\log\left(\frac{X}{Y}\right) \leq \log(t) \right) \\
&= \Pr(\log(X) - \log(Y) \leq \log(t)) \\
&\sim \mathcal{N}(0, 2 ... | Difference of two i.i.d. lognormal random variables | This does not strictly answer your question, but wouldn't it be easier to look at the ratio of the $X$ and $Y$? You then simply arrive at
$$
\begin{align}
\Pr\left(\frac{X}{Y} \leq t\right) &= \Pr\lef | Difference of two i.i.d. lognormal random variables
This does not strictly answer your question, but wouldn't it be easier to look at the ratio of the $X$ and $Y$? You then simply arrive at
$$
\begin{align}
\Pr\left(\frac{X}{Y} \leq t\right) &= \Pr\left(\log\left(\frac{X}{Y}\right) \leq \log(t) \right) \\
&= \Pr(\log(X... | Difference of two i.i.d. lognormal random variables
This does not strictly answer your question, but wouldn't it be easier to look at the ratio of the $X$ and $Y$? You then simply arrive at
$$
\begin{align}
\Pr\left(\frac{X}{Y} \leq t\right) &= \Pr\lef |
10,114 | Difference of two i.i.d. lognormal random variables | I have a feeling characteristic functions are the best option for handling this question.
For basics, kindly see:https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
now, kindly CTRL+F "independen". you'll see the answer
in brief:
The characteristic function approach is particularly useful in ana... | Difference of two i.i.d. lognormal random variables | I have a feeling characteristic functions are the best option for handling this question.
For basics, kindly see:https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
now, kindly | Difference of two i.i.d. lognormal random variables
I have a feeling characteristic functions are the best option for handling this question.
For basics, kindly see:https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
now, kindly CTRL+F "independen". you'll see the answer
in brief:
The characteri... | Difference of two i.i.d. lognormal random variables
I have a feeling characteristic functions are the best option for handling this question.
For basics, kindly see:https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)
now, kindly |
10,115 | Interpretation of Hartigans' dip test | Mr. Freeman (author of the paper I told you about) told me that he was actually looking only at the Pvalue of the dip test. This confusion comes from his sentence :
"HDS values range from 0 to 1 with values less than .05 indicating significant bimodality, and values greater than .05 but less than .10 suggesting bimodal... | Interpretation of Hartigans' dip test | Mr. Freeman (author of the paper I told you about) told me that he was actually looking only at the Pvalue of the dip test. This confusion comes from his sentence :
"HDS values range from 0 to 1 with | Interpretation of Hartigans' dip test
Mr. Freeman (author of the paper I told you about) told me that he was actually looking only at the Pvalue of the dip test. This confusion comes from his sentence :
"HDS values range from 0 to 1 with values less than .05 indicating significant bimodality, and values greater than .0... | Interpretation of Hartigans' dip test
Mr. Freeman (author of the paper I told you about) told me that he was actually looking only at the Pvalue of the dip test. This confusion comes from his sentence :
"HDS values range from 0 to 1 with |
10,116 | Consequences of modeling a non-stationary process using ARMA? | My impression is that this question does not have a unique, fully general answer, so I will only explore the simplest case, and in a bit informal way.
Assume that the true Data Generating Mechanism is
$$y_t = y_{t-1} + u_t,\;\; t=1,...,T,\;\; y_0 =0 \tag{1}$$
with $u_t$ a usual zero-mean i.i.d. white noise component,... | Consequences of modeling a non-stationary process using ARMA? | My impression is that this question does not have a unique, fully general answer, so I will only explore the simplest case, and in a bit informal way.
Assume that the true Data Generating Mechanism | Consequences of modeling a non-stationary process using ARMA?
My impression is that this question does not have a unique, fully general answer, so I will only explore the simplest case, and in a bit informal way.
Assume that the true Data Generating Mechanism is
$$y_t = y_{t-1} + u_t,\;\; t=1,...,T,\;\; y_0 =0 \tag{1... | Consequences of modeling a non-stationary process using ARMA?
My impression is that this question does not have a unique, fully general answer, so I will only explore the simplest case, and in a bit informal way.
Assume that the true Data Generating Mechanism |
10,117 | Consequences of modeling a non-stationary process using ARMA? | That is a good question.
As I realize, you just have considered pacf but that is not enough. ACF and PACF are both necessary to select the best model.
On the other hand, stationary tests are weak and sensitive and need a large amount of lags to be tested.
In addition, it is preferred to make time series stationary befo... | Consequences of modeling a non-stationary process using ARMA? | That is a good question.
As I realize, you just have considered pacf but that is not enough. ACF and PACF are both necessary to select the best model.
On the other hand, stationary tests are weak and | Consequences of modeling a non-stationary process using ARMA?
That is a good question.
As I realize, you just have considered pacf but that is not enough. ACF and PACF are both necessary to select the best model.
On the other hand, stationary tests are weak and sensitive and need a large amount of lags to be tested.
In... | Consequences of modeling a non-stationary process using ARMA?
That is a good question.
As I realize, you just have considered pacf but that is not enough. ACF and PACF are both necessary to select the best model.
On the other hand, stationary tests are weak and |
10,118 | Multivariate normal posterior | With the distributions on our random vectors:
$\mathbf x_i | \mathbf \mu \sim N(\mu , \mathbf \Sigma)$
$\mathbf \mu \sim N(\mathbf \mu_0, \mathbf \Sigma_0)$
By Bayes's rule the posterior distribution looks like:
$p(\mu| \{\mathbf x_i\}) \propto p(\mu) \prod_{i=1}^N p(\mathbf x_i | \mu)$
So:
$\ln p(\mu| \{\mathbf x_i\... | Multivariate normal posterior | With the distributions on our random vectors:
$\mathbf x_i | \mathbf \mu \sim N(\mu , \mathbf \Sigma)$
$\mathbf \mu \sim N(\mathbf \mu_0, \mathbf \Sigma_0)$
By Bayes's rule the posterior distribution | Multivariate normal posterior
With the distributions on our random vectors:
$\mathbf x_i | \mathbf \mu \sim N(\mu , \mathbf \Sigma)$
$\mathbf \mu \sim N(\mathbf \mu_0, \mathbf \Sigma_0)$
By Bayes's rule the posterior distribution looks like:
$p(\mu| \{\mathbf x_i\}) \propto p(\mu) \prod_{i=1}^N p(\mathbf x_i | \mu)$
... | Multivariate normal posterior
With the distributions on our random vectors:
$\mathbf x_i | \mathbf \mu \sim N(\mu , \mathbf \Sigma)$
$\mathbf \mu \sim N(\mathbf \mu_0, \mathbf \Sigma_0)$
By Bayes's rule the posterior distribution |
10,119 | What if your random sample is clearly not representative? | The answer given by MLS (use importance sampling) is only as good as the assumptions you can make about your distributions. The main strength of the finite population sampling paradigm is that it is non-parametric, as it does not make any assumptions about the distribution of the data to make (valid) inferences on the ... | What if your random sample is clearly not representative? | The answer given by MLS (use importance sampling) is only as good as the assumptions you can make about your distributions. The main strength of the finite population sampling paradigm is that it is n | What if your random sample is clearly not representative?
The answer given by MLS (use importance sampling) is only as good as the assumptions you can make about your distributions. The main strength of the finite population sampling paradigm is that it is non-parametric, as it does not make any assumptions about the d... | What if your random sample is clearly not representative?
The answer given by MLS (use importance sampling) is only as good as the assumptions you can make about your distributions. The main strength of the finite population sampling paradigm is that it is n |
10,120 | What if your random sample is clearly not representative? | I'm the Junior Member here, but I'd say that discarding and starting over is always the best answer, if you know that your sample is significantly unrepresentative, and if you have an idea of how the unrepresentative sampling arose in the first place and how to avoid it if possible the second time around.
What good wil... | What if your random sample is clearly not representative? | I'm the Junior Member here, but I'd say that discarding and starting over is always the best answer, if you know that your sample is significantly unrepresentative, and if you have an idea of how the | What if your random sample is clearly not representative?
I'm the Junior Member here, but I'd say that discarding and starting over is always the best answer, if you know that your sample is significantly unrepresentative, and if you have an idea of how the unrepresentative sampling arose in the first place and how to ... | What if your random sample is clearly not representative?
I'm the Junior Member here, but I'd say that discarding and starting over is always the best answer, if you know that your sample is significantly unrepresentative, and if you have an idea of how the |
10,121 | What if your random sample is clearly not representative? | This is a partial answer that assumes we know both the distribution $q$ from which was sampled, and the true (or desired) distribution $p$. Additionally, I assume that these distributions are different. If the samples were actually obtained through $p$, but they look wrong: the samples are still unbiased and any adapta... | What if your random sample is clearly not representative? | This is a partial answer that assumes we know both the distribution $q$ from which was sampled, and the true (or desired) distribution $p$. Additionally, I assume that these distributions are differen | What if your random sample is clearly not representative?
This is a partial answer that assumes we know both the distribution $q$ from which was sampled, and the true (or desired) distribution $p$. Additionally, I assume that these distributions are different. If the samples were actually obtained through $p$, but they... | What if your random sample is clearly not representative?
This is a partial answer that assumes we know both the distribution $q$ from which was sampled, and the true (or desired) distribution $p$. Additionally, I assume that these distributions are differen |
10,122 | What is behind Google Prediction API? | Google is using different machine learning techniques and algorithm for training and prediction. The strategies for large-scale supervised learning:
1. Sub-sample
2. Embarrassingly parallelize some algorithms
3. Distributed gradient descent
4. Majority Vote
5. Parameter mixture
6. Iterative parameter mixture
They shoul... | What is behind Google Prediction API? | Google is using different machine learning techniques and algorithm for training and prediction. The strategies for large-scale supervised learning:
1. Sub-sample
2. Embarrassingly parallelize some al | What is behind Google Prediction API?
Google is using different machine learning techniques and algorithm for training and prediction. The strategies for large-scale supervised learning:
1. Sub-sample
2. Embarrassingly parallelize some algorithms
3. Distributed gradient descent
4. Majority Vote
5. Parameter mixture
6. ... | What is behind Google Prediction API?
Google is using different machine learning techniques and algorithm for training and prediction. The strategies for large-scale supervised learning:
1. Sub-sample
2. Embarrassingly parallelize some al |
10,123 | What are some standard practices for creating synthetic data sets? | I'm not sure there are standard practices for generating synthetic data - it's used so heavily in so many different aspects of research that purpose-built data seems to be a more common and arguably more reasonable approach.
For me, my best standard practice is not to make the data set so it will work well with the mod... | What are some standard practices for creating synthetic data sets? | I'm not sure there are standard practices for generating synthetic data - it's used so heavily in so many different aspects of research that purpose-built data seems to be a more common and arguably m | What are some standard practices for creating synthetic data sets?
I'm not sure there are standard practices for generating synthetic data - it's used so heavily in so many different aspects of research that purpose-built data seems to be a more common and arguably more reasonable approach.
For me, my best standard pra... | What are some standard practices for creating synthetic data sets?
I'm not sure there are standard practices for generating synthetic data - it's used so heavily in so many different aspects of research that purpose-built data seems to be a more common and arguably m |
10,124 | What are some standard practices for creating synthetic data sets? | The R statistical package has a simulate function that will simulate data based on a model fit to existing data. This uses the fitted model as the "known" population relationship, then simulates new data based on that model. There is a method for this function in the lme4 package. These fitted objects can take into ... | What are some standard practices for creating synthetic data sets? | The R statistical package has a simulate function that will simulate data based on a model fit to existing data. This uses the fitted model as the "known" population relationship, then simulates new | What are some standard practices for creating synthetic data sets?
The R statistical package has a simulate function that will simulate data based on a model fit to existing data. This uses the fitted model as the "known" population relationship, then simulates new data based on that model. There is a method for this... | What are some standard practices for creating synthetic data sets?
The R statistical package has a simulate function that will simulate data based on a model fit to existing data. This uses the fitted model as the "known" population relationship, then simulates new |
10,125 | How to get hyper parameters in nested cross validation? | (I'm sure I wrote most of this already in some answer - but can't find it right now. If anyone stumbles across that answer, please link it).
I see 2 slightly different approaches here, which I think are both sensible.
But first some terminology:
Coming from an applied field, a (fitted/trained) model for me is a ready... | How to get hyper parameters in nested cross validation? | (I'm sure I wrote most of this already in some answer - but can't find it right now. If anyone stumbles across that answer, please link it).
I see 2 slightly different approaches here, which I think a | How to get hyper parameters in nested cross validation?
(I'm sure I wrote most of this already in some answer - but can't find it right now. If anyone stumbles across that answer, please link it).
I see 2 slightly different approaches here, which I think are both sensible.
But first some terminology:
Coming from an a... | How to get hyper parameters in nested cross validation?
(I'm sure I wrote most of this already in some answer - but can't find it right now. If anyone stumbles across that answer, please link it).
I see 2 slightly different approaches here, which I think a |
10,126 | How to get hyper parameters in nested cross validation? | I have read your question and the answer above 2 times (1st time 3 month ago). I'm interested and also want to find the absolute appropriate way to do cross-validation for my data. After a lot of thinking & reading, it seems that I find the holes and here is my fix:
To explain my confusion, let me try to walk through ... | How to get hyper parameters in nested cross validation? | I have read your question and the answer above 2 times (1st time 3 month ago). I'm interested and also want to find the absolute appropriate way to do cross-validation for my data. After a lot of thin | How to get hyper parameters in nested cross validation?
I have read your question and the answer above 2 times (1st time 3 month ago). I'm interested and also want to find the absolute appropriate way to do cross-validation for my data. After a lot of thinking & reading, it seems that I find the holes and here is my fi... | How to get hyper parameters in nested cross validation?
I have read your question and the answer above 2 times (1st time 3 month ago). I'm interested and also want to find the absolute appropriate way to do cross-validation for my data. After a lot of thin |
10,127 | How to get hyper parameters in nested cross validation? | You do not use nested cross validation to select the hyper parameters of the algorithm, this method is used for estimating the generalisation error of your model building procedure. Where by model building procedure I intend all the steps you applied to reach the final model you are going to be using on field.
A model ... | How to get hyper parameters in nested cross validation? | You do not use nested cross validation to select the hyper parameters of the algorithm, this method is used for estimating the generalisation error of your model building procedure. Where by model bui | How to get hyper parameters in nested cross validation?
You do not use nested cross validation to select the hyper parameters of the algorithm, this method is used for estimating the generalisation error of your model building procedure. Where by model building procedure I intend all the steps you applied to reach the ... | How to get hyper parameters in nested cross validation?
You do not use nested cross validation to select the hyper parameters of the algorithm, this method is used for estimating the generalisation error of your model building procedure. Where by model bui |
10,128 | How can one empirically demonstrate in R which cross-validation methods the AIC and BIC are equivalent to? | In an attempt to partially answer my own question, I read Wikipedia's description of leave-one-out cross validation
involves using a single observation
from the original sample as the
validation data, and the remaining
observations as the training data.
This is repeated such that each
observation in the sam... | How can one empirically demonstrate in R which cross-validation methods the AIC and BIC are equivale | In an attempt to partially answer my own question, I read Wikipedia's description of leave-one-out cross validation
involves using a single observation
from the original sample as the
validation | How can one empirically demonstrate in R which cross-validation methods the AIC and BIC are equivalent to?
In an attempt to partially answer my own question, I read Wikipedia's description of leave-one-out cross validation
involves using a single observation
from the original sample as the
validation data, and th... | How can one empirically demonstrate in R which cross-validation methods the AIC and BIC are equivale
In an attempt to partially answer my own question, I read Wikipedia's description of leave-one-out cross validation
involves using a single observation
from the original sample as the
validation |
10,129 | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s | Some related papers:
Wiki: `http://en.wikipedia.org/wiki/Ratio_distribution
http://www.jstatsoft.org/v16/i04/
http://link.springer.com/article/10.1007/s00362-012-0429-2
http://mrvar.fdv.uni-lj.si/pub/mz/mz1.1/cedilnik.pdf | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s | Some related papers:
Wiki: `http://en.wikipedia.org/wiki/Ratio_distribution
http://www.jstatsoft.org/v16/i04/
http://link.springer.com/article/10.1007/s00362-012-0429-2
http://mrvar.fdv.uni-lj.si/pub | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s
Some related papers:
Wiki: `http://en.wikipedia.org/wiki/Ratio_distribution
http://www.jstatsoft.org/v16/i04/
http://link.springer.com/article/10.1007/s00362-012-0429-2
http://mrvar.fdv.uni-lj.si/pub/mz/mz1.1/cedilnik.pdf | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s
Some related papers:
Wiki: `http://en.wikipedia.org/wiki/Ratio_distribution
http://www.jstatsoft.org/v16/i04/
http://link.springer.com/article/10.1007/s00362-012-0429-2
http://mrvar.fdv.uni-lj.si/pub |
10,130 | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s | Consider using a symbolic math package like Mathematica ,if you have a license, or Sage if you don't.
If your just doing numerical work, you might also just consider numerical differentiation.
While tedious, it does look straight forward. That is, all of the functions involved have easy to compute derivatives. You m... | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s | Consider using a symbolic math package like Mathematica ,if you have a license, or Sage if you don't.
If your just doing numerical work, you might also just consider numerical differentiation.
Whil | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s
Consider using a symbolic math package like Mathematica ,if you have a license, or Sage if you don't.
If your just doing numerical work, you might also just consider numerical differentiation.
While tedious, it does look straight forward... | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s
Consider using a symbolic math package like Mathematica ,if you have a license, or Sage if you don't.
If your just doing numerical work, you might also just consider numerical differentiation.
Whil |
10,131 | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s | This is the sort of problem that is very easy numerically, and less error prone as well. Since you say you only need the signs, I assume that accurate numerical approximations are more than sufficient for your needs. Here is some code with an example of the derivative against $\mu_x$:
pratio <- function(z, mu_x=1.0... | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s | This is the sort of problem that is very easy numerically, and less error prone as well. Since you say you only need the signs, I assume that accurate numerical approximations are more than sufficient | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s
This is the sort of problem that is very easy numerically, and less error prone as well. Since you say you only need the signs, I assume that accurate numerical approximations are more than sufficient for your needs. Here is some code with ... | Gaussian Ratio Distribution: Derivatives wrt underlying $\mu$'s and $\sigma^2$s
This is the sort of problem that is very easy numerically, and less error prone as well. Since you say you only need the signs, I assume that accurate numerical approximations are more than sufficient |
10,132 | How to understand degrees of freedom? | This is a subtle question. It takes a thoughtful person not to understand those quotations! Although they are suggestive, it turns out that none of them is exactly or generally correct. I haven't the time (and there isn't the space here) to give a full exposition, but I would like to share one approach and an insigh... | How to understand degrees of freedom? | This is a subtle question. It takes a thoughtful person not to understand those quotations! Although they are suggestive, it turns out that none of them is exactly or generally correct. I haven't t | How to understand degrees of freedom?
This is a subtle question. It takes a thoughtful person not to understand those quotations! Although they are suggestive, it turns out that none of them is exactly or generally correct. I haven't the time (and there isn't the space here) to give a full exposition, but I would li... | How to understand degrees of freedom?
This is a subtle question. It takes a thoughtful person not to understand those quotations! Although they are suggestive, it turns out that none of them is exactly or generally correct. I haven't t |
10,133 | How to understand degrees of freedom? | Or simply: the number of elements in a numerical array that you're allowed to change so that the value of the statistic remains unchanged.
# for instance if:
x + y + z = 10
you can change, for instance, x and y at random, but you cannot change z (you can, but not at random, therefore you're not free to change it - see... | How to understand degrees of freedom? | Or simply: the number of elements in a numerical array that you're allowed to change so that the value of the statistic remains unchanged.
# for instance if:
x + y + z = 10
you can change, for instan | How to understand degrees of freedom?
Or simply: the number of elements in a numerical array that you're allowed to change so that the value of the statistic remains unchanged.
# for instance if:
x + y + z = 10
you can change, for instance, x and y at random, but you cannot change z (you can, but not at random, theref... | How to understand degrees of freedom?
Or simply: the number of elements in a numerical array that you're allowed to change so that the value of the statistic remains unchanged.
# for instance if:
x + y + z = 10
you can change, for instan |
10,134 | How to understand degrees of freedom? | The concept is not at all difficult to make mathematical precise given a bit of general knowledge of $n$-dimensional Euclidean geometry, subspaces and orthogonal projections.
If $P$ is an orthogonal projection from $\mathbb{R}^n$ to a $p$-dimensional subspace $L$ and $x$ is an arbitrary $n$-vector then $Px$ is in $L$, ... | How to understand degrees of freedom? | The concept is not at all difficult to make mathematical precise given a bit of general knowledge of $n$-dimensional Euclidean geometry, subspaces and orthogonal projections.
If $P$ is an orthogonal p | How to understand degrees of freedom?
The concept is not at all difficult to make mathematical precise given a bit of general knowledge of $n$-dimensional Euclidean geometry, subspaces and orthogonal projections.
If $P$ is an orthogonal projection from $\mathbb{R}^n$ to a $p$-dimensional subspace $L$ and $x$ is an arbi... | How to understand degrees of freedom?
The concept is not at all difficult to make mathematical precise given a bit of general knowledge of $n$-dimensional Euclidean geometry, subspaces and orthogonal projections.
If $P$ is an orthogonal p |
10,135 | How to understand degrees of freedom? | It's really no different from the way the term "degrees of freedom" works in any other field. For example, suppose you have four variables: the length, the width, the area, and the perimeter of a rectangle. Do you really know four things? No, because there are only two degrees of freedom. If you know the length and the... | How to understand degrees of freedom? | It's really no different from the way the term "degrees of freedom" works in any other field. For example, suppose you have four variables: the length, the width, the area, and the perimeter of a rect | How to understand degrees of freedom?
It's really no different from the way the term "degrees of freedom" works in any other field. For example, suppose you have four variables: the length, the width, the area, and the perimeter of a rectangle. Do you really know four things? No, because there are only two degrees of f... | How to understand degrees of freedom?
It's really no different from the way the term "degrees of freedom" works in any other field. For example, suppose you have four variables: the length, the width, the area, and the perimeter of a rect |
10,136 | How to understand degrees of freedom? | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
I really like first sentence from
The Little Handbook... | How to understand degrees of freedom? | Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
| How to understand degrees of freedom?
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
I really like fi... | How to understand degrees of freedom?
Want to improve this post? Provide detailed answers to this question, including citations and an explanation of why your answer is correct. Answers without enough detail may be edited or deleted.
|
10,137 | How to understand degrees of freedom? | Wikipedia asserts that degrees of freedom of a random vector can be interpreted as the dimensions of the vector subspace. I want to go step-by-step, very basically through this as a partial answer and elaboration on the Wikipedia entry.
The example proposed is that of a random vector corresponding to the measurements o... | How to understand degrees of freedom? | Wikipedia asserts that degrees of freedom of a random vector can be interpreted as the dimensions of the vector subspace. I want to go step-by-step, very basically through this as a partial answer and | How to understand degrees of freedom?
Wikipedia asserts that degrees of freedom of a random vector can be interpreted as the dimensions of the vector subspace. I want to go step-by-step, very basically through this as a partial answer and elaboration on the Wikipedia entry.
The example proposed is that of a random vect... | How to understand degrees of freedom?
Wikipedia asserts that degrees of freedom of a random vector can be interpreted as the dimensions of the vector subspace. I want to go step-by-step, very basically through this as a partial answer and |
10,138 | How to understand degrees of freedom? | In my classes, I use one "simple" situation that might help you wonder and perhaps develop a gut feeling for what a degree of freedom may mean.
It is kind of a "Forrest Gump" approach to the subject, but it is worth the try.
Consider you have 10 independent observations $X_1, X_2, \ldots, X_{10}\sim N(\mu,\sigma^2)$ th... | How to understand degrees of freedom? | In my classes, I use one "simple" situation that might help you wonder and perhaps develop a gut feeling for what a degree of freedom may mean.
It is kind of a "Forrest Gump" approach to the subject, | How to understand degrees of freedom?
In my classes, I use one "simple" situation that might help you wonder and perhaps develop a gut feeling for what a degree of freedom may mean.
It is kind of a "Forrest Gump" approach to the subject, but it is worth the try.
Consider you have 10 independent observations $X_1, X_2, ... | How to understand degrees of freedom?
In my classes, I use one "simple" situation that might help you wonder and perhaps develop a gut feeling for what a degree of freedom may mean.
It is kind of a "Forrest Gump" approach to the subject, |
10,139 | How to understand degrees of freedom? | This particular issue is quite frustrating for students in statistics courses, since they often cannot get a straight answer on exactly what a degree-of-freedom is defined to be. I will try to clear that up here. Suppose we have a random vector $\mathbf{x} \in \mathbb{R}^n$ and we form a new random vector $\mathbf{t}... | How to understand degrees of freedom? | This particular issue is quite frustrating for students in statistics courses, since they often cannot get a straight answer on exactly what a degree-of-freedom is defined to be. I will try to clear | How to understand degrees of freedom?
This particular issue is quite frustrating for students in statistics courses, since they often cannot get a straight answer on exactly what a degree-of-freedom is defined to be. I will try to clear that up here. Suppose we have a random vector $\mathbf{x} \in \mathbb{R}^n$ and w... | How to understand degrees of freedom?
This particular issue is quite frustrating for students in statistics courses, since they often cannot get a straight answer on exactly what a degree-of-freedom is defined to be. I will try to clear |
10,140 | How to understand degrees of freedom? | You can see the degree of freedom as the number of observations minus the number of necessary relations among these observations. By exemple if you have $n$ sample of independant normal distribution observations $X_1,\dots,X_n$. The random variable $\sum_{i=1}^n (X_i-\overline{X}_n)^2\sim \mathcal{X}^2_{n-1}$, where $\... | How to understand degrees of freedom? | You can see the degree of freedom as the number of observations minus the number of necessary relations among these observations. By exemple if you have $n$ sample of independant normal distribution o | How to understand degrees of freedom?
You can see the degree of freedom as the number of observations minus the number of necessary relations among these observations. By exemple if you have $n$ sample of independant normal distribution observations $X_1,\dots,X_n$. The random variable $\sum_{i=1}^n (X_i-\overline{X}_n... | How to understand degrees of freedom?
You can see the degree of freedom as the number of observations minus the number of necessary relations among these observations. By exemple if you have $n$ sample of independant normal distribution o |
10,141 | How to understand degrees of freedom? | The clearest "formal" definition of degrees-of-freedom is that it is the dimension of the space of allowable values for a random vector. This generally arises in a context where we have a sample vector $\mathbf{x} \in \mathbb{R}^n$ and we form a new random vector $\mathbf{t} = T(\mathbf{x})$ via the linear function $T... | How to understand degrees of freedom? | The clearest "formal" definition of degrees-of-freedom is that it is the dimension of the space of allowable values for a random vector. This generally arises in a context where we have a sample vect | How to understand degrees of freedom?
The clearest "formal" definition of degrees-of-freedom is that it is the dimension of the space of allowable values for a random vector. This generally arises in a context where we have a sample vector $\mathbf{x} \in \mathbb{R}^n$ and we form a new random vector $\mathbf{t} = T(\... | How to understand degrees of freedom?
The clearest "formal" definition of degrees-of-freedom is that it is the dimension of the space of allowable values for a random vector. This generally arises in a context where we have a sample vect |
10,142 | How to understand degrees of freedom? | An intuitive explanation of degrees of freedom is that they represent
the number of independent pieces of information available in the data for estimating a parameter (i.e., unknown quantity) of interest.
As an example, in a simple linear regression model of the form:
$$
Y_i=\beta_0 + \beta_1\cdot X_i + \epsilon_i,\q... | How to understand degrees of freedom? | An intuitive explanation of degrees of freedom is that they represent
the number of independent pieces of information available in the data for estimating a parameter (i.e., unknown quantity) of inter | How to understand degrees of freedom?
An intuitive explanation of degrees of freedom is that they represent
the number of independent pieces of information available in the data for estimating a parameter (i.e., unknown quantity) of interest.
As an example, in a simple linear regression model of the form:
$$
Y_i=\bet... | How to understand degrees of freedom?
An intuitive explanation of degrees of freedom is that they represent
the number of independent pieces of information available in the data for estimating a parameter (i.e., unknown quantity) of inter |
10,143 | How to understand degrees of freedom? | For me the first explanation I understood was:
If you know some statistical value like mean or variation, how many
variables of data you need to know before you can know the value of
every variable?
This is the same as aL3xa said, but without giving any data point a special role and close to the third case given ... | How to understand degrees of freedom? | For me the first explanation I understood was:
If you know some statistical value like mean or variation, how many
variables of data you need to know before you can know the value of
every variab | How to understand degrees of freedom?
For me the first explanation I understood was:
If you know some statistical value like mean or variation, how many
variables of data you need to know before you can know the value of
every variable?
This is the same as aL3xa said, but without giving any data point a special r... | How to understand degrees of freedom?
For me the first explanation I understood was:
If you know some statistical value like mean or variation, how many
variables of data you need to know before you can know the value of
every variab |
10,144 | How to understand degrees of freedom? | Think of it this way. Variances are additive when independent. For example, suppose we are throwing darts at a board and we measure the standard deviations of the $x$ and $y$ displacements from the exact center of the board. Then $V_{x,y}=V_x+V_y$. But, $V_x=SD_x^2$ if we take the square root of the $V_{x,y}$ formula, ... | How to understand degrees of freedom? | Think of it this way. Variances are additive when independent. For example, suppose we are throwing darts at a board and we measure the standard deviations of the $x$ and $y$ displacements from the ex | How to understand degrees of freedom?
Think of it this way. Variances are additive when independent. For example, suppose we are throwing darts at a board and we measure the standard deviations of the $x$ and $y$ displacements from the exact center of the board. Then $V_{x,y}=V_x+V_y$. But, $V_x=SD_x^2$ if we take the ... | How to understand degrees of freedom?
Think of it this way. Variances are additive when independent. For example, suppose we are throwing darts at a board and we measure the standard deviations of the $x$ and $y$ displacements from the ex |
10,145 | Detecting outliers using standard deviations | Some outliers are clearly impossible. You mention 48 kg for baby weight. This is clearly an error. That's not a statistical issue, it's a substantive one. There are no 48 kg human babies. Any statistical method will identify such a point.
Personally, rather than rely on any test (even appropriate ones, as recommended ... | Detecting outliers using standard deviations | Some outliers are clearly impossible. You mention 48 kg for baby weight. This is clearly an error. That's not a statistical issue, it's a substantive one. There are no 48 kg human babies. Any statisti | Detecting outliers using standard deviations
Some outliers are clearly impossible. You mention 48 kg for baby weight. This is clearly an error. That's not a statistical issue, it's a substantive one. There are no 48 kg human babies. Any statistical method will identify such a point.
Personally, rather than rely on any... | Detecting outliers using standard deviations
Some outliers are clearly impossible. You mention 48 kg for baby weight. This is clearly an error. That's not a statistical issue, it's a substantive one. There are no 48 kg human babies. Any statisti |
10,146 | Detecting outliers using standard deviations | Yes. It is a bad way to "detect" oultiers. For normally distributed data, such a method would call 5% of the perfectly good (yet slightly extreme) observations "outliers". Also when you have a sample of size n and you look for extremely high or low observations to call them outliers, you are really looking at the extre... | Detecting outliers using standard deviations | Yes. It is a bad way to "detect" oultiers. For normally distributed data, such a method would call 5% of the perfectly good (yet slightly extreme) observations "outliers". Also when you have a sample | Detecting outliers using standard deviations
Yes. It is a bad way to "detect" oultiers. For normally distributed data, such a method would call 5% of the perfectly good (yet slightly extreme) observations "outliers". Also when you have a sample of size n and you look for extremely high or low observations to call them ... | Detecting outliers using standard deviations
Yes. It is a bad way to "detect" oultiers. For normally distributed data, such a method would call 5% of the perfectly good (yet slightly extreme) observations "outliers". Also when you have a sample |
10,147 | Detecting outliers using standard deviations | When you ask how many standard deviations from the mean a potential outlier is, don't forget that the outlier itself will raise the SD, and will also affect the value of the mean. If you have N values, the ratio of the distance from the mean divided by the SD can never exceed (N-1)/sqrt(N). This matters the most, of co... | Detecting outliers using standard deviations | When you ask how many standard deviations from the mean a potential outlier is, don't forget that the outlier itself will raise the SD, and will also affect the value of the mean. If you have N values | Detecting outliers using standard deviations
When you ask how many standard deviations from the mean a potential outlier is, don't forget that the outlier itself will raise the SD, and will also affect the value of the mean. If you have N values, the ratio of the distance from the mean divided by the SD can never excee... | Detecting outliers using standard deviations
When you ask how many standard deviations from the mean a potential outlier is, don't forget that the outlier itself will raise the SD, and will also affect the value of the mean. If you have N values |
10,148 | Detecting outliers using standard deviations | I think context is everything. For the example given, yes clearly a 48 kg baby is erroneous, and the use of 2 standard deviations would catch this case. However, there is no reason to think that the use of 2 standard deviations (or any other multiple of SD) is appropriate for other data. For example, if you are looking... | Detecting outliers using standard deviations | I think context is everything. For the example given, yes clearly a 48 kg baby is erroneous, and the use of 2 standard deviations would catch this case. However, there is no reason to think that the u | Detecting outliers using standard deviations
I think context is everything. For the example given, yes clearly a 48 kg baby is erroneous, and the use of 2 standard deviations would catch this case. However, there is no reason to think that the use of 2 standard deviations (or any other multiple of SD) is appropriate fo... | Detecting outliers using standard deviations
I think context is everything. For the example given, yes clearly a 48 kg baby is erroneous, and the use of 2 standard deviations would catch this case. However, there is no reason to think that the u |
10,149 | Can two random variables have the same distribution, yet be almost surely different? | Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.
But
$$
P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, .
$$
Hence, $X$ and $Y$ are different with probability one. | Can two random variables have the same distribution, yet be almost surely different? | Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.
But
$$
P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, | Can two random variables have the same distribution, yet be almost surely different?
Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.
But
$$
P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, .
$$
Hence, $X$ and $Y$ are differ... | Can two random variables have the same distribution, yet be almost surely different?
Let $X\sim N(0,1)$ and define $Y=-X$. It is easy to prove that $Y\sim N(0,1)$.
But
$$
P\{\omega : X(\omega)=Y(\omega)\} = P\{\omega : X(\omega)=0,Y(\omega)=0\} \leq P\{\omega : X(\omega)=0\} = 0 \, |
10,150 | Can two random variables have the same distribution, yet be almost surely different? | Any pair of independent random variables $X$ and $Y$ having the same continuous distribution provides a counterexample.
In fact, two random variables having the same distribution are not even necessarily defined on the same probability space, hence the question makes no sense in general. | Can two random variables have the same distribution, yet be almost surely different? | Any pair of independent random variables $X$ and $Y$ having the same continuous distribution provides a counterexample.
In fact, two random variables having the same distribution are not even necessar | Can two random variables have the same distribution, yet be almost surely different?
Any pair of independent random variables $X$ and $Y$ having the same continuous distribution provides a counterexample.
In fact, two random variables having the same distribution are not even necessarily defined on the same probability... | Can two random variables have the same distribution, yet be almost surely different?
Any pair of independent random variables $X$ and $Y$ having the same continuous distribution provides a counterexample.
In fact, two random variables having the same distribution are not even necessar |
10,151 | Can two random variables have the same distribution, yet be almost surely different? | Just consider $X(x)=x$ and $Y(x)=1-x$ with $x \in [0,1]$ with Borel or Lebesgue measure. For both the accumulated probability is $F(x)=x$ and the probability distibution is $f(x)=1$. For the sum $X+Y$ the distribution is a Dirac unit mass at $x=1$. | Can two random variables have the same distribution, yet be almost surely different? | Just consider $X(x)=x$ and $Y(x)=1-x$ with $x \in [0,1]$ with Borel or Lebesgue measure. For both the accumulated probability is $F(x)=x$ and the probability distibution is $f(x)=1$. For the sum $X+Y$ | Can two random variables have the same distribution, yet be almost surely different?
Just consider $X(x)=x$ and $Y(x)=1-x$ with $x \in [0,1]$ with Borel or Lebesgue measure. For both the accumulated probability is $F(x)=x$ and the probability distibution is $f(x)=1$. For the sum $X+Y$ the distribution is a Dirac unit m... | Can two random variables have the same distribution, yet be almost surely different?
Just consider $X(x)=x$ and $Y(x)=1-x$ with $x \in [0,1]$ with Borel or Lebesgue measure. For both the accumulated probability is $F(x)=x$ and the probability distibution is $f(x)=1$. For the sum $X+Y$ |
10,152 | "The total area underneath a probability density function is 1" - relative to what? | Probability density function is measured in percentages per unit of measure of your x-axis. Let's say at a given point $x_0$ your PDF is equal to 1000. This means that the probability of $x_0<x<x_0+dx$ is $1000\,dx$ where $dx$ is in meters. If you change the units to centimeters, then the probability should not change ... | "The total area underneath a probability density function is 1" - relative to what? | Probability density function is measured in percentages per unit of measure of your x-axis. Let's say at a given point $x_0$ your PDF is equal to 1000. This means that the probability of $x_0<x<x_0+dx | "The total area underneath a probability density function is 1" - relative to what?
Probability density function is measured in percentages per unit of measure of your x-axis. Let's say at a given point $x_0$ your PDF is equal to 1000. This means that the probability of $x_0<x<x_0+dx$ is $1000\,dx$ where $dx$ is in met... | "The total area underneath a probability density function is 1" - relative to what?
Probability density function is measured in percentages per unit of measure of your x-axis. Let's say at a given point $x_0$ your PDF is equal to 1000. This means that the probability of $x_0<x<x_0+dx |
10,153 | "The total area underneath a probability density function is 1" - relative to what? | It might help you to realise that the vertical axis is measured as a probability density. So if the horizontal axis is measured in km, then the vertical axis is measured as a probability density "per km". Suppose we draw a rectangular element on such a grid, which is 5 "km" wide and 0.1 "per km" high (which you might p... | "The total area underneath a probability density function is 1" - relative to what? | It might help you to realise that the vertical axis is measured as a probability density. So if the horizontal axis is measured in km, then the vertical axis is measured as a probability density "per | "The total area underneath a probability density function is 1" - relative to what?
It might help you to realise that the vertical axis is measured as a probability density. So if the horizontal axis is measured in km, then the vertical axis is measured as a probability density "per km". Suppose we draw a rectangular e... | "The total area underneath a probability density function is 1" - relative to what?
It might help you to realise that the vertical axis is measured as a probability density. So if the horizontal axis is measured in km, then the vertical axis is measured as a probability density "per |
10,154 | "The total area underneath a probability density function is 1" - relative to what? | You already got two answers, with an excellent one by Silverfish, however I feel that an illustration could be useful in here since you asked about geometry and "imagining" yourself those functions.
Lets start with a simple example of Bernoulli distribution:
$$
f(x) = \begin{cases}
p & \text{if }x=1, \\[6pt]
1-p & \tex... | "The total area underneath a probability density function is 1" - relative to what? | You already got two answers, with an excellent one by Silverfish, however I feel that an illustration could be useful in here since you asked about geometry and "imagining" yourself those functions.
L | "The total area underneath a probability density function is 1" - relative to what?
You already got two answers, with an excellent one by Silverfish, however I feel that an illustration could be useful in here since you asked about geometry and "imagining" yourself those functions.
Lets start with a simple example of B... | "The total area underneath a probability density function is 1" - relative to what?
You already got two answers, with an excellent one by Silverfish, however I feel that an illustration could be useful in here since you asked about geometry and "imagining" yourself those functions.
L |
10,155 | "The total area underneath a probability density function is 1" - relative to what? | The following key idea was mentioned in a comment, but not in an existing answer...
One way of intuiting about the properties of a PDF is to consider that the PDF and the CDF are related by integration (calculus) -- and that the CDF has a monotonic output representing a probability value between 0 and 1.
The unitless i... | "The total area underneath a probability density function is 1" - relative to what? | The following key idea was mentioned in a comment, but not in an existing answer...
One way of intuiting about the properties of a PDF is to consider that the PDF and the CDF are related by integratio | "The total area underneath a probability density function is 1" - relative to what?
The following key idea was mentioned in a comment, but not in an existing answer...
One way of intuiting about the properties of a PDF is to consider that the PDF and the CDF are related by integration (calculus) -- and that the CDF has... | "The total area underneath a probability density function is 1" - relative to what?
The following key idea was mentioned in a comment, but not in an existing answer...
One way of intuiting about the properties of a PDF is to consider that the PDF and the CDF are related by integratio |
10,156 | Amoeba Interview Question | Cute problem. This is the kind of stuff that probabilists do in their heads for fun.
The technique is to assume that there is such a probability of extinction, call it $P$. Then, looking at a one-deep decision tree for the possible outcomes we see--using the Law of Total Probability--that
$P=\frac{1}{4} + \frac{1}{4}P... | Amoeba Interview Question | Cute problem. This is the kind of stuff that probabilists do in their heads for fun.
The technique is to assume that there is such a probability of extinction, call it $P$. Then, looking at a one-dee | Amoeba Interview Question
Cute problem. This is the kind of stuff that probabilists do in their heads for fun.
The technique is to assume that there is such a probability of extinction, call it $P$. Then, looking at a one-deep decision tree for the possible outcomes we see--using the Law of Total Probability--that
$P=... | Amoeba Interview Question
Cute problem. This is the kind of stuff that probabilists do in their heads for fun.
The technique is to assume that there is such a probability of extinction, call it $P$. Then, looking at a one-dee |
10,157 | Amoeba Interview Question | Some back of the envelope calculation (litterally - I had an envelope lying around on my desk) gives me a probability of 42/111 (38%) of never reaching a population of 3.
I ran a quick Python simulation, seeing how many populations had died off by 20 generations (at which point they usually either died out or are in th... | Amoeba Interview Question | Some back of the envelope calculation (litterally - I had an envelope lying around on my desk) gives me a probability of 42/111 (38%) of never reaching a population of 3.
I ran a quick Python simulati | Amoeba Interview Question
Some back of the envelope calculation (litterally - I had an envelope lying around on my desk) gives me a probability of 42/111 (38%) of never reaching a population of 3.
I ran a quick Python simulation, seeing how many populations had died off by 20 generations (at which point they usually ei... | Amoeba Interview Question
Some back of the envelope calculation (litterally - I had an envelope lying around on my desk) gives me a probability of 42/111 (38%) of never reaching a population of 3.
I ran a quick Python simulati |
10,158 | Amoeba Interview Question | Like the answer from Mike Anderson says you can equate the probability for a lineage of an amoeba to become extinct to a sum of probabilities of the lineage of the children to become extinct.
$$p_{parent} = \frac{1}{4} p_{child}^3 + \frac{1}{4} p_{child}^2 + \frac{1}{4} p_{child} + \frac{1}{4}$$
Then when you set equal... | Amoeba Interview Question | Like the answer from Mike Anderson says you can equate the probability for a lineage of an amoeba to become extinct to a sum of probabilities of the lineage of the children to become extinct.
$$p_{par | Amoeba Interview Question
Like the answer from Mike Anderson says you can equate the probability for a lineage of an amoeba to become extinct to a sum of probabilities of the lineage of the children to become extinct.
$$p_{parent} = \frac{1}{4} p_{child}^3 + \frac{1}{4} p_{child}^2 + \frac{1}{4} p_{child} + \frac{1}{4}... | Amoeba Interview Question
Like the answer from Mike Anderson says you can equate the probability for a lineage of an amoeba to become extinct to a sum of probabilities of the lineage of the children to become extinct.
$$p_{par |
10,159 | Amoeba Interview Question | This sounds related to the Galton Watson process, originally formulated to study the survival of surnames. The probability depends on the expected number of sub-amoebas after a single division. In this case that expected number is $3/2,$ which is greater than the critical value of $1$, and thus the probability of extin... | Amoeba Interview Question | This sounds related to the Galton Watson process, originally formulated to study the survival of surnames. The probability depends on the expected number of sub-amoebas after a single division. In thi | Amoeba Interview Question
This sounds related to the Galton Watson process, originally formulated to study the survival of surnames. The probability depends on the expected number of sub-amoebas after a single division. In this case that expected number is $3/2,$ which is greater than the critical value of $1$, and thu... | Amoeba Interview Question
This sounds related to the Galton Watson process, originally formulated to study the survival of surnames. The probability depends on the expected number of sub-amoebas after a single division. In thi |
10,160 | Significance of categorical predictor in logistic regression | The following explanation is not limited to logistic regression but applies equally in normal linear regression and other GLMs. Usually, R excludes one level of the categorical and the coefficients denote the difference of each class to this reference class (or sometimes called baseline class) (this is called dummy cod... | Significance of categorical predictor in logistic regression | The following explanation is not limited to logistic regression but applies equally in normal linear regression and other GLMs. Usually, R excludes one level of the categorical and the coefficients de | Significance of categorical predictor in logistic regression
The following explanation is not limited to logistic regression but applies equally in normal linear regression and other GLMs. Usually, R excludes one level of the categorical and the coefficients denote the difference of each class to this reference class (... | Significance of categorical predictor in logistic regression
The following explanation is not limited to logistic regression but applies equally in normal linear regression and other GLMs. Usually, R excludes one level of the categorical and the coefficients de |
10,161 | Significance of categorical predictor in logistic regression | The $z$-value is just the test-statistic for a statistical test, so if you have trouble interpreting it your first step is to find out what the null hypothesis is. The null hypothesis for the test for CLASS0 is that its coefficient is 0. The coefficient for CLASS0 is the difference in log(odds) between CLASS0 and the r... | Significance of categorical predictor in logistic regression | The $z$-value is just the test-statistic for a statistical test, so if you have trouble interpreting it your first step is to find out what the null hypothesis is. The null hypothesis for the test for | Significance of categorical predictor in logistic regression
The $z$-value is just the test-statistic for a statistical test, so if you have trouble interpreting it your first step is to find out what the null hypothesis is. The null hypothesis for the test for CLASS0 is that its coefficient is 0. The coefficient for C... | Significance of categorical predictor in logistic regression
The $z$-value is just the test-statistic for a statistical test, so if you have trouble interpreting it your first step is to find out what the null hypothesis is. The null hypothesis for the test for |
10,162 | Why not just dump the neural networks and deep learning? [closed] | Not being able to know what solution generalizes best is an issue, but it shouldn't deter us from otherwise using a good solution. Humans themselves often do not known what generalizes best (consider, for example, competing unifying theories of physics), but that doesn't cause us too many problems.
It has been shown th... | Why not just dump the neural networks and deep learning? [closed] | Not being able to know what solution generalizes best is an issue, but it shouldn't deter us from otherwise using a good solution. Humans themselves often do not known what generalizes best (consider, | Why not just dump the neural networks and deep learning? [closed]
Not being able to know what solution generalizes best is an issue, but it shouldn't deter us from otherwise using a good solution. Humans themselves often do not known what generalizes best (consider, for example, competing unifying theories of physics),... | Why not just dump the neural networks and deep learning? [closed]
Not being able to know what solution generalizes best is an issue, but it shouldn't deter us from otherwise using a good solution. Humans themselves often do not known what generalizes best (consider, |
10,163 | Why not just dump the neural networks and deep learning? [closed] | As the comments to your question point out, there are a lot of people working on finding something better. I would though like to answer this question by expanding the comment left by @josh
All models are wrong but some are useful (Wiki)
The above statement is a general truth used to describe the nature of statistical... | Why not just dump the neural networks and deep learning? [closed] | As the comments to your question point out, there are a lot of people working on finding something better. I would though like to answer this question by expanding the comment left by @josh
All model | Why not just dump the neural networks and deep learning? [closed]
As the comments to your question point out, there are a lot of people working on finding something better. I would though like to answer this question by expanding the comment left by @josh
All models are wrong but some are useful (Wiki)
The above state... | Why not just dump the neural networks and deep learning? [closed]
As the comments to your question point out, there are a lot of people working on finding something better. I would though like to answer this question by expanding the comment left by @josh
All model |
10,164 | Why not just dump the neural networks and deep learning? [closed] | The global minimum may as well as be useless, so we don't really care if we find it or not. The reason is that, for deep networks, not only the time to find it becomes exponentially longer as the network size increases, but also the global minimum often corresponds to overfitting the training set. Thus the generalizati... | Why not just dump the neural networks and deep learning? [closed] | The global minimum may as well as be useless, so we don't really care if we find it or not. The reason is that, for deep networks, not only the time to find it becomes exponentially longer as the netw | Why not just dump the neural networks and deep learning? [closed]
The global minimum may as well as be useless, so we don't really care if we find it or not. The reason is that, for deep networks, not only the time to find it becomes exponentially longer as the network size increases, but also the global minimum often ... | Why not just dump the neural networks and deep learning? [closed]
The global minimum may as well as be useless, so we don't really care if we find it or not. The reason is that, for deep networks, not only the time to find it becomes exponentially longer as the netw |
10,165 | Why not just dump the neural networks and deep learning? [closed] | I think the best way to think about this question is through the competitive market place. If you dump deep learning, and your competitors use it, AND it happens to work better than what you used, then you'll be beaten on the market place.
I think that's what's happening, in part, today, i.e. deep learning seems to wo... | Why not just dump the neural networks and deep learning? [closed] | I think the best way to think about this question is through the competitive market place. If you dump deep learning, and your competitors use it, AND it happens to work better than what you used, the | Why not just dump the neural networks and deep learning? [closed]
I think the best way to think about this question is through the competitive market place. If you dump deep learning, and your competitors use it, AND it happens to work better than what you used, then you'll be beaten on the market place.
I think that'... | Why not just dump the neural networks and deep learning? [closed]
I think the best way to think about this question is through the competitive market place. If you dump deep learning, and your competitors use it, AND it happens to work better than what you used, the |
10,166 | Why not just dump the neural networks and deep learning? [closed] | There are excellent answers, mostly weighing in with the usefulness of DL and ANNs. But I would like to object the OP in a more fundamental way, since the question already takes for granted the mathematical inconsistency of neural networks.
First of all, there is a mathematical theory behind (most models of) Neural Net... | Why not just dump the neural networks and deep learning? [closed] | There are excellent answers, mostly weighing in with the usefulness of DL and ANNs. But I would like to object the OP in a more fundamental way, since the question already takes for granted the mathem | Why not just dump the neural networks and deep learning? [closed]
There are excellent answers, mostly weighing in with the usefulness of DL and ANNs. But I would like to object the OP in a more fundamental way, since the question already takes for granted the mathematical inconsistency of neural networks.
First of all,... | Why not just dump the neural networks and deep learning? [closed]
There are excellent answers, mostly weighing in with the usefulness of DL and ANNs. But I would like to object the OP in a more fundamental way, since the question already takes for granted the mathem |
10,167 | Why not just dump the neural networks and deep learning? [closed] | I guess for some problem we care less for the mathematical rigor and simplicity but more for its utility, current status is neural network is better in performing certain task like pattern recognition in image processing. | Why not just dump the neural networks and deep learning? [closed] | I guess for some problem we care less for the mathematical rigor and simplicity but more for its utility, current status is neural network is better in performing certain task like pattern recognition | Why not just dump the neural networks and deep learning? [closed]
I guess for some problem we care less for the mathematical rigor and simplicity but more for its utility, current status is neural network is better in performing certain task like pattern recognition in image processing. | Why not just dump the neural networks and deep learning? [closed]
I guess for some problem we care less for the mathematical rigor and simplicity but more for its utility, current status is neural network is better in performing certain task like pattern recognition |
10,168 | Why not just dump the neural networks and deep learning? [closed] | There is a lot in this question. Lets go over what you've wrote one by one.
The solutions that fit training data are infinite. We don't have precise mathematical equation that is satisfied by only a single one and that we can say generalizes best.
The fact that there are infinite many solutions comes from learning p... | Why not just dump the neural networks and deep learning? [closed] | There is a lot in this question. Lets go over what you've wrote one by one.
The solutions that fit training data are infinite. We don't have precise mathematical equation that is satisfied by only a | Why not just dump the neural networks and deep learning? [closed]
There is a lot in this question. Lets go over what you've wrote one by one.
The solutions that fit training data are infinite. We don't have precise mathematical equation that is satisfied by only a single one and that we can say generalizes best.
The... | Why not just dump the neural networks and deep learning? [closed]
There is a lot in this question. Lets go over what you've wrote one by one.
The solutions that fit training data are infinite. We don't have precise mathematical equation that is satisfied by only a |
10,169 | Why not just dump the neural networks and deep learning? [closed] | How about viewing neural networks from an experimental point of view? Just because we created them doesn't mean that we're obligued to understand them intuitively. Or that we're not allowed to play with them in order to have a better grasp of what they're doing.
Here's a couple of thoughts I have on them:
Structure: t... | Why not just dump the neural networks and deep learning? [closed] | How about viewing neural networks from an experimental point of view? Just because we created them doesn't mean that we're obligued to understand them intuitively. Or that we're not allowed to play wi | Why not just dump the neural networks and deep learning? [closed]
How about viewing neural networks from an experimental point of view? Just because we created them doesn't mean that we're obligued to understand them intuitively. Or that we're not allowed to play with them in order to have a better grasp of what they'r... | Why not just dump the neural networks and deep learning? [closed]
How about viewing neural networks from an experimental point of view? Just because we created them doesn't mean that we're obligued to understand them intuitively. Or that we're not allowed to play wi |
10,170 | Why not just dump the neural networks and deep learning? [closed] | Don't forget, there is a vast field of research that use LMs, GLM, multilevel modelling. Lately Bayesian techniques and Hamiltonian Monte Carlo(the STAN community is really at the forefront of this) have come of age and a number of problems that are solved by STAN really easily and don't really need NNs or deep nets. S... | Why not just dump the neural networks and deep learning? [closed] | Don't forget, there is a vast field of research that use LMs, GLM, multilevel modelling. Lately Bayesian techniques and Hamiltonian Monte Carlo(the STAN community is really at the forefront of this) h | Why not just dump the neural networks and deep learning? [closed]
Don't forget, there is a vast field of research that use LMs, GLM, multilevel modelling. Lately Bayesian techniques and Hamiltonian Monte Carlo(the STAN community is really at the forefront of this) have come of age and a number of problems that are solv... | Why not just dump the neural networks and deep learning? [closed]
Don't forget, there is a vast field of research that use LMs, GLM, multilevel modelling. Lately Bayesian techniques and Hamiltonian Monte Carlo(the STAN community is really at the forefront of this) h |
10,171 | Why not just dump the neural networks and deep learning? [closed] | What typically happens when there is no mathematical consistency (atleast in this case of neural networks)...when its not giving results as desired, on the test set, your boss will come back and say...Hey why don't you try Drop out (which weights,which layer, how many is your headache as there isn't mathematical way to... | Why not just dump the neural networks and deep learning? [closed] | What typically happens when there is no mathematical consistency (atleast in this case of neural networks)...when its not giving results as desired, on the test set, your boss will come back and say.. | Why not just dump the neural networks and deep learning? [closed]
What typically happens when there is no mathematical consistency (atleast in this case of neural networks)...when its not giving results as desired, on the test set, your boss will come back and say...Hey why don't you try Drop out (which weights,which l... | Why not just dump the neural networks and deep learning? [closed]
What typically happens when there is no mathematical consistency (atleast in this case of neural networks)...when its not giving results as desired, on the test set, your boss will come back and say.. |
10,172 | Why do residuals in linear regression always sum to zero when an intercept is included? | This follows directly from the normal equations, i.e. the equations that the OLS estimator solves,
$$\mathbf{X}^{\prime} \underbrace{\left( \mathbf{y} - \mathbf{X} \mathbf{b} \right)}_{\mathbf{e}} = 0 $$
The vector inside the parentheses is of course the residual vector or the projection of $\mathbf{y}$ onto the orthog... | Why do residuals in linear regression always sum to zero when an intercept is included? | This follows directly from the normal equations, i.e. the equations that the OLS estimator solves,
$$\mathbf{X}^{\prime} \underbrace{\left( \mathbf{y} - \mathbf{X} \mathbf{b} \right)}_{\mathbf{e}} = 0 | Why do residuals in linear regression always sum to zero when an intercept is included?
This follows directly from the normal equations, i.e. the equations that the OLS estimator solves,
$$\mathbf{X}^{\prime} \underbrace{\left( \mathbf{y} - \mathbf{X} \mathbf{b} \right)}_{\mathbf{e}} = 0 $$
The vector inside the parent... | Why do residuals in linear regression always sum to zero when an intercept is included?
This follows directly from the normal equations, i.e. the equations that the OLS estimator solves,
$$\mathbf{X}^{\prime} \underbrace{\left( \mathbf{y} - \mathbf{X} \mathbf{b} \right)}_{\mathbf{e}} = 0 |
10,173 | Why do residuals in linear regression always sum to zero when an intercept is included? | In case you are looking for a rather intuitive explanation.
In some sense, the linear regression model is nothing but a fancy mean. To find the arithmetic mean $\bar{x}$ over some values $x_1, x_2, \dots, x_n$, we find a value that is a measure of centrality in a sense that the sum of all deviations (where each deviati... | Why do residuals in linear regression always sum to zero when an intercept is included? | In case you are looking for a rather intuitive explanation.
In some sense, the linear regression model is nothing but a fancy mean. To find the arithmetic mean $\bar{x}$ over some values $x_1, x_2, \d | Why do residuals in linear regression always sum to zero when an intercept is included?
In case you are looking for a rather intuitive explanation.
In some sense, the linear regression model is nothing but a fancy mean. To find the arithmetic mean $\bar{x}$ over some values $x_1, x_2, \dots, x_n$, we find a value that ... | Why do residuals in linear regression always sum to zero when an intercept is included?
In case you are looking for a rather intuitive explanation.
In some sense, the linear regression model is nothing but a fancy mean. To find the arithmetic mean $\bar{x}$ over some values $x_1, x_2, \d |
10,174 | Why do residuals in linear regression always sum to zero when an intercept is included? | When an intercept is included in multiple linear regression,
$$
\hat{y}_i = \beta_0 + \beta_1x_{i,1} + \beta_2x_{i,2} +…+ \beta_px_{i,p}
$$
In Least squares regression, the sum of the squares of the errors is minimized.
$$
SSE=\displaystyle\sum\limits_{i=1}^n \left(e_i \right)^2= \sum_{i=1}^n\left(y_i - \hat{y_i} \righ... | Why do residuals in linear regression always sum to zero when an intercept is included? | When an intercept is included in multiple linear regression,
$$
\hat{y}_i = \beta_0 + \beta_1x_{i,1} + \beta_2x_{i,2} +…+ \beta_px_{i,p}
$$
In Least squares regression, the sum of the squares of the e | Why do residuals in linear regression always sum to zero when an intercept is included?
When an intercept is included in multiple linear regression,
$$
\hat{y}_i = \beta_0 + \beta_1x_{i,1} + \beta_2x_{i,2} +…+ \beta_px_{i,p}
$$
In Least squares regression, the sum of the squares of the errors is minimized.
$$
SSE=\disp... | Why do residuals in linear regression always sum to zero when an intercept is included?
When an intercept is included in multiple linear regression,
$$
\hat{y}_i = \beta_0 + \beta_1x_{i,1} + \beta_2x_{i,2} +…+ \beta_px_{i,p}
$$
In Least squares regression, the sum of the squares of the e |
10,175 | Why do residuals in linear regression always sum to zero when an intercept is included? | Write the linear model in matrical form as
\begin{align}
y = X\beta + \varepsilon,
\end{align}
where $y \in \mathbb{R}^n$ is the response vector, $X \in \mathbb{R}^{n \times p}$ is the design matrix, $\varepsilon \in \mathbb{R}^n$ is the error vector.
A key observation is that because the model has intercept, $1$, wh... | Why do residuals in linear regression always sum to zero when an intercept is included? | Write the linear model in matrical form as
\begin{align}
y = X\beta + \varepsilon,
\end{align}
where $y \in \mathbb{R}^n$ is the response vector, $X \in \mathbb{R}^{n \times p}$ is the design matrix | Why do residuals in linear regression always sum to zero when an intercept is included?
Write the linear model in matrical form as
\begin{align}
y = X\beta + \varepsilon,
\end{align}
where $y \in \mathbb{R}^n$ is the response vector, $X \in \mathbb{R}^{n \times p}$ is the design matrix, $\varepsilon \in \mathbb{R}^n$... | Why do residuals in linear regression always sum to zero when an intercept is included?
Write the linear model in matrical form as
\begin{align}
y = X\beta + \varepsilon,
\end{align}
where $y \in \mathbb{R}^n$ is the response vector, $X \in \mathbb{R}^{n \times p}$ is the design matrix |
10,176 | Why do residuals in linear regression always sum to zero when an intercept is included? | A simple derivation using matrix algebra:
$\sum e $ can be written as $1^Te$
Then
$1^Te = 1^T(M_x y)$ where $M_x$ is the orthogonal matrix. Since $M_x$ is symmetric we can rearrange so that
$(M_x1)^Ty$
which equals zero if $M_x$ and $1$ are orthogonal, which is the case if the matrix of the regressors $x$ contains the... | Why do residuals in linear regression always sum to zero when an intercept is included? | A simple derivation using matrix algebra:
$\sum e $ can be written as $1^Te$
Then
$1^Te = 1^T(M_x y)$ where $M_x$ is the orthogonal matrix. Since $M_x$ is symmetric we can rearrange so that
$(M_x1)^T | Why do residuals in linear regression always sum to zero when an intercept is included?
A simple derivation using matrix algebra:
$\sum e $ can be written as $1^Te$
Then
$1^Te = 1^T(M_x y)$ where $M_x$ is the orthogonal matrix. Since $M_x$ is symmetric we can rearrange so that
$(M_x1)^Ty$
which equals zero if $M_x$ an... | Why do residuals in linear regression always sum to zero when an intercept is included?
A simple derivation using matrix algebra:
$\sum e $ can be written as $1^Te$
Then
$1^Te = 1^T(M_x y)$ where $M_x$ is the orthogonal matrix. Since $M_x$ is symmetric we can rearrange so that
$(M_x1)^T |
10,177 | Why do residuals in linear regression always sum to zero when an intercept is included? | $e_i = y_i - [1, X] [a, b] = y_i - Xb - a = v_i - a$
$\frac{d}{da} \sum e_i^2 \propto \sum e_i\cdot 1 = \sum v_i - a = 0$ so $\hat{a} = \frac{1}{n}\sum v_i$
$\sum e_i = \sum_i v_i - a = \sum_i v_i - \frac{n}{n}\sum_i v_i = 0$
.. | Why do residuals in linear regression always sum to zero when an intercept is included? | $e_i = y_i - [1, X] [a, b] = y_i - Xb - a = v_i - a$
$\frac{d}{da} \sum e_i^2 \propto \sum e_i\cdot 1 = \sum v_i - a = 0$ so $\hat{a} = \frac{1}{n}\sum v_i$
$\sum e_i = \sum_i v_i - a = \sum_i v_i - \ | Why do residuals in linear regression always sum to zero when an intercept is included?
$e_i = y_i - [1, X] [a, b] = y_i - Xb - a = v_i - a$
$\frac{d}{da} \sum e_i^2 \propto \sum e_i\cdot 1 = \sum v_i - a = 0$ so $\hat{a} = \frac{1}{n}\sum v_i$
$\sum e_i = \sum_i v_i - a = \sum_i v_i - \frac{n}{n}\sum_i v_i = 0$
.. | Why do residuals in linear regression always sum to zero when an intercept is included?
$e_i = y_i - [1, X] [a, b] = y_i - Xb - a = v_i - a$
$\frac{d}{da} \sum e_i^2 \propto \sum e_i\cdot 1 = \sum v_i - a = 0$ so $\hat{a} = \frac{1}{n}\sum v_i$
$\sum e_i = \sum_i v_i - a = \sum_i v_i - \ |
10,178 | Is a model fitted to data or is data fitted to a model? | Pretty much every source or person I've ever interacted with except the Wolfram source you linked refers to the process as fitting a model to data. This makes sense, since the model is the dynamic object and the data is static (a.k.a. fixed and constant).
To put a point on it, I like Larry Wasserman's approach to this... | Is a model fitted to data or is data fitted to a model? | Pretty much every source or person I've ever interacted with except the Wolfram source you linked refers to the process as fitting a model to data. This makes sense, since the model is the dynamic ob | Is a model fitted to data or is data fitted to a model?
Pretty much every source or person I've ever interacted with except the Wolfram source you linked refers to the process as fitting a model to data. This makes sense, since the model is the dynamic object and the data is static (a.k.a. fixed and constant).
To put ... | Is a model fitted to data or is data fitted to a model?
Pretty much every source or person I've ever interacted with except the Wolfram source you linked refers to the process as fitting a model to data. This makes sense, since the model is the dynamic ob |
10,179 | Is a model fitted to data or is data fitted to a model? | In the field of Rasch modelling it is common to fit the data to the model. The model is assumed to be correct and it is the analyst's job to find data which conform to it. The Wikipedia article on Rasch contains more details about the how and the why.
But I agree with others that in general in statistics we fit the mod... | Is a model fitted to data or is data fitted to a model? | In the field of Rasch modelling it is common to fit the data to the model. The model is assumed to be correct and it is the analyst's job to find data which conform to it. The Wikipedia article on Ras | Is a model fitted to data or is data fitted to a model?
In the field of Rasch modelling it is common to fit the data to the model. The model is assumed to be correct and it is the analyst's job to find data which conform to it. The Wikipedia article on Rasch contains more details about the how and the why.
But I agree ... | Is a model fitted to data or is data fitted to a model?
In the field of Rasch modelling it is common to fit the data to the model. The model is assumed to be correct and it is the analyst's job to find data which conform to it. The Wikipedia article on Ras |
10,180 | Is a model fitted to data or is data fitted to a model? | Typically, the observed data are fixed while the model is mutable (e.g. because parameters are estimated), so it is the model that is made to fit the data, not the other way around. (Usually people mean this case when they say either expression.)
When people say they fit data to a model I find myself trying to figure o... | Is a model fitted to data or is data fitted to a model? | Typically, the observed data are fixed while the model is mutable (e.g. because parameters are estimated), so it is the model that is made to fit the data, not the other way around. (Usually people me | Is a model fitted to data or is data fitted to a model?
Typically, the observed data are fixed while the model is mutable (e.g. because parameters are estimated), so it is the model that is made to fit the data, not the other way around. (Usually people mean this case when they say either expression.)
When people say t... | Is a model fitted to data or is data fitted to a model?
Typically, the observed data are fixed while the model is mutable (e.g. because parameters are estimated), so it is the model that is made to fit the data, not the other way around. (Usually people me |
10,181 | Is a model fitted to data or is data fitted to a model? | Usually, we assume our data corresponds to the "real world" and making any modifications means we are moving away from modelling the "real world". For example, one needs to take care removing outliers since even if it makes computation nicer, outliers were still part of our data.
When testing a model or estimating pro... | Is a model fitted to data or is data fitted to a model? | Usually, we assume our data corresponds to the "real world" and making any modifications means we are moving away from modelling the "real world". For example, one needs to take care removing outliers | Is a model fitted to data or is data fitted to a model?
Usually, we assume our data corresponds to the "real world" and making any modifications means we are moving away from modelling the "real world". For example, one needs to take care removing outliers since even if it makes computation nicer, outliers were still p... | Is a model fitted to data or is data fitted to a model?
Usually, we assume our data corresponds to the "real world" and making any modifications means we are moving away from modelling the "real world". For example, one needs to take care removing outliers |
10,182 | Is a model fitted to data or is data fitted to a model? | I think 'fitting the model' sounds wrong to some because the model structure does not change as part of the fitting process. But the model parameters do, which is why it is correct.
Fitting is still an awkward term and it vaguely sounds like some underhand manipulation is being done. I much prefer to just talk about pa... | Is a model fitted to data or is data fitted to a model? | I think 'fitting the model' sounds wrong to some because the model structure does not change as part of the fitting process. But the model parameters do, which is why it is correct.
Fitting is still a | Is a model fitted to data or is data fitted to a model?
I think 'fitting the model' sounds wrong to some because the model structure does not change as part of the fitting process. But the model parameters do, which is why it is correct.
Fitting is still an awkward term and it vaguely sounds like some underhand manipul... | Is a model fitted to data or is data fitted to a model?
I think 'fitting the model' sounds wrong to some because the model structure does not change as part of the fitting process. But the model parameters do, which is why it is correct.
Fitting is still a |
10,183 | Is a model fitted to data or is data fitted to a model? | This could also be seen from Bayesian perspective and goodness of fit. We might interpret "model fitted to data" as in finding out probability of parameters fits the given data well i.e, $p(\theta|X)$ a posterior and "data fitted to a model" as in finding out probability of observing the data given the model i.e., $p... | Is a model fitted to data or is data fitted to a model? | This could also be seen from Bayesian perspective and goodness of fit. We might interpret "model fitted to data" as in finding out probability of parameters fits the given data well i.e, $p(\theta|X)$ | Is a model fitted to data or is data fitted to a model?
This could also be seen from Bayesian perspective and goodness of fit. We might interpret "model fitted to data" as in finding out probability of parameters fits the given data well i.e, $p(\theta|X)$ a posterior and "data fitted to a model" as in finding out pro... | Is a model fitted to data or is data fitted to a model?
This could also be seen from Bayesian perspective and goodness of fit. We might interpret "model fitted to data" as in finding out probability of parameters fits the given data well i.e, $p(\theta|X)$ |
10,184 | Is a model fitted to data or is data fitted to a model? | Interesting topic here and I could not let it pass me by without saying a few words. Here is my take on this. From a scientific point of view, a model by definition is a conceptual representation (universally accepted as a reference) of an event or process. With that in mind, it is my understanding that any data out th... | Is a model fitted to data or is data fitted to a model? | Interesting topic here and I could not let it pass me by without saying a few words. Here is my take on this. From a scientific point of view, a model by definition is a conceptual representation (uni | Is a model fitted to data or is data fitted to a model?
Interesting topic here and I could not let it pass me by without saying a few words. Here is my take on this. From a scientific point of view, a model by definition is a conceptual representation (universally accepted as a reference) of an event or process. With t... | Is a model fitted to data or is data fitted to a model?
Interesting topic here and I could not let it pass me by without saying a few words. Here is my take on this. From a scientific point of view, a model by definition is a conceptual representation (uni |
10,185 | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | Unbiasedness isn't necessarily especially important on its own.
Aside a very limited set of circumstances, most useful estimators are biased, however they're obtained.
If two estimators have the same variance, one can readily mount an argument for preferring an unbiased one to a biased one, but that's an unusual situat... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | Unbiasedness isn't necessarily especially important on its own.
Aside a very limited set of circumstances, most useful estimators are biased, however they're obtained.
If two estimators have the same | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
Unbiasedness isn't necessarily especially important on its own.
Aside a very limited set of circumstances, most useful estimators are biased, however they're obtained.
If two estimators have the same variance, one can readily mount an ar... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
Unbiasedness isn't necessarily especially important on its own.
Aside a very limited set of circumstances, most useful estimators are biased, however they're obtained.
If two estimators have the same |
10,186 | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | Maximum likelihood estimation (MLE) yields the most likely value of the model parameters, given the model and the data at hand -- which is a pretty attractive concept. Why would you choose parameter values that make the data observed less probable when you can choose the values that make the data observed the most prob... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | Maximum likelihood estimation (MLE) yields the most likely value of the model parameters, given the model and the data at hand -- which is a pretty attractive concept. Why would you choose parameter v | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
Maximum likelihood estimation (MLE) yields the most likely value of the model parameters, given the model and the data at hand -- which is a pretty attractive concept. Why would you choose parameter values that make the data observed les... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
Maximum likelihood estimation (MLE) yields the most likely value of the model parameters, given the model and the data at hand -- which is a pretty attractive concept. Why would you choose parameter v |
10,187 | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | To answer your question of why the MLE is so popular, consider that although it can be biased, it is consistent under standard conditions. In addition, it is asymptotically efficient, so at least for large samples, the MLE is likely to do as well or better as any other estimator you may cook up. Finally, the MLE is fou... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | To answer your question of why the MLE is so popular, consider that although it can be biased, it is consistent under standard conditions. In addition, it is asymptotically efficient, so at least for | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
To answer your question of why the MLE is so popular, consider that although it can be biased, it is consistent under standard conditions. In addition, it is asymptotically efficient, so at least for large samples, the MLE is likely to d... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
To answer your question of why the MLE is so popular, consider that although it can be biased, it is consistent under standard conditions. In addition, it is asymptotically efficient, so at least for |
10,188 | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | Actually, the scaling of the maximum likelihood estimates in order to obtain unbiased estimates is a standard procedure in many estimation problems. The reason for that is that the mle is a function of the sufficient statistics and so by the Rao-Blackwell theorem if you can find an unbiased estimator based on sufficien... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | Actually, the scaling of the maximum likelihood estimates in order to obtain unbiased estimates is a standard procedure in many estimation problems. The reason for that is that the mle is a function o | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
Actually, the scaling of the maximum likelihood estimates in order to obtain unbiased estimates is a standard procedure in many estimation problems. The reason for that is that the mle is a function of the sufficient statistics and so by... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
Actually, the scaling of the maximum likelihood estimates in order to obtain unbiased estimates is a standard procedure in many estimation problems. The reason for that is that the mle is a function o |
10,189 | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | I'd add that sometimes (often) we use an MLE estimator because that's what we got, even if in an ideal world it wouldn't be what we want. (I often think of statistics as being like engineering, where we use what we got, not what we want.) In many cases it's easy to define and solve for the MLE, and then get a value usi... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases | I'd add that sometimes (often) we use an MLE estimator because that's what we got, even if in an ideal world it wouldn't be what we want. (I often think of statistics as being like engineering, where | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
I'd add that sometimes (often) we use an MLE estimator because that's what we got, even if in an ideal world it wouldn't be what we want. (I often think of statistics as being like engineering, where we use what we got, not what we want.... | Maximum Likelihood Estimation -- why it is used despite being biased in many cases
I'd add that sometimes (often) we use an MLE estimator because that's what we got, even if in an ideal world it wouldn't be what we want. (I often think of statistics as being like engineering, where |
10,190 | Is a neural network essential for deep learning? | This is a good question.
Is a neural network essential for deep learning?
Yes, your teacher provided you with a correct definition of deep learning. You can still do machine learning (a broader category) without neural networks, but you need a neural network for it to qualify as 'deep learning'.
Isn't it possible to... | Is a neural network essential for deep learning? | This is a good question.
Is a neural network essential for deep learning?
Yes, your teacher provided you with a correct definition of deep learning. You can still do machine learning (a broader cate | Is a neural network essential for deep learning?
This is a good question.
Is a neural network essential for deep learning?
Yes, your teacher provided you with a correct definition of deep learning. You can still do machine learning (a broader category) without neural networks, but you need a neural network for it to ... | Is a neural network essential for deep learning?
This is a good question.
Is a neural network essential for deep learning?
Yes, your teacher provided you with a correct definition of deep learning. You can still do machine learning (a broader cate |
10,191 | Is a neural network essential for deep learning? | I'm going to disagree with the other answers. Fundamentally, I would say that deep learning is defined by a hierarchy of learned representations, and not by which particular model is used to define these representations. Indeed, this is how Goodfellow et al define it in the introductory section of their text Deep Learn... | Is a neural network essential for deep learning? | I'm going to disagree with the other answers. Fundamentally, I would say that deep learning is defined by a hierarchy of learned representations, and not by which particular model is used to define th | Is a neural network essential for deep learning?
I'm going to disagree with the other answers. Fundamentally, I would say that deep learning is defined by a hierarchy of learned representations, and not by which particular model is used to define these representations. Indeed, this is how Goodfellow et al define it in ... | Is a neural network essential for deep learning?
I'm going to disagree with the other answers. Fundamentally, I would say that deep learning is defined by a hierarchy of learned representations, and not by which particular model is used to define th |
10,192 | Is a neural network essential for deep learning? | This answer depends on the definition of artificial neural network (ANN) you take to be true.
See my question here: What *is* an Artificial Neural Network?.
Therefore, no objective answer can be given since:
To accommodate all definitions of ANNs, they are simply defined as arbitrary computational graphs, with tunable... | Is a neural network essential for deep learning? | This answer depends on the definition of artificial neural network (ANN) you take to be true.
See my question here: What *is* an Artificial Neural Network?.
Therefore, no objective answer can be given | Is a neural network essential for deep learning?
This answer depends on the definition of artificial neural network (ANN) you take to be true.
See my question here: What *is* an Artificial Neural Network?.
Therefore, no objective answer can be given since:
To accommodate all definitions of ANNs, they are simply define... | Is a neural network essential for deep learning?
This answer depends on the definition of artificial neural network (ANN) you take to be true.
See my question here: What *is* an Artificial Neural Network?.
Therefore, no objective answer can be given |
10,193 | Is a neural network essential for deep learning? | Deep learning is machine learning done using “deep” neural networks, i.e. such that have multiple (>2) layers. So you cannot do it without neural networks. For using other kinds of machine learning just use “machine learning” term, that includes neural networks as well. | Is a neural network essential for deep learning? | Deep learning is machine learning done using “deep” neural networks, i.e. such that have multiple (>2) layers. So you cannot do it without neural networks. For using other kinds of machine learning ju | Is a neural network essential for deep learning?
Deep learning is machine learning done using “deep” neural networks, i.e. such that have multiple (>2) layers. So you cannot do it without neural networks. For using other kinds of machine learning just use “machine learning” term, that includes neural networks as well. | Is a neural network essential for deep learning?
Deep learning is machine learning done using “deep” neural networks, i.e. such that have multiple (>2) layers. So you cannot do it without neural networks. For using other kinds of machine learning ju |
10,194 | Is a neural network essential for deep learning? | To answer from a different perspective, one philosophical point may help: the concept of learning in deep learning.
In deep learning, there are multiple learning steps. In the first step, the data input is 'converted' (or learned) into a synthetic intermediate output (a bit higher abstraction, loosely speaking). Then i... | Is a neural network essential for deep learning? | To answer from a different perspective, one philosophical point may help: the concept of learning in deep learning.
In deep learning, there are multiple learning steps. In the first step, the data inp | Is a neural network essential for deep learning?
To answer from a different perspective, one philosophical point may help: the concept of learning in deep learning.
In deep learning, there are multiple learning steps. In the first step, the data input is 'converted' (or learned) into a synthetic intermediate output (a ... | Is a neural network essential for deep learning?
To answer from a different perspective, one philosophical point may help: the concept of learning in deep learning.
In deep learning, there are multiple learning steps. In the first step, the data inp |
10,195 | Is a neural network essential for deep learning? | Deep learning is a subset of machine learning, which is a field dedicated to the study and development of machines that can learn, and the goal is of deep learning is to achieve eventually attain general artificial intelligence. Neural network is just one of the biological inspired model.
In simple terms I would defi... | Is a neural network essential for deep learning? | Deep learning is a subset of machine learning, which is a field dedicated to the study and development of machines that can learn, and the goal is of deep learning is to achieve eventually attain gen | Is a neural network essential for deep learning?
Deep learning is a subset of machine learning, which is a field dedicated to the study and development of machines that can learn, and the goal is of deep learning is to achieve eventually attain general artificial intelligence. Neural network is just one of the biolog... | Is a neural network essential for deep learning?
Deep learning is a subset of machine learning, which is a field dedicated to the study and development of machines that can learn, and the goal is of deep learning is to achieve eventually attain gen |
10,196 | How to easily determine the results distribution for multiple dice? | Exact solutions
The number of combinations in $n$ throws is of course $6^n$.
These calculations are most readily done using the probability generating function for one die,
$$p(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 = x \frac{1-x^6}{1-x}.$$
(Actually this is $6$ times the pgf--I'll take care of the factor of $6$ at the e... | How to easily determine the results distribution for multiple dice? | Exact solutions
The number of combinations in $n$ throws is of course $6^n$.
These calculations are most readily done using the probability generating function for one die,
$$p(x) = x + x^2 + x^3 + x^ | How to easily determine the results distribution for multiple dice?
Exact solutions
The number of combinations in $n$ throws is of course $6^n$.
These calculations are most readily done using the probability generating function for one die,
$$p(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 = x \frac{1-x^6}{1-x}.$$
(Actually thi... | How to easily determine the results distribution for multiple dice?
Exact solutions
The number of combinations in $n$ throws is of course $6^n$.
These calculations are most readily done using the probability generating function for one die,
$$p(x) = x + x^2 + x^3 + x^ |
10,197 | How to easily determine the results distribution for multiple dice? | Yet another way to quickly compute the probability distribution of a dice roll would be to use a specialized calculator designed just for that purpose.
Torben Mogensen, a CS professor at DIKU has an excellent dice roller called Troll.
The Troll dice roller and probability calculator prints out the probability distribut... | How to easily determine the results distribution for multiple dice? | Yet another way to quickly compute the probability distribution of a dice roll would be to use a specialized calculator designed just for that purpose.
Torben Mogensen, a CS professor at DIKU has an e | How to easily determine the results distribution for multiple dice?
Yet another way to quickly compute the probability distribution of a dice roll would be to use a specialized calculator designed just for that purpose.
Torben Mogensen, a CS professor at DIKU has an excellent dice roller called Troll.
The Troll dice ro... | How to easily determine the results distribution for multiple dice?
Yet another way to quickly compute the probability distribution of a dice roll would be to use a specialized calculator designed just for that purpose.
Torben Mogensen, a CS professor at DIKU has an e |
10,198 | How to easily determine the results distribution for multiple dice? | $\newcommand{red}{\color{red}}$
$\newcommand{blue}{\color{blue}}$
Let the first die be red and the second be black. Then there are 36 possible results:
\begin{array}{c|c|c|c|c|c|c}
&1&2&3&4&5&6\\\hline
\red{1}&\red{1},1&\red{1},2&\red{1},3&\red{1},4&\red{1},5&\red{1},6\\
&\blue{^2}&\blue{^3}&\blue{^4}&\blue{^5}&\blue{... | How to easily determine the results distribution for multiple dice? | $\newcommand{red}{\color{red}}$
$\newcommand{blue}{\color{blue}}$
Let the first die be red and the second be black. Then there are 36 possible results:
\begin{array}{c|c|c|c|c|c|c}
&1&2&3&4&5&6\\\hlin | How to easily determine the results distribution for multiple dice?
$\newcommand{red}{\color{red}}$
$\newcommand{blue}{\color{blue}}$
Let the first die be red and the second be black. Then there are 36 possible results:
\begin{array}{c|c|c|c|c|c|c}
&1&2&3&4&5&6\\\hline
\red{1}&\red{1},1&\red{1},2&\red{1},3&\red{1},4&\r... | How to easily determine the results distribution for multiple dice?
$\newcommand{red}{\color{red}}$
$\newcommand{blue}{\color{blue}}$
Let the first die be red and the second be black. Then there are 36 possible results:
\begin{array}{c|c|c|c|c|c|c}
&1&2&3&4&5&6\\\hlin |
10,199 | How to easily determine the results distribution for multiple dice? | There's a very neat way of computing the combinations or probabilities in a spreadsheet (such as excel) that computes the convolutions directly.
I'll do it in terms of probabilities and illustrate it for six sided dice but you can do it for dice with any number of sides (including adding different ones).
(btw it's als... | How to easily determine the results distribution for multiple dice? | There's a very neat way of computing the combinations or probabilities in a spreadsheet (such as excel) that computes the convolutions directly.
I'll do it in terms of probabilities and illustrate it | How to easily determine the results distribution for multiple dice?
There's a very neat way of computing the combinations or probabilities in a spreadsheet (such as excel) that computes the convolutions directly.
I'll do it in terms of probabilities and illustrate it for six sided dice but you can do it for dice with ... | How to easily determine the results distribution for multiple dice?
There's a very neat way of computing the combinations or probabilities in a spreadsheet (such as excel) that computes the convolutions directly.
I'll do it in terms of probabilities and illustrate it |
10,200 | How to easily determine the results distribution for multiple dice? | This is actually a suprisingly complicated question. Luckily for you, there exist an exact solution which is very well explained here:
http://mathworld.wolfram.com/Dice.html
The probability you are looking for is given by equation (10): "The probability of obtaining p points (a roll of p) on n s-sided dice".
In your c... | How to easily determine the results distribution for multiple dice? | This is actually a suprisingly complicated question. Luckily for you, there exist an exact solution which is very well explained here:
http://mathworld.wolfram.com/Dice.html
The probability you are lo | How to easily determine the results distribution for multiple dice?
This is actually a suprisingly complicated question. Luckily for you, there exist an exact solution which is very well explained here:
http://mathworld.wolfram.com/Dice.html
The probability you are looking for is given by equation (10): "The probabilit... | How to easily determine the results distribution for multiple dice?
This is actually a suprisingly complicated question. Luckily for you, there exist an exact solution which is very well explained here:
http://mathworld.wolfram.com/Dice.html
The probability you are lo |
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