idx int64 1 56k | question stringlengths 15 155 | answer stringlengths 2 29.2k ⌀ | question_cut stringlengths 15 100 | answer_cut stringlengths 2 200 ⌀ | conversation stringlengths 47 29.3k | conversation_cut stringlengths 47 301 |
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5,901 | Compendium of cross-validation techniques | K-fold cross-validation (CV) randomly breaks your data up into K partitions, and you in turn hold out one of those K parts as a test case, and lump the other K-1 parts together as your training data. Leave One Out (LOO) is the special case where you take your N data items and do N-fold CV. In some sense, Hold Out is an... | Compendium of cross-validation techniques | K-fold cross-validation (CV) randomly breaks your data up into K partitions, and you in turn hold out one of those K parts as a test case, and lump the other K-1 parts together as your training data. | Compendium of cross-validation techniques
K-fold cross-validation (CV) randomly breaks your data up into K partitions, and you in turn hold out one of those K parts as a test case, and lump the other K-1 parts together as your training data. Leave One Out (LOO) is the special case where you take your N data items and d... | Compendium of cross-validation techniques
K-fold cross-validation (CV) randomly breaks your data up into K partitions, and you in turn hold out one of those K parts as a test case, and lump the other K-1 parts together as your training data. |
5,902 | Compendium of cross-validation techniques | I found one of the references linked to in the Wikipedia article quite useful
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.529&rep=rep1&type=pdf
"A study of cross-validation and bootstrap for accuracy estimation and model selection", Ron Kohavi, IJCAI95
It contains an empirical comparison for a subset of... | Compendium of cross-validation techniques | I found one of the references linked to in the Wikipedia article quite useful
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.529&rep=rep1&type=pdf
"A study of cross-validation and bootstr | Compendium of cross-validation techniques
I found one of the references linked to in the Wikipedia article quite useful
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.529&rep=rep1&type=pdf
"A study of cross-validation and bootstrap for accuracy estimation and model selection", Ron Kohavi, IJCAI95
It contai... | Compendium of cross-validation techniques
I found one of the references linked to in the Wikipedia article quite useful
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.529&rep=rep1&type=pdf
"A study of cross-validation and bootstr |
5,903 | Compendium of cross-validation techniques | ...and a guide on when to use each of them...
Unfortunately that problem is harder than it gets credit for. There are at least 2 main uses of cross-validation: selecting a model, and evaluating model performance.
Roughly speaking, if you use a CV variant which splits the data using a high train-to-test ratio, this can... | Compendium of cross-validation techniques | ...and a guide on when to use each of them...
Unfortunately that problem is harder than it gets credit for. There are at least 2 main uses of cross-validation: selecting a model, and evaluating model | Compendium of cross-validation techniques
...and a guide on when to use each of them...
Unfortunately that problem is harder than it gets credit for. There are at least 2 main uses of cross-validation: selecting a model, and evaluating model performance.
Roughly speaking, if you use a CV variant which splits the data ... | Compendium of cross-validation techniques
...and a guide on when to use each of them...
Unfortunately that problem is harder than it gets credit for. There are at least 2 main uses of cross-validation: selecting a model, and evaluating model |
5,904 | What are correct values for precision and recall when the denominators equal 0? | The answers to the linked earlier question apply here too.
If (true positives + false negatives) = 0 then no positive cases in the input data, so any analysis of this case has no information, and so no conclusion about how positive cases are handled. You want N/A or something similar as the ratio result, avoiding a di... | What are correct values for precision and recall when the denominators equal 0? | The answers to the linked earlier question apply here too.
If (true positives + false negatives) = 0 then no positive cases in the input data, so any analysis of this case has no information, and so n | What are correct values for precision and recall when the denominators equal 0?
The answers to the linked earlier question apply here too.
If (true positives + false negatives) = 0 then no positive cases in the input data, so any analysis of this case has no information, and so no conclusion about how positive cases ar... | What are correct values for precision and recall when the denominators equal 0?
The answers to the linked earlier question apply here too.
If (true positives + false negatives) = 0 then no positive cases in the input data, so any analysis of this case has no information, and so n |
5,905 | What are correct values for precision and recall when the denominators equal 0? | An interesting answer is offered here:
https://github.com/dice-group/gerbil/wiki/Precision,-Recall-and-F1-measure
The authors of the module output different scores for precision and recall depending on whether true positives, false positives and false negatives are all 0. If they are, the outcome is ostensibly a good o... | What are correct values for precision and recall when the denominators equal 0? | An interesting answer is offered here:
https://github.com/dice-group/gerbil/wiki/Precision,-Recall-and-F1-measure
The authors of the module output different scores for precision and recall depending o | What are correct values for precision and recall when the denominators equal 0?
An interesting answer is offered here:
https://github.com/dice-group/gerbil/wiki/Precision,-Recall-and-F1-measure
The authors of the module output different scores for precision and recall depending on whether true positives, false positive... | What are correct values for precision and recall when the denominators equal 0?
An interesting answer is offered here:
https://github.com/dice-group/gerbil/wiki/Precision,-Recall-and-F1-measure
The authors of the module output different scores for precision and recall depending o |
5,906 | What are correct values for precision and recall when the denominators equal 0? | When evaluating a classifier at high thresholds, the precision might (often actually) not be 1 when recall is 0. It's usually N/A!
I think there is something wrong about how people plot the P/R curve. Avoiding N/A samples is a bias in the sense that you avoid singularity samples. I computed the average precision wrt to... | What are correct values for precision and recall when the denominators equal 0? | When evaluating a classifier at high thresholds, the precision might (often actually) not be 1 when recall is 0. It's usually N/A!
I think there is something wrong about how people plot the P/R curve. | What are correct values for precision and recall when the denominators equal 0?
When evaluating a classifier at high thresholds, the precision might (often actually) not be 1 when recall is 0. It's usually N/A!
I think there is something wrong about how people plot the P/R curve. Avoiding N/A samples is a bias in the s... | What are correct values for precision and recall when the denominators equal 0?
When evaluating a classifier at high thresholds, the precision might (often actually) not be 1 when recall is 0. It's usually N/A!
I think there is something wrong about how people plot the P/R curve. |
5,907 | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekutieli (2001) false discovery rate procedures? | Benjamini and Hochberg (1995) introduced the false discovery rate. Benjamini and Yekutieli (2001) proved that the estimator is valid under some forms of dependence. Dependence can arise as follows. Consider the continuous variable used in a t-test and another variable correlated with it; for example, testing if BMI ... | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekut | Benjamini and Hochberg (1995) introduced the false discovery rate. Benjamini and Yekutieli (2001) proved that the estimator is valid under some forms of dependence. Dependence can arise as follows. | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekutieli (2001) false discovery rate procedures?
Benjamini and Hochberg (1995) introduced the false discovery rate. Benjamini and Yekutieli (2001) proved that the estimator is valid under some forms of dependence. Depende... | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekut
Benjamini and Hochberg (1995) introduced the false discovery rate. Benjamini and Yekutieli (2001) proved that the estimator is valid under some forms of dependence. Dependence can arise as follows. |
5,908 | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekutieli (2001) false discovery rate procedures? | p.adjust is not misciting for BY. The reference is to Theorem 1.3 (proof in Section 5 on p.1182) in the paper:
Benjamini, Y., and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics 29, 1165–1188.
As this paper discusses several different adjustments... | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekut | p.adjust is not misciting for BY. The reference is to Theorem 1.3 (proof in Section 5 on p.1182) in the paper:
Benjamini, Y., and Yekutieli, D. (2001). The control of the false discovery rate in mult | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekutieli (2001) false discovery rate procedures?
p.adjust is not misciting for BY. The reference is to Theorem 1.3 (proof in Section 5 on p.1182) in the paper:
Benjamini, Y., and Yekutieli, D. (2001). The control of the fal... | What are the practical differences between the Benjamini & Hochberg (1995) and the Benjamini & Yekut
p.adjust is not misciting for BY. The reference is to Theorem 1.3 (proof in Section 5 on p.1182) in the paper:
Benjamini, Y., and Yekutieli, D. (2001). The control of the false discovery rate in mult |
5,909 | When and how to use standardized explanatory variables in linear regression | Although terminology is a contentious topic, I prefer to call "explanatory" variables, "predictor" variables.
When to standardise the predictors:
A lot of software for performing multiple linear regression will provide standardised coefficients which are equivalent to unstandardised coefficients where you manually sta... | When and how to use standardized explanatory variables in linear regression | Although terminology is a contentious topic, I prefer to call "explanatory" variables, "predictor" variables.
When to standardise the predictors:
A lot of software for performing multiple linear regr | When and how to use standardized explanatory variables in linear regression
Although terminology is a contentious topic, I prefer to call "explanatory" variables, "predictor" variables.
When to standardise the predictors:
A lot of software for performing multiple linear regression will provide standardised coefficient... | When and how to use standardized explanatory variables in linear regression
Although terminology is a contentious topic, I prefer to call "explanatory" variables, "predictor" variables.
When to standardise the predictors:
A lot of software for performing multiple linear regr |
5,910 | Intuitive explanation of convergence in distribution and convergence in probability | How can a random number converge to a constant?
Let's say you have $N$ balls in the box. You can pick them one by one. After you picked $k$ balls, I ask you: what's the mean weight of the balls in the box? Your best answer would be $\bar x_k=\frac{1}{k}\sum_{i=1}^kx_i$. You realize that $\bar x_k$ itself is the random... | Intuitive explanation of convergence in distribution and convergence in probability | How can a random number converge to a constant?
Let's say you have $N$ balls in the box. You can pick them one by one. After you picked $k$ balls, I ask you: what's the mean weight of the balls in th | Intuitive explanation of convergence in distribution and convergence in probability
How can a random number converge to a constant?
Let's say you have $N$ balls in the box. You can pick them one by one. After you picked $k$ balls, I ask you: what's the mean weight of the balls in the box? Your best answer would be $\b... | Intuitive explanation of convergence in distribution and convergence in probability
How can a random number converge to a constant?
Let's say you have $N$ balls in the box. You can pick them one by one. After you picked $k$ balls, I ask you: what's the mean weight of the balls in th |
5,911 | Intuitive explanation of convergence in distribution and convergence in probability | It's not clear how much intuition a reader of this question might have about convergence of anything, let alone of random variables, so I will write as if the answer is "very little". Something that might help: rather than thinking "how can a random variable converge", ask how a sequence of random variables can converg... | Intuitive explanation of convergence in distribution and convergence in probability | It's not clear how much intuition a reader of this question might have about convergence of anything, let alone of random variables, so I will write as if the answer is "very little". Something that m | Intuitive explanation of convergence in distribution and convergence in probability
It's not clear how much intuition a reader of this question might have about convergence of anything, let alone of random variables, so I will write as if the answer is "very little". Something that might help: rather than thinking "how... | Intuitive explanation of convergence in distribution and convergence in probability
It's not clear how much intuition a reader of this question might have about convergence of anything, let alone of random variables, so I will write as if the answer is "very little". Something that m |
5,912 | Intuitive explanation of convergence in distribution and convergence in probability | In my mind, the existing answers all convey useful points, but they do not make an important distinction clear between the two modes of convergence.
Let $X_n$, $n=1,2,\dots$, and $Y$ be random variables. For intuition, imagine $X_n$ are assigned their values by some random experiment that changes a little bit for each ... | Intuitive explanation of convergence in distribution and convergence in probability | In my mind, the existing answers all convey useful points, but they do not make an important distinction clear between the two modes of convergence.
Let $X_n$, $n=1,2,\dots$, and $Y$ be random variabl | Intuitive explanation of convergence in distribution and convergence in probability
In my mind, the existing answers all convey useful points, but they do not make an important distinction clear between the two modes of convergence.
Let $X_n$, $n=1,2,\dots$, and $Y$ be random variables. For intuition, imagine $X_n$ are... | Intuitive explanation of convergence in distribution and convergence in probability
In my mind, the existing answers all convey useful points, but they do not make an important distinction clear between the two modes of convergence.
Let $X_n$, $n=1,2,\dots$, and $Y$ be random variabl |
5,913 | Intuitive explanation of convergence in distribution and convergence in probability | What I don't understand is how can a random variable converge to a
single number but also converge to a distribution?
If you're learning econometrics, you're probably wondering about this in the context of a regression model. It converges to a degenerate distribution, to a constant. But something else does have a no... | Intuitive explanation of convergence in distribution and convergence in probability | What I don't understand is how can a random variable converge to a
single number but also converge to a distribution?
If you're learning econometrics, you're probably wondering about this in the co | Intuitive explanation of convergence in distribution and convergence in probability
What I don't understand is how can a random variable converge to a
single number but also converge to a distribution?
If you're learning econometrics, you're probably wondering about this in the context of a regression model. It conv... | Intuitive explanation of convergence in distribution and convergence in probability
What I don't understand is how can a random variable converge to a
single number but also converge to a distribution?
If you're learning econometrics, you're probably wondering about this in the co |
5,914 | Intuitive explanation of convergence in distribution and convergence in probability | Let me try to give a very short answer, using some very simple examples.
Convergence in distribution
Let $X_n \sim N\left(\frac{1}{n}, 1 \right)$, for all n, then $X_n$ converges to $X \sim N(0, 1)$ in distribution. However, the randomness in the realization of $X_n$ does not change over time. If we have to predict the... | Intuitive explanation of convergence in distribution and convergence in probability | Let me try to give a very short answer, using some very simple examples.
Convergence in distribution
Let $X_n \sim N\left(\frac{1}{n}, 1 \right)$, for all n, then $X_n$ converges to $X \sim N(0, 1)$ i | Intuitive explanation of convergence in distribution and convergence in probability
Let me try to give a very short answer, using some very simple examples.
Convergence in distribution
Let $X_n \sim N\left(\frac{1}{n}, 1 \right)$, for all n, then $X_n$ converges to $X \sim N(0, 1)$ in distribution. However, the randomn... | Intuitive explanation of convergence in distribution and convergence in probability
Let me try to give a very short answer, using some very simple examples.
Convergence in distribution
Let $X_n \sim N\left(\frac{1}{n}, 1 \right)$, for all n, then $X_n$ converges to $X \sim N(0, 1)$ i |
5,915 | Intuitive explanation of convergence in distribution and convergence in probability | Convergence in probability to a constant: a larger and larger share of the PDF get restrained to a band around a fixed value as the sequence progress (note that nothing is said on the values whose probabilities are not yet in the band as the sequence progress)
Convergence in distribution: consider the PDF as a function... | Intuitive explanation of convergence in distribution and convergence in probability | Convergence in probability to a constant: a larger and larger share of the PDF get restrained to a band around a fixed value as the sequence progress (note that nothing is said on the values whose pro | Intuitive explanation of convergence in distribution and convergence in probability
Convergence in probability to a constant: a larger and larger share of the PDF get restrained to a band around a fixed value as the sequence progress (note that nothing is said on the values whose probabilities are not yet in the band a... | Intuitive explanation of convergence in distribution and convergence in probability
Convergence in probability to a constant: a larger and larger share of the PDF get restrained to a band around a fixed value as the sequence progress (note that nothing is said on the values whose pro |
5,916 | Why do I get a 100% accuracy decision tree? | Your test sample is a subset of your training sample:
x_train = x[0:2635]
x_test = x[0:658]
y_train = y[0:2635]
y_test = y[0:658]
This means that you evaluate your model on a part of your training data, i.e., you are doing in-sample evaluation. In-sample accuracy is a notoriously poor indicator to out-of-sample accura... | Why do I get a 100% accuracy decision tree? | Your test sample is a subset of your training sample:
x_train = x[0:2635]
x_test = x[0:658]
y_train = y[0:2635]
y_test = y[0:658]
This means that you evaluate your model on a part of your training da | Why do I get a 100% accuracy decision tree?
Your test sample is a subset of your training sample:
x_train = x[0:2635]
x_test = x[0:658]
y_train = y[0:2635]
y_test = y[0:658]
This means that you evaluate your model on a part of your training data, i.e., you are doing in-sample evaluation. In-sample accuracy is a notori... | Why do I get a 100% accuracy decision tree?
Your test sample is a subset of your training sample:
x_train = x[0:2635]
x_test = x[0:658]
y_train = y[0:2635]
y_test = y[0:658]
This means that you evaluate your model on a part of your training da |
5,917 | Why do I get a 100% accuracy decision tree? | You are getting 100% accuracy because you are using a part of training data for testing. At the time of training, decision tree gained the knowledge about that data, and now if you give same data to predict it will give exactly same value. That's why decision tree producing correct results every time.
For any machine l... | Why do I get a 100% accuracy decision tree? | You are getting 100% accuracy because you are using a part of training data for testing. At the time of training, decision tree gained the knowledge about that data, and now if you give same data to p | Why do I get a 100% accuracy decision tree?
You are getting 100% accuracy because you are using a part of training data for testing. At the time of training, decision tree gained the knowledge about that data, and now if you give same data to predict it will give exactly same value. That's why decision tree producing c... | Why do I get a 100% accuracy decision tree?
You are getting 100% accuracy because you are using a part of training data for testing. At the time of training, decision tree gained the knowledge about that data, and now if you give same data to p |
5,918 | Why do I get a 100% accuracy decision tree? | As other users have told you, you are using as test set a subset of the train set, and a decision tree is very prone to overfitting.
You almost had it when you imported
from sklearn.cross_validation import train_test_split
But then you don't use the function. You should have done:
x_train, x_test, y_train, y_test = t... | Why do I get a 100% accuracy decision tree? | As other users have told you, you are using as test set a subset of the train set, and a decision tree is very prone to overfitting.
You almost had it when you imported
from sklearn.cross_validation | Why do I get a 100% accuracy decision tree?
As other users have told you, you are using as test set a subset of the train set, and a decision tree is very prone to overfitting.
You almost had it when you imported
from sklearn.cross_validation import train_test_split
But then you don't use the function. You should hav... | Why do I get a 100% accuracy decision tree?
As other users have told you, you are using as test set a subset of the train set, and a decision tree is very prone to overfitting.
You almost had it when you imported
from sklearn.cross_validation |
5,919 | Why do I get a 100% accuracy decision tree? | As pointed by @Stephan Kolassa and @Sanjay Chandlekar, this is due to the fact that your test sample is a subset of your training sample.
However, for the selection of those samples, random sampling would be more appropriate to ensure that both samples are representative. Depending on your data structure, you might als... | Why do I get a 100% accuracy decision tree? | As pointed by @Stephan Kolassa and @Sanjay Chandlekar, this is due to the fact that your test sample is a subset of your training sample.
However, for the selection of those samples, random sampling w | Why do I get a 100% accuracy decision tree?
As pointed by @Stephan Kolassa and @Sanjay Chandlekar, this is due to the fact that your test sample is a subset of your training sample.
However, for the selection of those samples, random sampling would be more appropriate to ensure that both samples are representative. Dep... | Why do I get a 100% accuracy decision tree?
As pointed by @Stephan Kolassa and @Sanjay Chandlekar, this is due to the fact that your test sample is a subset of your training sample.
However, for the selection of those samples, random sampling w |
5,920 | Why do I get a 100% accuracy decision tree? | Just want to chime in on the intuition for why you need to split training and test samples explicitly.
If you have $n$ observations and make $n$ (actually, $n-1$, and possibly far less) splits on your data, you will perfectly classify every point (if this isn't immediately clear, write down some small-scale examples, e... | Why do I get a 100% accuracy decision tree? | Just want to chime in on the intuition for why you need to split training and test samples explicitly.
If you have $n$ observations and make $n$ (actually, $n-1$, and possibly far less) splits on your | Why do I get a 100% accuracy decision tree?
Just want to chime in on the intuition for why you need to split training and test samples explicitly.
If you have $n$ observations and make $n$ (actually, $n-1$, and possibly far less) splits on your data, you will perfectly classify every point (if this isn't immediately cl... | Why do I get a 100% accuracy decision tree?
Just want to chime in on the intuition for why you need to split training and test samples explicitly.
If you have $n$ observations and make $n$ (actually, $n-1$, and possibly far less) splits on your |
5,921 | Why do I get a 100% accuracy decision tree? | You don't need 100% accuracy to get overfitting. With enough buckets, you can get irreproducible results (something that would look terrible out-of-sample).
See this excerpted article from the Lancet, describing the method of chopping a sample into buckets which are far too fine.
Munchausen's Statistical Grid
It is al... | Why do I get a 100% accuracy decision tree? | You don't need 100% accuracy to get overfitting. With enough buckets, you can get irreproducible results (something that would look terrible out-of-sample).
See this excerpted article from the Lancet, | Why do I get a 100% accuracy decision tree?
You don't need 100% accuracy to get overfitting. With enough buckets, you can get irreproducible results (something that would look terrible out-of-sample).
See this excerpted article from the Lancet, describing the method of chopping a sample into buckets which are far too f... | Why do I get a 100% accuracy decision tree?
You don't need 100% accuracy to get overfitting. With enough buckets, you can get irreproducible results (something that would look terrible out-of-sample).
See this excerpted article from the Lancet, |
5,922 | Bound for the correlation of three random variables | The common correlation $\rho$ can have value $+1$ but not $-1$. If $\rho_{X,Y}= \rho_{X,Z}=-1$, then $\rho_{Y,Z}$ cannot equal $-1$ but is in fact $+1$.
The smallest value of the common correlation of three random variables
is $-\frac{1}{2}$. More generally,
the minimum common correlation of $n$ random variables is $-\... | Bound for the correlation of three random variables | The common correlation $\rho$ can have value $+1$ but not $-1$. If $\rho_{X,Y}= \rho_{X,Z}=-1$, then $\rho_{Y,Z}$ cannot equal $-1$ but is in fact $+1$.
The smallest value of the common correlation of | Bound for the correlation of three random variables
The common correlation $\rho$ can have value $+1$ but not $-1$. If $\rho_{X,Y}= \rho_{X,Z}=-1$, then $\rho_{Y,Z}$ cannot equal $-1$ but is in fact $+1$.
The smallest value of the common correlation of three random variables
is $-\frac{1}{2}$. More generally,
the minim... | Bound for the correlation of three random variables
The common correlation $\rho$ can have value $+1$ but not $-1$. If $\rho_{X,Y}= \rho_{X,Z}=-1$, then $\rho_{Y,Z}$ cannot equal $-1$ but is in fact $+1$.
The smallest value of the common correlation of |
5,923 | Bound for the correlation of three random variables | The tightest possible bound is $-1/2 \le \rho \le 1$. All such values can actually appear--none are impossible.
To show there is nothing especially deep or mysterious about the result, this answer first presents a completely elementary solution, requiring only the obvious fact that variances--being the expected values... | Bound for the correlation of three random variables | The tightest possible bound is $-1/2 \le \rho \le 1$. All such values can actually appear--none are impossible.
To show there is nothing especially deep or mysterious about the result, this answer fi | Bound for the correlation of three random variables
The tightest possible bound is $-1/2 \le \rho \le 1$. All such values can actually appear--none are impossible.
To show there is nothing especially deep or mysterious about the result, this answer first presents a completely elementary solution, requiring only the ob... | Bound for the correlation of three random variables
The tightest possible bound is $-1/2 \le \rho \le 1$. All such values can actually appear--none are impossible.
To show there is nothing especially deep or mysterious about the result, this answer fi |
5,924 | Bound for the correlation of three random variables | Your correlation matrix is
$$ \begin{pmatrix} 1&\rho&\rho\\ \rho&1&\rho\\ \rho&\rho&1 \end{pmatrix}$$
The matrix is positive semidefinite if the leading principal minors are all non-negative. The principal minors are the determinants of the "north-west" blocks of the matrix, i.e. 1, the determinant of
$$ \begin{pmatrix... | Bound for the correlation of three random variables | Your correlation matrix is
$$ \begin{pmatrix} 1&\rho&\rho\\ \rho&1&\rho\\ \rho&\rho&1 \end{pmatrix}$$
The matrix is positive semidefinite if the leading principal minors are all non-negative. The prin | Bound for the correlation of three random variables
Your correlation matrix is
$$ \begin{pmatrix} 1&\rho&\rho\\ \rho&1&\rho\\ \rho&\rho&1 \end{pmatrix}$$
The matrix is positive semidefinite if the leading principal minors are all non-negative. The principal minors are the determinants of the "north-west" blocks of the ... | Bound for the correlation of three random variables
Your correlation matrix is
$$ \begin{pmatrix} 1&\rho&\rho\\ \rho&1&\rho\\ \rho&\rho&1 \end{pmatrix}$$
The matrix is positive semidefinite if the leading principal minors are all non-negative. The prin |
5,925 | Bound for the correlation of three random variables | 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.
There exist random variables $X$, $Y$ and $Z$ with pai... | Bound for the correlation of three random variables | 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.
| Bound for the correlation of three random variables
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.
Th... | Bound for the correlation of three random variables
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.
|
5,926 | How do DAGs help to reduce bias in causal inference? | A DAG is a Directed Acyclic Graph.
A “Graph” is a structure with nodes (which are usually variables in statistics) and arcs (lines) connecting nodes to other nodes. “Directed” means that all the arcs have a direction, where one end of the arc has an arrow head, and the other does not, which usually refers to causation.... | How do DAGs help to reduce bias in causal inference? | A DAG is a Directed Acyclic Graph.
A “Graph” is a structure with nodes (which are usually variables in statistics) and arcs (lines) connecting nodes to other nodes. “Directed” means that all the arcs | How do DAGs help to reduce bias in causal inference?
A DAG is a Directed Acyclic Graph.
A “Graph” is a structure with nodes (which are usually variables in statistics) and arcs (lines) connecting nodes to other nodes. “Directed” means that all the arcs have a direction, where one end of the arc has an arrow head, and t... | How do DAGs help to reduce bias in causal inference?
A DAG is a Directed Acyclic Graph.
A “Graph” is a structure with nodes (which are usually variables in statistics) and arcs (lines) connecting nodes to other nodes. “Directed” means that all the arcs |
5,927 | How do DAGs help to reduce bias in causal inference? | This is generally a fairly elaborate topic, and may require more reading on your part for better understanding, but I will try to answer a couple of your questions in isolation and leave references for further reading.
Confounding
Consider the example below:
Controlling for the confounding variable "Gender" gives us m... | How do DAGs help to reduce bias in causal inference? | This is generally a fairly elaborate topic, and may require more reading on your part for better understanding, but I will try to answer a couple of your questions in isolation and leave references fo | How do DAGs help to reduce bias in causal inference?
This is generally a fairly elaborate topic, and may require more reading on your part for better understanding, but I will try to answer a couple of your questions in isolation and leave references for further reading.
Confounding
Consider the example below:
Control... | How do DAGs help to reduce bias in causal inference?
This is generally a fairly elaborate topic, and may require more reading on your part for better understanding, but I will try to answer a couple of your questions in isolation and leave references fo |
5,928 | How do DAGs help to reduce bias in causal inference? | A different angle.
It constraints the hypothesis space to hypothesis that imply the causation graph. It is like injecting domain knowledge in your model, so that the learning phase is more informed.
This can be modelled (in a very abstract way) by the VC-dimension of the model, or the PAC framework in general | How do DAGs help to reduce bias in causal inference? | A different angle.
It constraints the hypothesis space to hypothesis that imply the causation graph. It is like injecting domain knowledge in your model, so that the learning phase is more informed.
T | How do DAGs help to reduce bias in causal inference?
A different angle.
It constraints the hypothesis space to hypothesis that imply the causation graph. It is like injecting domain knowledge in your model, so that the learning phase is more informed.
This can be modelled (in a very abstract way) by the VC-dimension of... | How do DAGs help to reduce bias in causal inference?
A different angle.
It constraints the hypothesis space to hypothesis that imply the causation graph. It is like injecting domain knowledge in your model, so that the learning phase is more informed.
T |
5,929 | Tiny (real) datasets for giving examples in class? | The data and story library is an " online library of datafiles and stories that illustrate the use of basic statistics methods".
This site seems to have what you need, and you can search it for particular data sets. | Tiny (real) datasets for giving examples in class? | The data and story library is an " online library of datafiles and stories that illustrate the use of basic statistics methods".
This site seems to have what you need, and you can search it for partic | Tiny (real) datasets for giving examples in class?
The data and story library is an " online library of datafiles and stories that illustrate the use of basic statistics methods".
This site seems to have what you need, and you can search it for particular data sets. | Tiny (real) datasets for giving examples in class?
The data and story library is an " online library of datafiles and stories that illustrate the use of basic statistics methods".
This site seems to have what you need, and you can search it for partic |
5,930 | Tiny (real) datasets for giving examples in class? | There's a book called "A Handbook of Small Datasets" by D.J. Hand, F. Daly, A.D. Lunn, K.J. McConway and E. Ostrowski. The Statistics department at NCSU have electronically posted the datasets from this book here.
The website above gives only the data; you would need to read the book to get the story behind the numbe... | Tiny (real) datasets for giving examples in class? | There's a book called "A Handbook of Small Datasets" by D.J. Hand, F. Daly, A.D. Lunn, K.J. McConway and E. Ostrowski. The Statistics department at NCSU have electronically posted the datasets from t | Tiny (real) datasets for giving examples in class?
There's a book called "A Handbook of Small Datasets" by D.J. Hand, F. Daly, A.D. Lunn, K.J. McConway and E. Ostrowski. The Statistics department at NCSU have electronically posted the datasets from this book here.
The website above gives only the data; you would need... | Tiny (real) datasets for giving examples in class?
There's a book called "A Handbook of Small Datasets" by D.J. Hand, F. Daly, A.D. Lunn, K.J. McConway and E. Ostrowski. The Statistics department at NCSU have electronically posted the datasets from t |
5,931 | Tiny (real) datasets for giving examples in class? | For two-way tables, I like the data on gender and survival of the titanic passengers:
| Alive Dead | Total
-------+-------------+------
Female | 308 154 | 462
Male | 142 709 | 851
-------+-------------+------
Total | 450 863 | 1313
With this data, one can discuss things like the chi-square tes... | Tiny (real) datasets for giving examples in class? | For two-way tables, I like the data on gender and survival of the titanic passengers:
| Alive Dead | Total
-------+-------------+------
Female | 308 154 | 462
Male | 142 709 | 851
| Tiny (real) datasets for giving examples in class?
For two-way tables, I like the data on gender and survival of the titanic passengers:
| Alive Dead | Total
-------+-------------+------
Female | 308 154 | 462
Male | 142 709 | 851
-------+-------------+------
Total | 450 863 | 1313
With this d... | Tiny (real) datasets for giving examples in class?
For two-way tables, I like the data on gender and survival of the titanic passengers:
| Alive Dead | Total
-------+-------------+------
Female | 308 154 | 462
Male | 142 709 | 851
|
5,932 | Tiny (real) datasets for giving examples in class? | The Journal of Statistical Education has an archive of educational data sets. | Tiny (real) datasets for giving examples in class? | The Journal of Statistical Education has an archive of educational data sets. | Tiny (real) datasets for giving examples in class?
The Journal of Statistical Education has an archive of educational data sets. | Tiny (real) datasets for giving examples in class?
The Journal of Statistical Education has an archive of educational data sets. |
5,933 | Tiny (real) datasets for giving examples in class? | CAUSEweb has data sets as well as lots of other teaching resources.
See http://www.causeweb.org/resources/datasets/ for the datasets.
CAUSE stands for Consortium for the Advancement of Undergraduate Statistics Education. | Tiny (real) datasets for giving examples in class? | CAUSEweb has data sets as well as lots of other teaching resources.
See http://www.causeweb.org/resources/datasets/ for the datasets.
CAUSE stands for Consortium for the Advancement of Undergraduate S | Tiny (real) datasets for giving examples in class?
CAUSEweb has data sets as well as lots of other teaching resources.
See http://www.causeweb.org/resources/datasets/ for the datasets.
CAUSE stands for Consortium for the Advancement of Undergraduate Statistics Education. | Tiny (real) datasets for giving examples in class?
CAUSEweb has data sets as well as lots of other teaching resources.
See http://www.causeweb.org/resources/datasets/ for the datasets.
CAUSE stands for Consortium for the Advancement of Undergraduate S |
5,934 | Tiny (real) datasets for giving examples in class? | Probably such an obvious answer that it does not really need to be mentioned, but for correlation or linear regression Anscombe's quartet is a logical choice. Although it is not a real story with real data I think it is such a simple example it would reasonably fit into your criteria. | Tiny (real) datasets for giving examples in class? | Probably such an obvious answer that it does not really need to be mentioned, but for correlation or linear regression Anscombe's quartet is a logical choice. Although it is not a real story with real | Tiny (real) datasets for giving examples in class?
Probably such an obvious answer that it does not really need to be mentioned, but for correlation or linear regression Anscombe's quartet is a logical choice. Although it is not a real story with real data I think it is such a simple example it would reasonably fit int... | Tiny (real) datasets for giving examples in class?
Probably such an obvious answer that it does not really need to be mentioned, but for correlation or linear regression Anscombe's quartet is a logical choice. Although it is not a real story with real |
5,935 | Tiny (real) datasets for giving examples in class? | StatSci.org is a nice source for datasets. | Tiny (real) datasets for giving examples in class? | StatSci.org is a nice source for datasets. | Tiny (real) datasets for giving examples in class?
StatSci.org is a nice source for datasets. | Tiny (real) datasets for giving examples in class?
StatSci.org is a nice source for datasets. |
5,936 | Tiny (real) datasets for giving examples in class? | A nice article entitled Resource Discovery for Teaching Statistics has shed light on this this topic. | Tiny (real) datasets for giving examples in class? | A nice article entitled Resource Discovery for Teaching Statistics has shed light on this this topic. | Tiny (real) datasets for giving examples in class?
A nice article entitled Resource Discovery for Teaching Statistics has shed light on this this topic. | Tiny (real) datasets for giving examples in class?
A nice article entitled Resource Discovery for Teaching Statistics has shed light on this this topic. |
5,937 | Tiny (real) datasets for giving examples in class? | https://tuvalabs.com
I am sure you have found what you were looking for long back, but for anyone else who come across thread - TuvaLabs is nice source for the datasets for Classrooms. It curates datasets, story, description, small exercise and visualization capability also you can requests datasets on it. | Tiny (real) datasets for giving examples in class? | https://tuvalabs.com
I am sure you have found what you were looking for long back, but for anyone else who come across thread - TuvaLabs is nice source for the datasets for Classrooms. It curates data | Tiny (real) datasets for giving examples in class?
https://tuvalabs.com
I am sure you have found what you were looking for long back, but for anyone else who come across thread - TuvaLabs is nice source for the datasets for Classrooms. It curates datasets, story, description, small exercise and visualization capability... | Tiny (real) datasets for giving examples in class?
https://tuvalabs.com
I am sure you have found what you were looking for long back, but for anyone else who come across thread - TuvaLabs is nice source for the datasets for Classrooms. It curates data |
5,938 | Functions of Independent Random Variables | The most general and abstract definition of independence makes this assertion trivial while supplying an important qualifying condition: that two random variables are independent means the sigma-algebras they generate are independent. Because the sigma-algebra generated by a measurable function of a sigma-algebra is a... | Functions of Independent Random Variables | The most general and abstract definition of independence makes this assertion trivial while supplying an important qualifying condition: that two random variables are independent means the sigma-algeb | Functions of Independent Random Variables
The most general and abstract definition of independence makes this assertion trivial while supplying an important qualifying condition: that two random variables are independent means the sigma-algebras they generate are independent. Because the sigma-algebra generated by a m... | Functions of Independent Random Variables
The most general and abstract definition of independence makes this assertion trivial while supplying an important qualifying condition: that two random variables are independent means the sigma-algeb |
5,939 | Functions of Independent Random Variables | Consider this "less advanced" proof:
Let $X:\Omega_X\to\mathbb{R}^n,Y:\Omega_Y\to\mathbb{R}^m,f:\mathbb{R}^n\to\mathbb{R}^k,g:\mathbb{R}^m\to\mathbb{R}^p$, where $X,Y$ are independent random variables and $f,g$ are measurable functions. Then:
$$
P\{f(X)\leq x \text{ and } g(Y)\leq y\}\\=P(\{f(X)\leq x\}\cap\{g(Y)\leq y... | Functions of Independent Random Variables | Consider this "less advanced" proof:
Let $X:\Omega_X\to\mathbb{R}^n,Y:\Omega_Y\to\mathbb{R}^m,f:\mathbb{R}^n\to\mathbb{R}^k,g:\mathbb{R}^m\to\mathbb{R}^p$, where $X,Y$ are independent random variables | Functions of Independent Random Variables
Consider this "less advanced" proof:
Let $X:\Omega_X\to\mathbb{R}^n,Y:\Omega_Y\to\mathbb{R}^m,f:\mathbb{R}^n\to\mathbb{R}^k,g:\mathbb{R}^m\to\mathbb{R}^p$, where $X,Y$ are independent random variables and $f,g$ are measurable functions. Then:
$$
P\{f(X)\leq x \text{ and } g(Y)\... | Functions of Independent Random Variables
Consider this "less advanced" proof:
Let $X:\Omega_X\to\mathbb{R}^n,Y:\Omega_Y\to\mathbb{R}^m,f:\mathbb{R}^n\to\mathbb{R}^k,g:\mathbb{R}^m\to\mathbb{R}^p$, where $X,Y$ are independent random variables |
5,940 | Functions of Independent Random Variables | Yes, $g(X)$ and $h(Y)$ are independent for any functions $g$ and $h$ so long as $X$ and $Y$ are independent. It's a very well known results, which is studied in probability theory courses. I'm sure you can find it in any standard text like Billingsley's. | Functions of Independent Random Variables | Yes, $g(X)$ and $h(Y)$ are independent for any functions $g$ and $h$ so long as $X$ and $Y$ are independent. It's a very well known results, which is studied in probability theory courses. I'm sure yo | Functions of Independent Random Variables
Yes, $g(X)$ and $h(Y)$ are independent for any functions $g$ and $h$ so long as $X$ and $Y$ are independent. It's a very well known results, which is studied in probability theory courses. I'm sure you can find it in any standard text like Billingsley's. | Functions of Independent Random Variables
Yes, $g(X)$ and $h(Y)$ are independent for any functions $g$ and $h$ so long as $X$ and $Y$ are independent. It's a very well known results, which is studied in probability theory courses. I'm sure yo |
5,941 | Functions of Independent Random Variables | Not as an alternative, but as an addition, to the previous brilliant answers: the independence of functions of independent random variables is, in fact, quite intuitive.
Usually, we think that $X$ and $Y$ being independent means that knowing the value of $X$ gives no information about the value of $Y$ and vice versa. T... | Functions of Independent Random Variables | Not as an alternative, but as an addition, to the previous brilliant answers: the independence of functions of independent random variables is, in fact, quite intuitive.
Usually, we think that $X$ and | Functions of Independent Random Variables
Not as an alternative, but as an addition, to the previous brilliant answers: the independence of functions of independent random variables is, in fact, quite intuitive.
Usually, we think that $X$ and $Y$ being independent means that knowing the value of $X$ gives no informatio... | Functions of Independent Random Variables
Not as an alternative, but as an addition, to the previous brilliant answers: the independence of functions of independent random variables is, in fact, quite intuitive.
Usually, we think that $X$ and |
5,942 | Ridge, lasso and elastic net | In The Elements of Statistical Learning book, Hastie et al. provide a very insightful and thorough comparison of these shrinkage techniques. The book is available online (pdf). The comparison is done in section 3.4.3, page 69.
The main difference between Lasso and Ridge is the penalty term they use. Ridge uses $L_2$ pe... | Ridge, lasso and elastic net | In The Elements of Statistical Learning book, Hastie et al. provide a very insightful and thorough comparison of these shrinkage techniques. The book is available online (pdf). The comparison is done | Ridge, lasso and elastic net
In The Elements of Statistical Learning book, Hastie et al. provide a very insightful and thorough comparison of these shrinkage techniques. The book is available online (pdf). The comparison is done in section 3.4.3, page 69.
The main difference between Lasso and Ridge is the penalty term ... | Ridge, lasso and elastic net
In The Elements of Statistical Learning book, Hastie et al. provide a very insightful and thorough comparison of these shrinkage techniques. The book is available online (pdf). The comparison is done |
5,943 | Ridge, lasso and elastic net | To summarize, here are some salient differences between Lasso, Ridge and Elastic-net:
Lasso does a sparse selection, while Ridge does not.
When you have highly-correlated variables, Ridge regression shrinks the two coefficients towards one another. Lasso is somewhat indifferent and generally picks one over the other.... | Ridge, lasso and elastic net | To summarize, here are some salient differences between Lasso, Ridge and Elastic-net:
Lasso does a sparse selection, while Ridge does not.
When you have highly-correlated variables, Ridge regression | Ridge, lasso and elastic net
To summarize, here are some salient differences between Lasso, Ridge and Elastic-net:
Lasso does a sparse selection, while Ridge does not.
When you have highly-correlated variables, Ridge regression shrinks the two coefficients towards one another. Lasso is somewhat indifferent and genera... | Ridge, lasso and elastic net
To summarize, here are some salient differences between Lasso, Ridge and Elastic-net:
Lasso does a sparse selection, while Ridge does not.
When you have highly-correlated variables, Ridge regression |
5,944 | Ridge, lasso and elastic net | I highly recommended you to have a look at An introduction to statistical learning book (Tibshirani et. al, 2013).
The reason for this is that Elements of statistical learning book is intended for individuals with advanced
training in the mathematical sciences. In the foreword to ISL, the authors write:
An Introducti... | Ridge, lasso and elastic net | I highly recommended you to have a look at An introduction to statistical learning book (Tibshirani et. al, 2013).
The reason for this is that Elements of statistical learning book is intended for in | Ridge, lasso and elastic net
I highly recommended you to have a look at An introduction to statistical learning book (Tibshirani et. al, 2013).
The reason for this is that Elements of statistical learning book is intended for individuals with advanced
training in the mathematical sciences. In the foreword to ISL, the ... | Ridge, lasso and elastic net
I highly recommended you to have a look at An introduction to statistical learning book (Tibshirani et. al, 2013).
The reason for this is that Elements of statistical learning book is intended for in |
5,945 | Ridge, lasso and elastic net | The above answers are very clear and informative. I would like to add one minor point from the statistic perspective. Take the ridge regression as an example. It is an extension of the ordinal least square regression to solve the multicollinearity problems when there are many correlated features. If the linear regressi... | Ridge, lasso and elastic net | The above answers are very clear and informative. I would like to add one minor point from the statistic perspective. Take the ridge regression as an example. It is an extension of the ordinal least s | Ridge, lasso and elastic net
The above answers are very clear and informative. I would like to add one minor point from the statistic perspective. Take the ridge regression as an example. It is an extension of the ordinal least square regression to solve the multicollinearity problems when there are many correlated fea... | Ridge, lasso and elastic net
The above answers are very clear and informative. I would like to add one minor point from the statistic perspective. Take the ridge regression as an example. It is an extension of the ordinal least s |
5,946 | What is an instrumental variable? | [The following perhaps seems a little technical because of the use of equations but it builds mainly on the arrow charts to provide the intuition which only requires very basic understanding of OLS - so don't be repulsed.]
Suppose you want to estimate the causal effect of $x_i$ on $y_i$ given by the estimated coefficie... | What is an instrumental variable? | [The following perhaps seems a little technical because of the use of equations but it builds mainly on the arrow charts to provide the intuition which only requires very basic understanding of OLS - | What is an instrumental variable?
[The following perhaps seems a little technical because of the use of equations but it builds mainly on the arrow charts to provide the intuition which only requires very basic understanding of OLS - so don't be repulsed.]
Suppose you want to estimate the causal effect of $x_i$ on $y_i... | What is an instrumental variable?
[The following perhaps seems a little technical because of the use of equations but it builds mainly on the arrow charts to provide the intuition which only requires very basic understanding of OLS - |
5,947 | What is an instrumental variable? | As a medical statistician with no previous knowledge of econom(etr)ics, I struggled to get to grips with instrumental variables as I often struggled to follow their examples and didn't understand their rather different terminology (e.g. 'endogeneity', 'reduced form', 'structural equation', 'omitted variables'). Here's ... | What is an instrumental variable? | As a medical statistician with no previous knowledge of econom(etr)ics, I struggled to get to grips with instrumental variables as I often struggled to follow their examples and didn't understand thei | What is an instrumental variable?
As a medical statistician with no previous knowledge of econom(etr)ics, I struggled to get to grips with instrumental variables as I often struggled to follow their examples and didn't understand their rather different terminology (e.g. 'endogeneity', 'reduced form', 'structural equati... | What is an instrumental variable?
As a medical statistician with no previous knowledge of econom(etr)ics, I struggled to get to grips with instrumental variables as I often struggled to follow their examples and didn't understand thei |
5,948 | What is an instrumental variable? | 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.
Here are some slides that I prepared for an econometri... | What is an instrumental variable? | 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.
| What is an instrumental variable?
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.
Here are some slides... | What is an instrumental variable?
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.
|
5,949 | What is an instrumental variable? | Non-technical (usually that's all I'm good for anyway): There are times when not only does X cause Y, but Y causes X as well. An instrumental variable is a device that can "clean up" this messy, inconvenient relationship so that the best estimates can be made of X's effect on Y.
The instrumental variable is chosen ... | What is an instrumental variable? | Non-technical (usually that's all I'm good for anyway): There are times when not only does X cause Y, but Y causes X as well. An instrumental variable is a device that can "clean up" this messy, inc | What is an instrumental variable?
Non-technical (usually that's all I'm good for anyway): There are times when not only does X cause Y, but Y causes X as well. An instrumental variable is a device that can "clean up" this messy, inconvenient relationship so that the best estimates can be made of X's effect on Y.
Th... | What is an instrumental variable?
Non-technical (usually that's all I'm good for anyway): There are times when not only does X cause Y, but Y causes X as well. An instrumental variable is a device that can "clean up" this messy, inc |
5,950 | Were generative adversarial networks introduced by Jürgen Schmidhuber? | I self-published the basic idea of a deterministic variety of generative adversarial networks (GANs) in a 2010 blog post (archive.org). I had searched for but could not find anything similar anywhere, and had no time to try implementing it. I was not and still am not a neural network researcher and have no connections ... | Were generative adversarial networks introduced by Jürgen Schmidhuber? | I self-published the basic idea of a deterministic variety of generative adversarial networks (GANs) in a 2010 blog post (archive.org). I had searched for but could not find anything similar anywhere, | Were generative adversarial networks introduced by Jürgen Schmidhuber?
I self-published the basic idea of a deterministic variety of generative adversarial networks (GANs) in a 2010 blog post (archive.org). I had searched for but could not find anything similar anywhere, and had no time to try implementing it. I was no... | Were generative adversarial networks introduced by Jürgen Schmidhuber?
I self-published the basic idea of a deterministic variety of generative adversarial networks (GANs) in a 2010 blog post (archive.org). I had searched for but could not find anything similar anywhere, |
5,951 | Were generative adversarial networks introduced by Jürgen Schmidhuber? | An answer from Ian Goodfellow on Was Jürgen Schmidhuber right when he claimed credit for GANs at NIPS 2016? posted on 2017-03-21:
He isn’t claiming credit for GANs, exactly. It’s more complicated.
You can see what he wrote in his own words when he was a reviewer of
the NIPS 2014 submission on GANs: Export Reviews,... | Were generative adversarial networks introduced by Jürgen Schmidhuber? | An answer from Ian Goodfellow on Was Jürgen Schmidhuber right when he claimed credit for GANs at NIPS 2016? posted on 2017-03-21:
He isn’t claiming credit for GANs, exactly. It’s more complicated.
| Were generative adversarial networks introduced by Jürgen Schmidhuber?
An answer from Ian Goodfellow on Was Jürgen Schmidhuber right when he claimed credit for GANs at NIPS 2016? posted on 2017-03-21:
He isn’t claiming credit for GANs, exactly. It’s more complicated.
You can see what he wrote in his own words when h... | Were generative adversarial networks introduced by Jürgen Schmidhuber?
An answer from Ian Goodfellow on Was Jürgen Schmidhuber right when he claimed credit for GANs at NIPS 2016? posted on 2017-03-21:
He isn’t claiming credit for GANs, exactly. It’s more complicated.
|
5,952 | Were generative adversarial networks introduced by Jürgen Schmidhuber? | This is taken straight out from the 1991 original paper of Schmidhuber in 1991:
"I propose a novel general principle for unsupervised learning of distributed non-redundant internal representations of input patterns. The principle is based on two opposing forces. For each representational unit there is an adaptive predi... | Were generative adversarial networks introduced by Jürgen Schmidhuber? | This is taken straight out from the 1991 original paper of Schmidhuber in 1991:
"I propose a novel general principle for unsupervised learning of distributed non-redundant internal representations of | Were generative adversarial networks introduced by Jürgen Schmidhuber?
This is taken straight out from the 1991 original paper of Schmidhuber in 1991:
"I propose a novel general principle for unsupervised learning of distributed non-redundant internal representations of input patterns. The principle is based on two opp... | Were generative adversarial networks introduced by Jürgen Schmidhuber?
This is taken straight out from the 1991 original paper of Schmidhuber in 1991:
"I propose a novel general principle for unsupervised learning of distributed non-redundant internal representations of |
5,953 | Can you explain Parzen window (kernel) density estimation in layman's terms? | 1) My understanding is that users have a choice of functions to use for $\phi$, and that the Gaussian function is a very common choice.
2) The density at $x$ is the mean of the different values of $\phi_h(x_i - x)$ at $x$. For example, you might have $x_1=1$, $x_2 = 2$, and a Gaussian distribution with $\sigma=1$ for $... | Can you explain Parzen window (kernel) density estimation in layman's terms? | 1) My understanding is that users have a choice of functions to use for $\phi$, and that the Gaussian function is a very common choice.
2) The density at $x$ is the mean of the different values of $\p | Can you explain Parzen window (kernel) density estimation in layman's terms?
1) My understanding is that users have a choice of functions to use for $\phi$, and that the Gaussian function is a very common choice.
2) The density at $x$ is the mean of the different values of $\phi_h(x_i - x)$ at $x$. For example, you mig... | Can you explain Parzen window (kernel) density estimation in layman's terms?
1) My understanding is that users have a choice of functions to use for $\phi$, and that the Gaussian function is a very common choice.
2) The density at $x$ is the mean of the different values of $\p |
5,954 | Can you explain Parzen window (kernel) density estimation in layman's terms? | Parzen window density estimation is another name for kernel density estimation. It is a nonparametric method for estimating continuous density function from the data.
Imagine that you have some datapoints $x_1,\dots,x_n$ that come from common unknown, presumably continuous, distribution $f$. You are interested in estim... | Can you explain Parzen window (kernel) density estimation in layman's terms? | Parzen window density estimation is another name for kernel density estimation. It is a nonparametric method for estimating continuous density function from the data.
Imagine that you have some datapo | Can you explain Parzen window (kernel) density estimation in layman's terms?
Parzen window density estimation is another name for kernel density estimation. It is a nonparametric method for estimating continuous density function from the data.
Imagine that you have some datapoints $x_1,\dots,x_n$ that come from common ... | Can you explain Parzen window (kernel) density estimation in layman's terms?
Parzen window density estimation is another name for kernel density estimation. It is a nonparametric method for estimating continuous density function from the data.
Imagine that you have some datapo |
5,955 | Calculating confidence intervals for a logistic regression | Your question may come from the fact that you are dealing with Odds Ratios and Probabilities which is confusing at first. Since the logistic model is a non linear transformation of $\beta^Tx$ computing the confidence intervals is not as straightforward.
Background
Recall that for the Logistic regression model
Probabil... | Calculating confidence intervals for a logistic regression | Your question may come from the fact that you are dealing with Odds Ratios and Probabilities which is confusing at first. Since the logistic model is a non linear transformation of $\beta^Tx$ computin | Calculating confidence intervals for a logistic regression
Your question may come from the fact that you are dealing with Odds Ratios and Probabilities which is confusing at first. Since the logistic model is a non linear transformation of $\beta^Tx$ computing the confidence intervals is not as straightforward.
Backgro... | Calculating confidence intervals for a logistic regression
Your question may come from the fact that you are dealing with Odds Ratios and Probabilities which is confusing at first. Since the logistic model is a non linear transformation of $\beta^Tx$ computin |
5,956 | Calculating confidence intervals for a logistic regression | To get the 95% confidence interval of the prediction you can calculate on the logit scale and then convert those back to the probability scale 0-1. Here is an example using the titanic dataset.
library(titanic)
data("titanic_train")
titanic_train$Pclass = factor(titanic_train$Pclass, levels = c(1,2,3), labels = c('Fi... | Calculating confidence intervals for a logistic regression | To get the 95% confidence interval of the prediction you can calculate on the logit scale and then convert those back to the probability scale 0-1. Here is an example using the titanic dataset.
libra | Calculating confidence intervals for a logistic regression
To get the 95% confidence interval of the prediction you can calculate on the logit scale and then convert those back to the probability scale 0-1. Here is an example using the titanic dataset.
library(titanic)
data("titanic_train")
titanic_train$Pclass = fac... | Calculating confidence intervals for a logistic regression
To get the 95% confidence interval of the prediction you can calculate on the logit scale and then convert those back to the probability scale 0-1. Here is an example using the titanic dataset.
libra |
5,957 | Calculating confidence intervals for a logistic regression | The explanations above are very nice and detailed.
Here is the simple way to just get the result:
Random data frame:
dat <- data.frame(
trial = factor(c(rep("noise", 340), rep("signal", 340))),
rating = c(rep(1, 70), rep(2, 67), rep(3, 28),
rep(4, 45), rep(5, 53), rep(6, 77), rep(1... | Calculating confidence intervals for a logistic regression | The explanations above are very nice and detailed.
Here is the simple way to just get the result:
Random data frame:
dat <- data.frame(
trial = factor(c(rep("noise", 340), rep("signal", 340) | Calculating confidence intervals for a logistic regression
The explanations above are very nice and detailed.
Here is the simple way to just get the result:
Random data frame:
dat <- data.frame(
trial = factor(c(rep("noise", 340), rep("signal", 340))),
rating = c(rep(1, 70), rep(2, 67), rep(3, 28), ... | Calculating confidence intervals for a logistic regression
The explanations above are very nice and detailed.
Here is the simple way to just get the result:
Random data frame:
dat <- data.frame(
trial = factor(c(rep("noise", 340), rep("signal", 340) |
5,958 | Do Bayesian priors become irrelevant with large sample size? | It is not that easy. Information in your data overwhelms prior information not only when your sample size is large, but also when your data provides enough information to overwhelm the prior information. Uninformative priors get easily persuaded by data, while strongly informative ones may be more resistant. In extreme... | Do Bayesian priors become irrelevant with large sample size? | It is not that easy. Information in your data overwhelms prior information not only when your sample size is large, but also when your data provides enough information to overwhelm the prior informati | Do Bayesian priors become irrelevant with large sample size?
It is not that easy. Information in your data overwhelms prior information not only when your sample size is large, but also when your data provides enough information to overwhelm the prior information. Uninformative priors get easily persuaded by data, whil... | Do Bayesian priors become irrelevant with large sample size?
It is not that easy. Information in your data overwhelms prior information not only when your sample size is large, but also when your data provides enough information to overwhelm the prior informati |
5,959 | Do Bayesian priors become irrelevant with large sample size? | When performing Bayesian inference, we operate by maximizing our likelihood function in combination with the priors we have about the parameters.
This is actually not what most practitioners consider to be Bayesian inference. It is possible to estimate parameters this way, but I would not call it Bayesian inference.
... | Do Bayesian priors become irrelevant with large sample size? | When performing Bayesian inference, we operate by maximizing our likelihood function in combination with the priors we have about the parameters.
This is actually not what most practitioners consider | Do Bayesian priors become irrelevant with large sample size?
When performing Bayesian inference, we operate by maximizing our likelihood function in combination with the priors we have about the parameters.
This is actually not what most practitioners consider to be Bayesian inference. It is possible to estimate para... | Do Bayesian priors become irrelevant with large sample size?
When performing Bayesian inference, we operate by maximizing our likelihood function in combination with the priors we have about the parameters.
This is actually not what most practitioners consider |
5,960 | Do Bayesian priors become irrelevant with large sample size? | Another issue to keep in mind is you can have a lot of data, but still have very little information about certain parameters in your model. In such cases, even a mildly informative prior can be extremely helpful when performing inference.
As a silly example, suppose you were comparing means of two groups and you had 1... | Do Bayesian priors become irrelevant with large sample size? | Another issue to keep in mind is you can have a lot of data, but still have very little information about certain parameters in your model. In such cases, even a mildly informative prior can be extrem | Do Bayesian priors become irrelevant with large sample size?
Another issue to keep in mind is you can have a lot of data, but still have very little information about certain parameters in your model. In such cases, even a mildly informative prior can be extremely helpful when performing inference.
As a silly example,... | Do Bayesian priors become irrelevant with large sample size?
Another issue to keep in mind is you can have a lot of data, but still have very little information about certain parameters in your model. In such cases, even a mildly informative prior can be extrem |
5,961 | What is the significance of logistic regression coefficients? | That the author has forced someone as thoughtful as you to have ask a question like this is compelling illustration of why the practice -- still way too common -- of confining reporting of regression model results to a table like this is so unacceptable.
You can, as pointed out, try to transform the logit coefficient... | What is the significance of logistic regression coefficients? | That the author has forced someone as thoughtful as you to have ask a question like this is compelling illustration of why the practice -- still way too common -- of confining reporting of regression | What is the significance of logistic regression coefficients?
That the author has forced someone as thoughtful as you to have ask a question like this is compelling illustration of why the practice -- still way too common -- of confining reporting of regression model results to a table like this is so unacceptable.
Y... | What is the significance of logistic regression coefficients?
That the author has forced someone as thoughtful as you to have ask a question like this is compelling illustration of why the practice -- still way too common -- of confining reporting of regression |
5,962 | What is the significance of logistic regression coefficients? | The idea here is that in logistic regression, we predict not the actual probability that, say, a southerner votes Republican, but a transformed version of it, the "log odds". Instead of the probability $p$, we deal with $\log p/(1-p)$ and find linear regression coefficients for the log odds.
So for example, let's assum... | What is the significance of logistic regression coefficients? | The idea here is that in logistic regression, we predict not the actual probability that, say, a southerner votes Republican, but a transformed version of it, the "log odds". Instead of the probabilit | What is the significance of logistic regression coefficients?
The idea here is that in logistic regression, we predict not the actual probability that, say, a southerner votes Republican, but a transformed version of it, the "log odds". Instead of the probability $p$, we deal with $\log p/(1-p)$ and find linear regress... | What is the significance of logistic regression coefficients?
The idea here is that in logistic regression, we predict not the actual probability that, say, a southerner votes Republican, but a transformed version of it, the "log odds". Instead of the probabilit |
5,963 | What is the significance of logistic regression coefficients? | The coefficients in the logistic regression represent the tendency for a given region/demographic to vote Republican, compared to a reference category. A positive coefficent means that region is more likely to vote Republican, and vice-versa for a negative coefficient; a larger absolute value means a stronger tendency ... | What is the significance of logistic regression coefficients? | The coefficients in the logistic regression represent the tendency for a given region/demographic to vote Republican, compared to a reference category. A positive coefficent means that region is more | What is the significance of logistic regression coefficients?
The coefficients in the logistic regression represent the tendency for a given region/demographic to vote Republican, compared to a reference category. A positive coefficent means that region is more likely to vote Republican, and vice-versa for a negative c... | What is the significance of logistic regression coefficients?
The coefficients in the logistic regression represent the tendency for a given region/demographic to vote Republican, compared to a reference category. A positive coefficent means that region is more |
5,964 | What is the significance of logistic regression coefficients? | You also asked "how do I know what is significant and what is not." (I assume you mean statistically significant, since practical or substantive significance is another matter.) The asterisks in the table refer to the footnote: some effects are noted as having small p-values. These are obtained using a Wald test of... | What is the significance of logistic regression coefficients? | You also asked "how do I know what is significant and what is not." (I assume you mean statistically significant, since practical or substantive significance is another matter.) The asterisks in th | What is the significance of logistic regression coefficients?
You also asked "how do I know what is significant and what is not." (I assume you mean statistically significant, since practical or substantive significance is another matter.) The asterisks in the table refer to the footnote: some effects are noted as ... | What is the significance of logistic regression coefficients?
You also asked "how do I know what is significant and what is not." (I assume you mean statistically significant, since practical or substantive significance is another matter.) The asterisks in th |
5,965 | What is the significance of logistic regression coefficients? | Let me just stress the importance of what rolando2 and dmk38 both noted: significance is commonly misread, and there is a high risk of that happening with that tabular presentation of results.
Paul Schrodt recently offered a nice description of the issue:
Researchers find it nearly impossible to adhere to the correct ... | What is the significance of logistic regression coefficients? | Let me just stress the importance of what rolando2 and dmk38 both noted: significance is commonly misread, and there is a high risk of that happening with that tabular presentation of results.
Paul Sc | What is the significance of logistic regression coefficients?
Let me just stress the importance of what rolando2 and dmk38 both noted: significance is commonly misread, and there is a high risk of that happening with that tabular presentation of results.
Paul Schrodt recently offered a nice description of the issue:
R... | What is the significance of logistic regression coefficients?
Let me just stress the importance of what rolando2 and dmk38 both noted: significance is commonly misread, and there is a high risk of that happening with that tabular presentation of results.
Paul Sc |
5,966 | What is the purpose of characteristic functions? | Back in the day, people used logarithm tables to multiply numbers faster. Why is this? Logarithms convert multiplication to addition, since $\log(ab) = \log(a) + \log(b)$. So in order to multiply two large numbers $a$ and $b$, you found their logarithms, added the logarithms, $z = \log(a) + \log(b)$, and then looked... | What is the purpose of characteristic functions? | Back in the day, people used logarithm tables to multiply numbers faster. Why is this? Logarithms convert multiplication to addition, since $\log(ab) = \log(a) + \log(b)$. So in order to multiply t | What is the purpose of characteristic functions?
Back in the day, people used logarithm tables to multiply numbers faster. Why is this? Logarithms convert multiplication to addition, since $\log(ab) = \log(a) + \log(b)$. So in order to multiply two large numbers $a$ and $b$, you found their logarithms, added the log... | What is the purpose of characteristic functions?
Back in the day, people used logarithm tables to multiply numbers faster. Why is this? Logarithms convert multiplication to addition, since $\log(ab) = \log(a) + \log(b)$. So in order to multiply t |
5,967 | What is the purpose of characteristic functions? | @charles.y.zheng and @cardinal gave very good answers, I will add my two cents. Yes the characteristic function might look like unnecessary complication, but it is a powerful tool which can get you results. If you are trying to prove something with cumulative distribution function it is always advisable to check whethe... | What is the purpose of characteristic functions? | @charles.y.zheng and @cardinal gave very good answers, I will add my two cents. Yes the characteristic function might look like unnecessary complication, but it is a powerful tool which can get you re | What is the purpose of characteristic functions?
@charles.y.zheng and @cardinal gave very good answers, I will add my two cents. Yes the characteristic function might look like unnecessary complication, but it is a powerful tool which can get you results. If you are trying to prove something with cumulative distributio... | What is the purpose of characteristic functions?
@charles.y.zheng and @cardinal gave very good answers, I will add my two cents. Yes the characteristic function might look like unnecessary complication, but it is a powerful tool which can get you re |
5,968 | What is the purpose of characteristic functions? | The purpose of characteristic functions is that they can be used to derive the properties of distributions in probability theory. If you're not interested in such derivations you do not need to learn about characteristic functions. | What is the purpose of characteristic functions? | The purpose of characteristic functions is that they can be used to derive the properties of distributions in probability theory. If you're not interested in such derivations you do not need to learn | What is the purpose of characteristic functions?
The purpose of characteristic functions is that they can be used to derive the properties of distributions in probability theory. If you're not interested in such derivations you do not need to learn about characteristic functions. | What is the purpose of characteristic functions?
The purpose of characteristic functions is that they can be used to derive the properties of distributions in probability theory. If you're not interested in such derivations you do not need to learn |
5,969 | What is the purpose of characteristic functions? | The characteristic function is the Fourier transform of the density function of the distribution. If you have any intuition regarding Fourier transforms, this fact may be enlightening. The common story about Fourier transforms is that they describe the function 'in frequency space.' Since a probability density is usua... | What is the purpose of characteristic functions? | The characteristic function is the Fourier transform of the density function of the distribution. If you have any intuition regarding Fourier transforms, this fact may be enlightening. The common sto | What is the purpose of characteristic functions?
The characteristic function is the Fourier transform of the density function of the distribution. If you have any intuition regarding Fourier transforms, this fact may be enlightening. The common story about Fourier transforms is that they describe the function 'in freq... | What is the purpose of characteristic functions?
The characteristic function is the Fourier transform of the density function of the distribution. If you have any intuition regarding Fourier transforms, this fact may be enlightening. The common sto |
5,970 | What is the purpose of characteristic functions? | The Fourier transformation is a decomposition of the function (non-periodic) in its frequencies. Interpretation for densities?
Fourier transformation is the continuous version of a Fourier series since no density is periodic no expression like "characteristic series". | What is the purpose of characteristic functions? | The Fourier transformation is a decomposition of the function (non-periodic) in its frequencies. Interpretation for densities?
Fourier transformation is the continuous version of a Fourier series sinc | What is the purpose of characteristic functions?
The Fourier transformation is a decomposition of the function (non-periodic) in its frequencies. Interpretation for densities?
Fourier transformation is the continuous version of a Fourier series since no density is periodic no expression like "characteristic series". | What is the purpose of characteristic functions?
The Fourier transformation is a decomposition of the function (non-periodic) in its frequencies. Interpretation for densities?
Fourier transformation is the continuous version of a Fourier series sinc |
5,971 | What are best practices in identifying interaction effects? | Cox and Wermuth (1996) or Cox (1984) discussed some methods for detecting interactions. The problem is usually how general the interaction terms should be. Basically, we
(a) fit (and test) all second-order interaction terms, one at a time, and (b) plot their corresponding p-values (i.e., the No. terms as a function of... | What are best practices in identifying interaction effects? | Cox and Wermuth (1996) or Cox (1984) discussed some methods for detecting interactions. The problem is usually how general the interaction terms should be. Basically, we
(a) fit (and test) all second | What are best practices in identifying interaction effects?
Cox and Wermuth (1996) or Cox (1984) discussed some methods for detecting interactions. The problem is usually how general the interaction terms should be. Basically, we
(a) fit (and test) all second-order interaction terms, one at a time, and (b) plot their ... | What are best practices in identifying interaction effects?
Cox and Wermuth (1996) or Cox (1984) discussed some methods for detecting interactions. The problem is usually how general the interaction terms should be. Basically, we
(a) fit (and test) all second |
5,972 | What are best practices in identifying interaction effects? | My best practice would be to think about the problem to hand before fitting the model. What is a plausible model given the phenomenon you are studying? Fitting all possible combinations of variables and interactions sounds like data dredging to me. | What are best practices in identifying interaction effects? | My best practice would be to think about the problem to hand before fitting the model. What is a plausible model given the phenomenon you are studying? Fitting all possible combinations of variables a | What are best practices in identifying interaction effects?
My best practice would be to think about the problem to hand before fitting the model. What is a plausible model given the phenomenon you are studying? Fitting all possible combinations of variables and interactions sounds like data dredging to me. | What are best practices in identifying interaction effects?
My best practice would be to think about the problem to hand before fitting the model. What is a plausible model given the phenomenon you are studying? Fitting all possible combinations of variables a |
5,973 | What are best practices in identifying interaction effects? | Fitting a tree model (i.e. using R), will help you identify complex interactions between the explanatory variables. Read the example on page 30 here. | What are best practices in identifying interaction effects? | Fitting a tree model (i.e. using R), will help you identify complex interactions between the explanatory variables. Read the example on page 30 here. | What are best practices in identifying interaction effects?
Fitting a tree model (i.e. using R), will help you identify complex interactions between the explanatory variables. Read the example on page 30 here. | What are best practices in identifying interaction effects?
Fitting a tree model (i.e. using R), will help you identify complex interactions between the explanatory variables. Read the example on page 30 here. |
5,974 | What are best practices in identifying interaction effects? | I'll preface this response as I entirely agree with Gavin, and if you're interested in fitting any type of model it should be reflective of the phenomenon under study. What the problem is with the logic of identifying any and all effects (and what Gavin refers to when he says data dredging) is that you could fit an inf... | What are best practices in identifying interaction effects? | I'll preface this response as I entirely agree with Gavin, and if you're interested in fitting any type of model it should be reflective of the phenomenon under study. What the problem is with the log | What are best practices in identifying interaction effects?
I'll preface this response as I entirely agree with Gavin, and if you're interested in fitting any type of model it should be reflective of the phenomenon under study. What the problem is with the logic of identifying any and all effects (and what Gavin refers... | What are best practices in identifying interaction effects?
I'll preface this response as I entirely agree with Gavin, and if you're interested in fitting any type of model it should be reflective of the phenomenon under study. What the problem is with the log |
5,975 | What are best practices in identifying interaction effects? | How large is $n$ ? how many observations do you have ? this is crucial ...
Sobol indices will tell you the proportion of variance explained by interaction if you have a lot of observations and a few $n$, otherwise you will have to do modelling (linear to start with). You have a nice R package for that called sensitivi... | What are best practices in identifying interaction effects? | How large is $n$ ? how many observations do you have ? this is crucial ...
Sobol indices will tell you the proportion of variance explained by interaction if you have a lot of observations and a few | What are best practices in identifying interaction effects?
How large is $n$ ? how many observations do you have ? this is crucial ...
Sobol indices will tell you the proportion of variance explained by interaction if you have a lot of observations and a few $n$, otherwise you will have to do modelling (linear to star... | What are best practices in identifying interaction effects?
How large is $n$ ? how many observations do you have ? this is crucial ...
Sobol indices will tell you the proportion of variance explained by interaction if you have a lot of observations and a few |
5,976 | What are best practices in identifying interaction effects? | I think this is a good use case of LASSO. You can throw all the interaction terms, and LASSO can find the ones that matter, by using cross-validation to select the best regularization parameter. This doesn't need to be confined to linear interaction, as we can add much richer class of interactions (e.g. $x_m^2 * x_n$ t... | What are best practices in identifying interaction effects? | I think this is a good use case of LASSO. You can throw all the interaction terms, and LASSO can find the ones that matter, by using cross-validation to select the best regularization parameter. This | What are best practices in identifying interaction effects?
I think this is a good use case of LASSO. You can throw all the interaction terms, and LASSO can find the ones that matter, by using cross-validation to select the best regularization parameter. This doesn't need to be confined to linear interaction, as we can... | What are best practices in identifying interaction effects?
I think this is a good use case of LASSO. You can throw all the interaction terms, and LASSO can find the ones that matter, by using cross-validation to select the best regularization parameter. This |
5,977 | Why is a likelihood-ratio test distributed chi-squared? | As mentioned by @Nick this is a consequence of Wilks' theorem. But note that the test statistic is asymptotically $\chi^2$-distributed, not $\chi^2$-distributed.
I am very impressed by this theorem because it holds in a very wide context. Consider a statistical model with likelihood $l(\theta \mid y)$ where $y$ is the ... | Why is a likelihood-ratio test distributed chi-squared? | As mentioned by @Nick this is a consequence of Wilks' theorem. But note that the test statistic is asymptotically $\chi^2$-distributed, not $\chi^2$-distributed.
I am very impressed by this theorem be | Why is a likelihood-ratio test distributed chi-squared?
As mentioned by @Nick this is a consequence of Wilks' theorem. But note that the test statistic is asymptotically $\chi^2$-distributed, not $\chi^2$-distributed.
I am very impressed by this theorem because it holds in a very wide context. Consider a statistical mo... | Why is a likelihood-ratio test distributed chi-squared?
As mentioned by @Nick this is a consequence of Wilks' theorem. But note that the test statistic is asymptotically $\chi^2$-distributed, not $\chi^2$-distributed.
I am very impressed by this theorem be |
5,978 | Why is a likelihood-ratio test distributed chi-squared? | I second Nick Sabbe's harsh comment, and my short answer is, It is not. I mean, it only is in the normal linear model. For absolutely any other sort of circumstances, the exact distribution is not a $\chi^2$. In many situations, you can hope that Wilks' theorem conditions are satisfied, and then asymptotically the log-... | Why is a likelihood-ratio test distributed chi-squared? | I second Nick Sabbe's harsh comment, and my short answer is, It is not. I mean, it only is in the normal linear model. For absolutely any other sort of circumstances, the exact distribution is not a $ | Why is a likelihood-ratio test distributed chi-squared?
I second Nick Sabbe's harsh comment, and my short answer is, It is not. I mean, it only is in the normal linear model. For absolutely any other sort of circumstances, the exact distribution is not a $\chi^2$. In many situations, you can hope that Wilks' theorem co... | Why is a likelihood-ratio test distributed chi-squared?
I second Nick Sabbe's harsh comment, and my short answer is, It is not. I mean, it only is in the normal linear model. For absolutely any other sort of circumstances, the exact distribution is not a $ |
5,979 | Why is a likelihood-ratio test distributed chi-squared? | As other commentators have pointed out, Wilks' theorem (Wilks 1938) only shows that, under various regularity conditions, this statistic is asymptotically chi-squared distributed. The asymptotic result follows from taking a multivariate Taylor expansion of the log-likelihood function and looking at what happens when t... | Why is a likelihood-ratio test distributed chi-squared? | As other commentators have pointed out, Wilks' theorem (Wilks 1938) only shows that, under various regularity conditions, this statistic is asymptotically chi-squared distributed. The asymptotic resu | Why is a likelihood-ratio test distributed chi-squared?
As other commentators have pointed out, Wilks' theorem (Wilks 1938) only shows that, under various regularity conditions, this statistic is asymptotically chi-squared distributed. The asymptotic result follows from taking a multivariate Taylor expansion of the lo... | Why is a likelihood-ratio test distributed chi-squared?
As other commentators have pointed out, Wilks' theorem (Wilks 1938) only shows that, under various regularity conditions, this statistic is asymptotically chi-squared distributed. The asymptotic resu |
5,980 | 10-fold Cross-validation vs leave-one-out cross-validation | Just to add slightly to the answer of @SubravetiSuraj (+1)
Cross-validation gives a pessimistically biased estimate of performance because most statistical models will improve if the training set is made larger. This means that k-fold cross-validation estimates the performance of a model trained on a dataset $100\time... | 10-fold Cross-validation vs leave-one-out cross-validation | Just to add slightly to the answer of @SubravetiSuraj (+1)
Cross-validation gives a pessimistically biased estimate of performance because most statistical models will improve if the training set is m | 10-fold Cross-validation vs leave-one-out cross-validation
Just to add slightly to the answer of @SubravetiSuraj (+1)
Cross-validation gives a pessimistically biased estimate of performance because most statistical models will improve if the training set is made larger. This means that k-fold cross-validation estimate... | 10-fold Cross-validation vs leave-one-out cross-validation
Just to add slightly to the answer of @SubravetiSuraj (+1)
Cross-validation gives a pessimistically biased estimate of performance because most statistical models will improve if the training set is m |
5,981 | 10-fold Cross-validation vs leave-one-out cross-validation | In my opinion, leave one out cross validation is better when you have a small set of training data. In this case, you can't really make 10 folds to make predictions on using the rest of your data to train the model.
If you have a large amount of training data on the other hand, 10-fold cross validation would be a bett... | 10-fold Cross-validation vs leave-one-out cross-validation | In my opinion, leave one out cross validation is better when you have a small set of training data. In this case, you can't really make 10 folds to make predictions on using the rest of your data to t | 10-fold Cross-validation vs leave-one-out cross-validation
In my opinion, leave one out cross validation is better when you have a small set of training data. In this case, you can't really make 10 folds to make predictions on using the rest of your data to train the model.
If you have a large amount of training data ... | 10-fold Cross-validation vs leave-one-out cross-validation
In my opinion, leave one out cross validation is better when you have a small set of training data. In this case, you can't really make 10 folds to make predictions on using the rest of your data to t |
5,982 | 10-fold Cross-validation vs leave-one-out cross-validation | The existing answers focus on getting good estimates of the out of sample prediction error. This is not the only perspective on the LOOCV versus K-fold CV decision. In particular, some readers may believe there is a true model and may wish to recover it.
In this case, Shao 1993 famously showed that for linear models, L... | 10-fold Cross-validation vs leave-one-out cross-validation | The existing answers focus on getting good estimates of the out of sample prediction error. This is not the only perspective on the LOOCV versus K-fold CV decision. In particular, some readers may bel | 10-fold Cross-validation vs leave-one-out cross-validation
The existing answers focus on getting good estimates of the out of sample prediction error. This is not the only perspective on the LOOCV versus K-fold CV decision. In particular, some readers may believe there is a true model and may wish to recover it.
In thi... | 10-fold Cross-validation vs leave-one-out cross-validation
The existing answers focus on getting good estimates of the out of sample prediction error. This is not the only perspective on the LOOCV versus K-fold CV decision. In particular, some readers may bel |
5,983 | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ($L^1$)? | 1. Which method is preferred?
Yes, elastic net is always preferred over lasso & ridge regression because it solves the limitations of both methods, while also including each as special cases. So if the ridge or lasso solution is, indeed, the best, then any good model selection routine will identify that as part of the ... | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ( | 1. Which method is preferred?
Yes, elastic net is always preferred over lasso & ridge regression because it solves the limitations of both methods, while also including each as special cases. So if th | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ($L^1$)?
1. Which method is preferred?
Yes, elastic net is always preferred over lasso & ridge regression because it solves the limitations of both methods, while also including each as special cases. So if the ridge or l... | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso (
1. Which method is preferred?
Yes, elastic net is always preferred over lasso & ridge regression because it solves the limitations of both methods, while also including each as special cases. So if th |
5,984 | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ($L^1$)? | I generally agree with the answer by @Sycorax, but I would like to add some qualification.
Saying that "elastic net is always preferred over lasso & ridge regression" may be a little too strong. In small or medium samples elastic net may not select pure LASSO or pure ridge solution even if the former or the latter is a... | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ( | I generally agree with the answer by @Sycorax, but I would like to add some qualification.
Saying that "elastic net is always preferred over lasso & ridge regression" may be a little too strong. In sm | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ($L^1$)?
I generally agree with the answer by @Sycorax, but I would like to add some qualification.
Saying that "elastic net is always preferred over lasso & ridge regression" may be a little too strong. In small or mediu... | What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso (
I generally agree with the answer by @Sycorax, but I would like to add some qualification.
Saying that "elastic net is always preferred over lasso & ridge regression" may be a little too strong. In sm |
5,985 | Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data | The reason that the $R^2$ values are not matching is because randomForest is reporting variation explained as opposed to variance explained. I think this is a common misunderstanding about $R^2$ that is perpetuated in textbooks. I even mentioned this on another thread the other day. If you want an example, see the (oth... | Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data | The reason that the $R^2$ values are not matching is because randomForest is reporting variation explained as opposed to variance explained. I think this is a common misunderstanding about $R^2$ that | Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data
The reason that the $R^2$ values are not matching is because randomForest is reporting variation explained as opposed to variance explained. I think this is a common misunderstanding about $R^2$ that is perpetuated in textbooks. I... | Manually calculated $R^2$ doesn't match up with randomForest() $R^2$ for testing new data
The reason that the $R^2$ values are not matching is because randomForest is reporting variation explained as opposed to variance explained. I think this is a common misunderstanding about $R^2$ that |
5,986 | Why is logistic regression a linear model? | The logistic regression model is of the form
$$
\mathrm{logit}(p_i) = \mathrm{ln}\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_p x_{p,i}.
$$
It is called a generalized linear model not because the estimated probability of the response event is linear, but because the logi... | Why is logistic regression a linear model? | The logistic regression model is of the form
$$
\mathrm{logit}(p_i) = \mathrm{ln}\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_p x_{p,i}.
$$
It is calle | Why is logistic regression a linear model?
The logistic regression model is of the form
$$
\mathrm{logit}(p_i) = \mathrm{ln}\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_p x_{p,i}.
$$
It is called a generalized linear model not because the estimated probability of the res... | Why is logistic regression a linear model?
The logistic regression model is of the form
$$
\mathrm{logit}(p_i) = \mathrm{ln}\left(\frac{p_i}{1-p_i}\right) = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \cdots + \beta_p x_{p,i}.
$$
It is calle |
5,987 | Why is logistic regression a linear model? | Logistic regression uses the general linear equation $Y=b_0+∑(b_i X_i)+\epsilon$. In linear regression $Y$ is a continuous dependent variable, but in logistic regression it is regressing for the probability of a categorical outcome (for example 0 and 1).
The probability of $Y=1$ is:
$$
P(Y=1) = {1 \over 1+e^{-(b_0+\sum... | Why is logistic regression a linear model? | Logistic regression uses the general linear equation $Y=b_0+∑(b_i X_i)+\epsilon$. In linear regression $Y$ is a continuous dependent variable, but in logistic regression it is regressing for the proba | Why is logistic regression a linear model?
Logistic regression uses the general linear equation $Y=b_0+∑(b_i X_i)+\epsilon$. In linear regression $Y$ is a continuous dependent variable, but in logistic regression it is regressing for the probability of a categorical outcome (for example 0 and 1).
The probability of $Y=... | Why is logistic regression a linear model?
Logistic regression uses the general linear equation $Y=b_0+∑(b_i X_i)+\epsilon$. In linear regression $Y$ is a continuous dependent variable, but in logistic regression it is regressing for the proba |
5,988 | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model | Below are those residual plots with the approximate mean and spread of points (limits that include most of the values) at each value of fitted (and hence of $x$) marked in - to a rough approximation indicating the conditional mean (red) and conditional mean $\pm$ (roughly!) twice the conditional standard deviation (pur... | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model | Below are those residual plots with the approximate mean and spread of points (limits that include most of the values) at each value of fitted (and hence of $x$) marked in - to a rough approximation i | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model
Below are those residual plots with the approximate mean and spread of points (limits that include most of the values) at each value of fitted (and hence of $x$) marked in - to a rough approximation indicating the conditio... | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model
Below are those residual plots with the approximate mean and spread of points (limits that include most of the values) at each value of fitted (and hence of $x$) marked in - to a rough approximation i |
5,989 | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model | You wrote
The second plot seems to indicate that the absolute value of the
residuals is strongly positively correlated with the fitted values,
It doesn't "seem" to, it does. And that's what heteroskedastic means.
Then you give a matrix of all 1s, which is irrelevant; correlation can exist and be less than 1.
Then... | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model | You wrote
The second plot seems to indicate that the absolute value of the
residuals is strongly positively correlated with the fitted values,
It doesn't "seem" to, it does. And that's what hetero | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model
You wrote
The second plot seems to indicate that the absolute value of the
residuals is strongly positively correlated with the fitted values,
It doesn't "seem" to, it does. And that's what heteroskedastic means.
Then... | Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model
You wrote
The second plot seems to indicate that the absolute value of the
residuals is strongly positively correlated with the fitted values,
It doesn't "seem" to, it does. And that's what hetero |
5,990 | Measures of variable importance in random forests | The first one can be 'interpreted' as follows: if a predictor is important in your current model, then assigning other values for that predictor randomly but 'realistically' (i.e.: permuting this predictor's values over your dataset), should have a negative influence on prediction, i.e.: using the same model to predict... | Measures of variable importance in random forests | The first one can be 'interpreted' as follows: if a predictor is important in your current model, then assigning other values for that predictor randomly but 'realistically' (i.e.: permuting this pred | Measures of variable importance in random forests
The first one can be 'interpreted' as follows: if a predictor is important in your current model, then assigning other values for that predictor randomly but 'realistically' (i.e.: permuting this predictor's values over your dataset), should have a negative influence on... | Measures of variable importance in random forests
The first one can be 'interpreted' as follows: if a predictor is important in your current model, then assigning other values for that predictor randomly but 'realistically' (i.e.: permuting this pred |
5,991 | Measures of variable importance in random forests | Random Forest importance metrics as implemented in the randomForest package in R have quirks in that correlated predictors get low importance values.
http://bioinformatics.oxfordjournals.org/content/early/2010/04/12/bioinformatics.btq134.full.pdf
I have a modified implementation of random forests out on CRAN which imp... | Measures of variable importance in random forests | Random Forest importance metrics as implemented in the randomForest package in R have quirks in that correlated predictors get low importance values.
http://bioinformatics.oxfordjournals.org/content/ | Measures of variable importance in random forests
Random Forest importance metrics as implemented in the randomForest package in R have quirks in that correlated predictors get low importance values.
http://bioinformatics.oxfordjournals.org/content/early/2010/04/12/bioinformatics.btq134.full.pdf
I have a modified impl... | Measures of variable importance in random forests
Random Forest importance metrics as implemented in the randomForest package in R have quirks in that correlated predictors get low importance values.
http://bioinformatics.oxfordjournals.org/content/ |
5,992 | Standard error clustering in R (either manually or in plm) | Edit as of December 2021:
Probably the easiest way to get clustered standard errors in R now is via the the feols function in the fixest package or felm function in the lfe package:
feols in fixest: Clustering syntax and standard error computational procedure
felm in lfe: CRAN documentation
Original answers and som... | Standard error clustering in R (either manually or in plm) | Edit as of December 2021:
Probably the easiest way to get clustered standard errors in R now is via the the feols function in the fixest package or felm function in the lfe package:
feols in fixest: | Standard error clustering in R (either manually or in plm)
Edit as of December 2021:
Probably the easiest way to get clustered standard errors in R now is via the the feols function in the fixest package or felm function in the lfe package:
feols in fixest: Clustering syntax and standard error computational procedure
... | Standard error clustering in R (either manually or in plm)
Edit as of December 2021:
Probably the easiest way to get clustered standard errors in R now is via the the feols function in the fixest package or felm function in the lfe package:
feols in fixest: |
5,993 | Standard error clustering in R (either manually or in plm) | After a lot of reading, I found the solution for doing clustering within the lm framework.
There's an excellent white paper by Mahmood Arai that provides a tutorial on clustering in the lm framework, which he does with degrees-of-freedom corrections instead of my messy attempts above. He provides his functions for both... | Standard error clustering in R (either manually or in plm) | After a lot of reading, I found the solution for doing clustering within the lm framework.
There's an excellent white paper by Mahmood Arai that provides a tutorial on clustering in the lm framework, | Standard error clustering in R (either manually or in plm)
After a lot of reading, I found the solution for doing clustering within the lm framework.
There's an excellent white paper by Mahmood Arai that provides a tutorial on clustering in the lm framework, which he does with degrees-of-freedom corrections instead of ... | Standard error clustering in R (either manually or in plm)
After a lot of reading, I found the solution for doing clustering within the lm framework.
There's an excellent white paper by Mahmood Arai that provides a tutorial on clustering in the lm framework, |
5,994 | Standard error clustering in R (either manually or in plm) | The easiest way to compute clustered standard errors in R is to use the modified summary function.
lm.object <- lm(y ~ x, data = data)
summary(lm.object, cluster=c("c"))
There's an excellent post on clustering within the lm framework. The site also provides the modified summary function for both one- and two-way clus... | Standard error clustering in R (either manually or in plm) | The easiest way to compute clustered standard errors in R is to use the modified summary function.
lm.object <- lm(y ~ x, data = data)
summary(lm.object, cluster=c("c"))
There's an excellent post on | Standard error clustering in R (either manually or in plm)
The easiest way to compute clustered standard errors in R is to use the modified summary function.
lm.object <- lm(y ~ x, data = data)
summary(lm.object, cluster=c("c"))
There's an excellent post on clustering within the lm framework. The site also provides t... | Standard error clustering in R (either manually or in plm)
The easiest way to compute clustered standard errors in R is to use the modified summary function.
lm.object <- lm(y ~ x, data = data)
summary(lm.object, cluster=c("c"))
There's an excellent post on |
5,995 | Standard error clustering in R (either manually or in plm) | If you don't have a time index, you don't need one: plm will add a fictitious one by itself, and it won't be used unless you ask for it. So this call should work:
> x <- plm(price ~ carat, data = diamonds, index = "cut")
Error in pdim.default(index[[1]], index[[2]]) :
duplicate couples (time-id)
Except that it d... | Standard error clustering in R (either manually or in plm) | If you don't have a time index, you don't need one: plm will add a fictitious one by itself, and it won't be used unless you ask for it. So this call should work:
> x <- plm(price ~ carat, data = dia | Standard error clustering in R (either manually or in plm)
If you don't have a time index, you don't need one: plm will add a fictitious one by itself, and it won't be used unless you ask for it. So this call should work:
> x <- plm(price ~ carat, data = diamonds, index = "cut")
Error in pdim.default(index[[1]], inde... | Standard error clustering in R (either manually or in plm)
If you don't have a time index, you don't need one: plm will add a fictitious one by itself, and it won't be used unless you ask for it. So this call should work:
> x <- plm(price ~ carat, data = dia |
5,996 | What exactly is a Bayesian model? | In essence, one where inference is based on using Bayes theorem to obtain a posterior distribution for a quantity or quantities of interest form some model (such as parameter values) based on some prior distribution for the relevant unknown parameters and the likelihood from the model.
i.e. from a distributional model ... | What exactly is a Bayesian model? | In essence, one where inference is based on using Bayes theorem to obtain a posterior distribution for a quantity or quantities of interest form some model (such as parameter values) based on some pri | What exactly is a Bayesian model?
In essence, one where inference is based on using Bayes theorem to obtain a posterior distribution for a quantity or quantities of interest form some model (such as parameter values) based on some prior distribution for the relevant unknown parameters and the likelihood from the model.... | What exactly is a Bayesian model?
In essence, one where inference is based on using Bayes theorem to obtain a posterior distribution for a quantity or quantities of interest form some model (such as parameter values) based on some pri |
5,997 | What exactly is a Bayesian model? | A Bayesian model is just a model that draws its inferences from the posterior distribution, i.e. utilizes a prior distribution and a likelihood which are related by Bayes' theorem. | What exactly is a Bayesian model? | A Bayesian model is just a model that draws its inferences from the posterior distribution, i.e. utilizes a prior distribution and a likelihood which are related by Bayes' theorem. | What exactly is a Bayesian model?
A Bayesian model is just a model that draws its inferences from the posterior distribution, i.e. utilizes a prior distribution and a likelihood which are related by Bayes' theorem. | What exactly is a Bayesian model?
A Bayesian model is just a model that draws its inferences from the posterior distribution, i.e. utilizes a prior distribution and a likelihood which are related by Bayes' theorem. |
5,998 | What exactly is a Bayesian model? | Can I call a model wherein Bayes' Theorem is used a "Bayesian model"?
No
I am afraid such a definition might be too broad.
You are right. Bayes' theorem is a legitimate relation between marginal event probabilities and conditional probabilities. It holds regardless of your interpretation of probability.
So what ex... | What exactly is a Bayesian model? | Can I call a model wherein Bayes' Theorem is used a "Bayesian model"?
No
I am afraid such a definition might be too broad.
You are right. Bayes' theorem is a legitimate relation between marginal e | What exactly is a Bayesian model?
Can I call a model wherein Bayes' Theorem is used a "Bayesian model"?
No
I am afraid such a definition might be too broad.
You are right. Bayes' theorem is a legitimate relation between marginal event probabilities and conditional probabilities. It holds regardless of your interpre... | What exactly is a Bayesian model?
Can I call a model wherein Bayes' Theorem is used a "Bayesian model"?
No
I am afraid such a definition might be too broad.
You are right. Bayes' theorem is a legitimate relation between marginal e |
5,999 | What exactly is a Bayesian model? | A statistical model can be seen as a procedure/story describing how some data came to be. A Bayesian model is a statistical model where you use probability to represent all uncertainty within the model, both the uncertainty regarding the output but also the uncertainty regarding the input (aka parameters) to the model.... | What exactly is a Bayesian model? | A statistical model can be seen as a procedure/story describing how some data came to be. A Bayesian model is a statistical model where you use probability to represent all uncertainty within the mode | What exactly is a Bayesian model?
A statistical model can be seen as a procedure/story describing how some data came to be. A Bayesian model is a statistical model where you use probability to represent all uncertainty within the model, both the uncertainty regarding the output but also the uncertainty regarding the in... | What exactly is a Bayesian model?
A statistical model can be seen as a procedure/story describing how some data came to be. A Bayesian model is a statistical model where you use probability to represent all uncertainty within the mode |
6,000 | What exactly is a Bayesian model? | Your question is more on the semantic side: when can I call a model "Bayesian"?
Drawing conclusions from this excellent paper:
Fienberg, S. E. (2006). When did bayesian inference become "bayesian"? Bayesian Analysis,
1(1):1-40.
there are 2 answers:
Your model is first Bayesian if it uses Bayes' rule (that's the "alg... | What exactly is a Bayesian model? | Your question is more on the semantic side: when can I call a model "Bayesian"?
Drawing conclusions from this excellent paper:
Fienberg, S. E. (2006). When did bayesian inference become "bayesian"? B | What exactly is a Bayesian model?
Your question is more on the semantic side: when can I call a model "Bayesian"?
Drawing conclusions from this excellent paper:
Fienberg, S. E. (2006). When did bayesian inference become "bayesian"? Bayesian Analysis,
1(1):1-40.
there are 2 answers:
Your model is first Bayesian if it... | What exactly is a Bayesian model?
Your question is more on the semantic side: when can I call a model "Bayesian"?
Drawing conclusions from this excellent paper:
Fienberg, S. E. (2006). When did bayesian inference become "bayesian"? B |
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