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 |
|---|---|---|---|---|---|---|
35,601 | Scaling dataset in Random Forest | Any monotonic injective transformation of the features won't change the model wrt how it splits the data. The reason is the same as for why scaling is unnecessary: the random forest looks for partitions, and partitions only depend on how the data are sorted. If there is an optimal split on some scale, then by the defin... | Scaling dataset in Random Forest | Any monotonic injective transformation of the features won't change the model wrt how it splits the data. The reason is the same as for why scaling is unnecessary: the random forest looks for partitio | Scaling dataset in Random Forest
Any monotonic injective transformation of the features won't change the model wrt how it splits the data. The reason is the same as for why scaling is unnecessary: the random forest looks for partitions, and partitions only depend on how the data are sorted. If there is an optimal split... | Scaling dataset in Random Forest
Any monotonic injective transformation of the features won't change the model wrt how it splits the data. The reason is the same as for why scaling is unnecessary: the random forest looks for partitio |
35,602 | History of the term "early stopping" | It depends on how deep you want to go. Abraham Wald was studying sequential testing and stopping criteria in 1945. It was further studied in the 1960 and 1970s (cf Ferguson, 1989). In 1989 Pocock and Hughes described using early stopping for clinical trials (I'm not sure if this is the earliest paper like this). As abo... | History of the term "early stopping" | It depends on how deep you want to go. Abraham Wald was studying sequential testing and stopping criteria in 1945. It was further studied in the 1960 and 1970s (cf Ferguson, 1989). In 1989 Pocock and | History of the term "early stopping"
It depends on how deep you want to go. Abraham Wald was studying sequential testing and stopping criteria in 1945. It was further studied in the 1960 and 1970s (cf Ferguson, 1989). In 1989 Pocock and Hughes described using early stopping for clinical trials (I'm not sure if this is ... | History of the term "early stopping"
It depends on how deep you want to go. Abraham Wald was studying sequential testing and stopping criteria in 1945. It was further studied in the 1960 and 1970s (cf Ferguson, 1989). In 1989 Pocock and |
35,603 | History of the term "early stopping" | The earliest reference I can find is a 1997 paper as shown - PubMed is a powerful tool to analyze trends in academic litearture. I obviously can't search "early stopping" apropos of nothing because it's apparently quite intuitive to terminate any laborious procedure early when desired precision is achieved, whether in ... | History of the term "early stopping" | The earliest reference I can find is a 1997 paper as shown - PubMed is a powerful tool to analyze trends in academic litearture. I obviously can't search "early stopping" apropos of nothing because it | History of the term "early stopping"
The earliest reference I can find is a 1997 paper as shown - PubMed is a powerful tool to analyze trends in academic litearture. I obviously can't search "early stopping" apropos of nothing because it's apparently quite intuitive to terminate any laborious procedure early when desir... | History of the term "early stopping"
The earliest reference I can find is a 1997 paper as shown - PubMed is a powerful tool to analyze trends in academic litearture. I obviously can't search "early stopping" apropos of nothing because it |
35,604 | Bayes risk estimator under quartic loss | Unless the posterior is symmetric around $\mathbb{E}[\theta|y]$ there is no reason for the estimator to be equal to $\mathbb{E}[\theta|y]$. It is one of the solutions to a third degree polynomial equation.
Take the following counter-example of a Gamma $\mathfrak G(\alpha,\beta)$ posterior. The posterior risk function i... | Bayes risk estimator under quartic loss | Unless the posterior is symmetric around $\mathbb{E}[\theta|y]$ there is no reason for the estimator to be equal to $\mathbb{E}[\theta|y]$. It is one of the solutions to a third degree polynomial equa | Bayes risk estimator under quartic loss
Unless the posterior is symmetric around $\mathbb{E}[\theta|y]$ there is no reason for the estimator to be equal to $\mathbb{E}[\theta|y]$. It is one of the solutions to a third degree polynomial equation.
Take the following counter-example of a Gamma $\mathfrak G(\alpha,\beta)$ ... | Bayes risk estimator under quartic loss
Unless the posterior is symmetric around $\mathbb{E}[\theta|y]$ there is no reason for the estimator to be equal to $\mathbb{E}[\theta|y]$. It is one of the solutions to a third degree polynomial equa |
35,605 | Prove that Kurtosis is at least one more than the square of the skewness | This demonstration uses the clever (yet elementary) method employed in Fisher's reference Burnside & Panton vol. II, section 142. It assumes a familiarity with the basics of determinants: their definition, computing them with row-reduction, and expanding them by minors. I also make liberal use of block-matrix notatio... | Prove that Kurtosis is at least one more than the square of the skewness | This demonstration uses the clever (yet elementary) method employed in Fisher's reference Burnside & Panton vol. II, section 142. It assumes a familiarity with the basics of determinants: their defin | Prove that Kurtosis is at least one more than the square of the skewness
This demonstration uses the clever (yet elementary) method employed in Fisher's reference Burnside & Panton vol. II, section 142. It assumes a familiarity with the basics of determinants: their definition, computing them with row-reduction, and e... | Prove that Kurtosis is at least one more than the square of the skewness
This demonstration uses the clever (yet elementary) method employed in Fisher's reference Burnside & Panton vol. II, section 142. It assumes a familiarity with the basics of determinants: their defin |
35,606 | Prove that Kurtosis is at least one more than the square of the skewness | Consider the quadratic form
\begin{align} Q(a, ~b,~c) & := \frac1n\sum \left(a + xb+ x^2 c\right)^2 \\ &= a^2\cdot v_0 + b^2\cdot v_2 + c^2\cdot v_4+ 2ab\cdot v_1+ 2bc\cdot v_3+ 2ca\cdot v_2,\tag 1\label 1\end{align}
that is, $v_i:= \frac{\sum x^i}n,~i\in \{0,~1,~2,~3,~4\}.$
$\eqref{1}$ is positive semi-definite.
Then ... | Prove that Kurtosis is at least one more than the square of the skewness | Consider the quadratic form
\begin{align} Q(a, ~b,~c) & := \frac1n\sum \left(a + xb+ x^2 c\right)^2 \\ &= a^2\cdot v_0 + b^2\cdot v_2 + c^2\cdot v_4+ 2ab\cdot v_1+ 2bc\cdot v_3+ 2ca\cdot v_2,\tag 1\la | Prove that Kurtosis is at least one more than the square of the skewness
Consider the quadratic form
\begin{align} Q(a, ~b,~c) & := \frac1n\sum \left(a + xb+ x^2 c\right)^2 \\ &= a^2\cdot v_0 + b^2\cdot v_2 + c^2\cdot v_4+ 2ab\cdot v_1+ 2bc\cdot v_3+ 2ca\cdot v_2,\tag 1\label 1\end{align}
that is, $v_i:= \frac{\sum x^i... | Prove that Kurtosis is at least one more than the square of the skewness
Consider the quadratic form
\begin{align} Q(a, ~b,~c) & := \frac1n\sum \left(a + xb+ x^2 c\right)^2 \\ &= a^2\cdot v_0 + b^2\cdot v_2 + c^2\cdot v_4+ 2ab\cdot v_1+ 2bc\cdot v_3+ 2ca\cdot v_2,\tag 1\la |
35,607 | Maximum Likelihood Estimator on birthday paradox | This is a problem amenable to maximum likelihood estimation, provided one can write down the likelihood or design a latent variable representation to run an EM algorithm.
Let us denote $\bar N$ for the number of days/boxes (like $\bar N=365$ in the original problem) and $M$ for the number of individuals. Borrowing from... | Maximum Likelihood Estimator on birthday paradox | This is a problem amenable to maximum likelihood estimation, provided one can write down the likelihood or design a latent variable representation to run an EM algorithm.
Let us denote $\bar N$ for th | Maximum Likelihood Estimator on birthday paradox
This is a problem amenable to maximum likelihood estimation, provided one can write down the likelihood or design a latent variable representation to run an EM algorithm.
Let us denote $\bar N$ for the number of days/boxes (like $\bar N=365$ in the original problem) and ... | Maximum Likelihood Estimator on birthday paradox
This is a problem amenable to maximum likelihood estimation, provided one can write down the likelihood or design a latent variable representation to run an EM algorithm.
Let us denote $\bar N$ for th |
35,608 | heteroskedasticity and logistic regression | Except in a very technical sense (which @BigBendRegion's answer gets at) heteroskedasticity isn't a "thing' in a logistic regression model.
Heteroskedasticity is when the standard deviation of the errors around the regression line (that is the average distance between the predicted Y value at a given X value and the ac... | heteroskedasticity and logistic regression | Except in a very technical sense (which @BigBendRegion's answer gets at) heteroskedasticity isn't a "thing' in a logistic regression model.
Heteroskedasticity is when the standard deviation of the err | heteroskedasticity and logistic regression
Except in a very technical sense (which @BigBendRegion's answer gets at) heteroskedasticity isn't a "thing' in a logistic regression model.
Heteroskedasticity is when the standard deviation of the errors around the regression line (that is the average distance between the pred... | heteroskedasticity and logistic regression
Except in a very technical sense (which @BigBendRegion's answer gets at) heteroskedasticity isn't a "thing' in a logistic regression model.
Heteroskedasticity is when the standard deviation of the err |
35,609 | heteroskedasticity and logistic regression | With the logistic regression model, heteroscedasticity is automatically assumed to exist. The conditional distribution of $Y$ given $X=x$ is assumed to be Bernoulli with parameter $\pi(x)$, a probability. The variance of this distribution is $\pi(x)\times (1-\pi(x))$, a nonconstant function of $x$. Likewise, you do no... | heteroskedasticity and logistic regression | With the logistic regression model, heteroscedasticity is automatically assumed to exist. The conditional distribution of $Y$ given $X=x$ is assumed to be Bernoulli with parameter $\pi(x)$, a probabi | heteroskedasticity and logistic regression
With the logistic regression model, heteroscedasticity is automatically assumed to exist. The conditional distribution of $Y$ given $X=x$ is assumed to be Bernoulli with parameter $\pi(x)$, a probability. The variance of this distribution is $\pi(x)\times (1-\pi(x))$, a nonco... | heteroskedasticity and logistic regression
With the logistic regression model, heteroscedasticity is automatically assumed to exist. The conditional distribution of $Y$ given $X=x$ is assumed to be Bernoulli with parameter $\pi(x)$, a probabi |
35,610 | Which loss function has a less optimal answer for w? | Since the data is linearly separable, other loss functions can be made $0$ via a suitable choice of weights, $w$. However, second one has a regularisation term, so the loss function doesn't only favor the linear separability, but the norm of the weights as well.
Notes:
Given a loss function, I assumed the chosen optimi... | Which loss function has a less optimal answer for w? | Since the data is linearly separable, other loss functions can be made $0$ via a suitable choice of weights, $w$. However, second one has a regularisation term, so the loss function doesn't only favor | Which loss function has a less optimal answer for w?
Since the data is linearly separable, other loss functions can be made $0$ via a suitable choice of weights, $w$. However, second one has a regularisation term, so the loss function doesn't only favor the linear separability, but the norm of the weights as well.
Note... | Which loss function has a less optimal answer for w?
Since the data is linearly separable, other loss functions can be made $0$ via a suitable choice of weights, $w$. However, second one has a regularisation term, so the loss function doesn't only favor |
35,611 | Which loss function has a less optimal answer for w? | Your data are lineary separable, this means that all linear classifier will make accurate separation of the data. A linear regression is a linear classifier, i.e it uses a line to separate data. So either you use a linear regression with no penalty or a linear regression with penalty you are still going to separate cor... | Which loss function has a less optimal answer for w? | Your data are lineary separable, this means that all linear classifier will make accurate separation of the data. A linear regression is a linear classifier, i.e it uses a line to separate data. So ei | Which loss function has a less optimal answer for w?
Your data are lineary separable, this means that all linear classifier will make accurate separation of the data. A linear regression is a linear classifier, i.e it uses a line to separate data. So either you use a linear regression with no penalty or a linear regres... | Which loss function has a less optimal answer for w?
Your data are lineary separable, this means that all linear classifier will make accurate separation of the data. A linear regression is a linear classifier, i.e it uses a line to separate data. So ei |
35,612 | How to prove $s^2$ is a consistent estimator of $\sigma^2$? | It's a very well known result that :
If $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$ , then $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$
Thus, $ \mathbb{E}(Z_n) = n-1 $ and $ \text{var}(Z_n) = 2(n-1)$ .
If you wish to see a proof of the above result, please... | How to prove $s^2$ is a consistent estimator of $\sigma^2$? | It's a very well known result that :
If $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$ , then $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$
Thu | How to prove $s^2$ is a consistent estimator of $\sigma^2$?
It's a very well known result that :
If $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$ , then $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$
Thus, $ \mathbb{E}(Z_n) = n-1 $ and $ \text{var}(Z_n) = 2(n-1... | How to prove $s^2$ is a consistent estimator of $\sigma^2$?
It's a very well known result that :
If $X_1, X_2, \cdots, X_n \stackrel{\text{iid}}{\sim} N(\mu,\sigma^2)$ , then $$Z_n = \dfrac{\displaystyle\sum(X_i - \bar{X})^2}{\sigma^2} \sim \chi^2_{n-1}$$
Thu |
35,613 | Hacking linear regression | You decreased the (total) variance in the data making it easier to explain.
Thanks to the same $y$-axis limits, it can be easily seen that the data of your first example is much more spread in this direction than in the $x$-axis direction.
Your linear regression model does capture the slight trend of increasing $y$ wit... | Hacking linear regression | You decreased the (total) variance in the data making it easier to explain.
Thanks to the same $y$-axis limits, it can be easily seen that the data of your first example is much more spread in this di | Hacking linear regression
You decreased the (total) variance in the data making it easier to explain.
Thanks to the same $y$-axis limits, it can be easily seen that the data of your first example is much more spread in this direction than in the $x$-axis direction.
Your linear regression model does capture the slight t... | Hacking linear regression
You decreased the (total) variance in the data making it easier to explain.
Thanks to the same $y$-axis limits, it can be easily seen that the data of your first example is much more spread in this di |
35,614 | Hacking linear regression | The edited question asks: "When (if ever) is this a valid statistical approach?"
I can't think of a good reason.
One COULD argue that, by taking averages by binning, you've "de-noised" the data. However, that implicitly assumes that all of the observed variance is noise. That seems like a highly dubious assumption.
By ... | Hacking linear regression | The edited question asks: "When (if ever) is this a valid statistical approach?"
I can't think of a good reason.
One COULD argue that, by taking averages by binning, you've "de-noised" the data. Howev | Hacking linear regression
The edited question asks: "When (if ever) is this a valid statistical approach?"
I can't think of a good reason.
One COULD argue that, by taking averages by binning, you've "de-noised" the data. However, that implicitly assumes that all of the observed variance is noise. That seems like a high... | Hacking linear regression
The edited question asks: "When (if ever) is this a valid statistical approach?"
I can't think of a good reason.
One COULD argue that, by taking averages by binning, you've "de-noised" the data. Howev |
35,615 | Clarification on the definition of "population" | It is true that populations are usually broad but they do not have to be. For example, according to Newbold, Carlson, & Thorne (2013) Statistics for business and economics, textbook:
A population is the complete set of all items that interest an investigator. Population size, $N$, can be very large or even infinite. A... | Clarification on the definition of "population" | It is true that populations are usually broad but they do not have to be. For example, according to Newbold, Carlson, & Thorne (2013) Statistics for business and economics, textbook:
A population is | Clarification on the definition of "population"
It is true that populations are usually broad but they do not have to be. For example, according to Newbold, Carlson, & Thorne (2013) Statistics for business and economics, textbook:
A population is the complete set of all items that interest an investigator. Population ... | Clarification on the definition of "population"
It is true that populations are usually broad but they do not have to be. For example, according to Newbold, Carlson, & Thorne (2013) Statistics for business and economics, textbook:
A population is |
35,616 | Clarification on the definition of "population" | The gist is that the instances of the game show are a sample of the results that conceivable could have happened.
Think about flipping a coin and trying to figure out the probability of getting H or T. You’ve flipped 50 times and gotten 30 H and 20 T. It is inarguable that you got H on 60% of the flips. However, it is... | Clarification on the definition of "population" | The gist is that the instances of the game show are a sample of the results that conceivable could have happened.
Think about flipping a coin and trying to figure out the probability of getting H or T | Clarification on the definition of "population"
The gist is that the instances of the game show are a sample of the results that conceivable could have happened.
Think about flipping a coin and trying to figure out the probability of getting H or T. You’ve flipped 50 times and gotten 30 H and 20 T. It is inarguable th... | Clarification on the definition of "population"
The gist is that the instances of the game show are a sample of the results that conceivable could have happened.
Think about flipping a coin and trying to figure out the probability of getting H or T |
35,617 | Clarification on the definition of "population" | Others have answered perfectly to the first question, I'll consider the part about random samples:
No, your sample is not random.
But don't worry, it's fine.
Statisticians use non-random samples all the time! For instance, typically, when you do research about some disease progression, you decide a date and you pick ev... | Clarification on the definition of "population" | Others have answered perfectly to the first question, I'll consider the part about random samples:
No, your sample is not random.
But don't worry, it's fine.
Statisticians use non-random samples all t | Clarification on the definition of "population"
Others have answered perfectly to the first question, I'll consider the part about random samples:
No, your sample is not random.
But don't worry, it's fine.
Statisticians use non-random samples all the time! For instance, typically, when you do research about some diseas... | Clarification on the definition of "population"
Others have answered perfectly to the first question, I'll consider the part about random samples:
No, your sample is not random.
But don't worry, it's fine.
Statisticians use non-random samples all t |
35,618 | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}(p)$, or for some other reason? | In principle, any continuous distribution can serve as a starting point for a random number generator. Nevertheless, the standard continuous uniform distribution is a natural starting place for a pseudo-random number generator (PRNG) for a few main reasons:
The method of inverse transformation sampling allows us to g... | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1} | In principle, any continuous distribution can serve as a starting point for a random number generator. Nevertheless, the standard continuous uniform distribution is a natural starting place for a pse | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}(p)$, or for some other reason?
In principle, any continuous distribution can serve as a starting point for a random number generator. Nevertheless, the standard continuous uniform distribution is a natural starting pla... | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}
In principle, any continuous distribution can serve as a starting point for a random number generator. Nevertheless, the standard continuous uniform distribution is a natural starting place for a pse |
35,619 | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}(p)$, or for some other reason? | Vetting a PRNG. First, the standard uniform distribution $\mathsf{Unif}(0,1)$ is mathematically simple. So given pseudorandom numbers that are purported to be indistiguishable, for practical purposes, from a random sample from a standard uniform population, it is easy to test whether the claim is true.
For example, clo... | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1} | Vetting a PRNG. First, the standard uniform distribution $\mathsf{Unif}(0,1)$ is mathematically simple. So given pseudorandom numbers that are purported to be indistiguishable, for practical purposes, | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}(p)$, or for some other reason?
Vetting a PRNG. First, the standard uniform distribution $\mathsf{Unif}(0,1)$ is mathematically simple. So given pseudorandom numbers that are purported to be indistiguishable, for practic... | Historically, did pseudo RNGs have range [0,1] because of the ease of drawing from $f$ using $F^{-1}
Vetting a PRNG. First, the standard uniform distribution $\mathsf{Unif}(0,1)$ is mathematically simple. So given pseudorandom numbers that are purported to be indistiguishable, for practical purposes, |
35,620 | Why are there many lines in the graph of random forest? | Unfortunately, ?plot.randomForest is not helpful at all. However, it does tell us that plot.randomForest invisibly returns the data that is plotted (try (plot(rf)) with the brackets), which is a three column matrix with columns called OOB, M and R. The first is pretty obviously the overall out-of-bag error. The other t... | Why are there many lines in the graph of random forest? | Unfortunately, ?plot.randomForest is not helpful at all. However, it does tell us that plot.randomForest invisibly returns the data that is plotted (try (plot(rf)) with the brackets), which is a three | Why are there many lines in the graph of random forest?
Unfortunately, ?plot.randomForest is not helpful at all. However, it does tell us that plot.randomForest invisibly returns the data that is plotted (try (plot(rf)) with the brackets), which is a three column matrix with columns called OOB, M and R. The first is pr... | Why are there many lines in the graph of random forest?
Unfortunately, ?plot.randomForest is not helpful at all. However, it does tell us that plot.randomForest invisibly returns the data that is plotted (try (plot(rf)) with the brackets), which is a three |
35,621 | ( 3 level ) random intercept at level 2 & random slope and random intercept at level 3 | In this type of scenario it is useful to recall that
(1 | A / B)
is the same as
(1 | A) + (1 | A:B)
which also generalises to models with more levels.
Thus, in the case mentioned in the question, where we want random slopes only on the higher level, we would have:
(time | therapist) + (1 | therapist:subject)
In t... | ( 3 level ) random intercept at level 2 & random slope and random intercept at level 3 | In this type of scenario it is useful to recall that
(1 | A / B)
is the same as
(1 | A) + (1 | A:B)
which also generalises to models with more levels.
Thus, in the case mentioned in the question, | ( 3 level ) random intercept at level 2 & random slope and random intercept at level 3
In this type of scenario it is useful to recall that
(1 | A / B)
is the same as
(1 | A) + (1 | A:B)
which also generalises to models with more levels.
Thus, in the case mentioned in the question, where we want random slopes only... | ( 3 level ) random intercept at level 2 & random slope and random intercept at level 3
In this type of scenario it is useful to recall that
(1 | A / B)
is the same as
(1 | A) + (1 | A:B)
which also generalises to models with more levels.
Thus, in the case mentioned in the question, |
35,622 | How to test for the best parameters for transformed independent variable in linear model | This is a standard least squares problem and, as such, is solved by finding parameters that minimize the sum of squared residuals.
A general approach is first to fit the model
$$E[Y] = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k.$$
Replace the $y_i$ by their residuals
$$e_i = y_i - (\hat\beta_0 + \hat\beta_1 x_1 + \cd... | How to test for the best parameters for transformed independent variable in linear model | This is a standard least squares problem and, as such, is solved by finding parameters that minimize the sum of squared residuals.
A general approach is first to fit the model
$$E[Y] = \beta_0 + \beta | How to test for the best parameters for transformed independent variable in linear model
This is a standard least squares problem and, as such, is solved by finding parameters that minimize the sum of squared residuals.
A general approach is first to fit the model
$$E[Y] = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k.$... | How to test for the best parameters for transformed independent variable in linear model
This is a standard least squares problem and, as such, is solved by finding parameters that minimize the sum of squared residuals.
A general approach is first to fit the model
$$E[Y] = \beta_0 + \beta |
35,623 | How to test for the best parameters for transformed independent variable in linear model | Thanks for the data. Although this is just your graph with a smoother added, I suggest that -- beyond eyeballing the data -- the graph isn't supportive of any idea that there is an underlying sigmoid. There is nothing magic about the bandwidth used, and I tried lower and higher values with the same conclusion.
Natura... | How to test for the best parameters for transformed independent variable in linear model | Thanks for the data. Although this is just your graph with a smoother added, I suggest that -- beyond eyeballing the data -- the graph isn't supportive of any idea that there is an underlying sigmoid. | How to test for the best parameters for transformed independent variable in linear model
Thanks for the data. Although this is just your graph with a smoother added, I suggest that -- beyond eyeballing the data -- the graph isn't supportive of any idea that there is an underlying sigmoid. There is nothing magic about t... | How to test for the best parameters for transformed independent variable in linear model
Thanks for the data. Although this is just your graph with a smoother added, I suggest that -- beyond eyeballing the data -- the graph isn't supportive of any idea that there is an underlying sigmoid. |
35,624 | How to test for the best parameters for transformed independent variable in linear model | A possible mistake that you're doing is not displaying the effect of $x_{k+1}$ after the effect of all other $x_i$. For this, rather than directly plotting $y$ against $x_{k+1}$ you should Partial Regression Plots (aka added variable plots; Wiki, or here). The three steps are (from Wiki):
Computing the residuals of re... | How to test for the best parameters for transformed independent variable in linear model | A possible mistake that you're doing is not displaying the effect of $x_{k+1}$ after the effect of all other $x_i$. For this, rather than directly plotting $y$ against $x_{k+1}$ you should Partial Reg | How to test for the best parameters for transformed independent variable in linear model
A possible mistake that you're doing is not displaying the effect of $x_{k+1}$ after the effect of all other $x_i$. For this, rather than directly plotting $y$ against $x_{k+1}$ you should Partial Regression Plots (aka added variab... | How to test for the best parameters for transformed independent variable in linear model
A possible mistake that you're doing is not displaying the effect of $x_{k+1}$ after the effect of all other $x_i$. For this, rather than directly plotting $y$ against $x_{k+1}$ you should Partial Reg |
35,625 | Is this a valid argument why null hypothesis should always be equality? | No, that isn't valid.
For instance, suppose you are interested in the relative heights of adult men and women in different parts of the world. You could find the usual difference across a broad population (say it's 9 cm, just for a guess). Then you look at (say) a tribe in New Guinea. Your null could be "Men in New Gu... | Is this a valid argument why null hypothesis should always be equality? | No, that isn't valid.
For instance, suppose you are interested in the relative heights of adult men and women in different parts of the world. You could find the usual difference across a broad popula | Is this a valid argument why null hypothesis should always be equality?
No, that isn't valid.
For instance, suppose you are interested in the relative heights of adult men and women in different parts of the world. You could find the usual difference across a broad population (say it's 9 cm, just for a guess). Then yo... | Is this a valid argument why null hypothesis should always be equality?
No, that isn't valid.
For instance, suppose you are interested in the relative heights of adult men and women in different parts of the world. You could find the usual difference across a broad popula |
35,626 | The difference of Standard Error between glm(y~x, family=poisson(link=identity)) and optim() in R | In statistical likelihood theory, minus the second derivative of the log-likelihood function is called the observed information. We might write this as
$$
I = -\ddot \ell(y; \theta)
$$
where the dots indicate differentiation with respect to $\theta$. The expected value of the observed information
$$
{\cal I} = E(I)
$$
... | The difference of Standard Error between glm(y~x, family=poisson(link=identity)) and optim() in R | In statistical likelihood theory, minus the second derivative of the log-likelihood function is called the observed information. We might write this as
$$
I = -\ddot \ell(y; \theta)
$$
where the dots | The difference of Standard Error between glm(y~x, family=poisson(link=identity)) and optim() in R
In statistical likelihood theory, minus the second derivative of the log-likelihood function is called the observed information. We might write this as
$$
I = -\ddot \ell(y; \theta)
$$
where the dots indicate differentiati... | The difference of Standard Error between glm(y~x, family=poisson(link=identity)) and optim() in R
In statistical likelihood theory, minus the second derivative of the log-likelihood function is called the observed information. We might write this as
$$
I = -\ddot \ell(y; \theta)
$$
where the dots |
35,627 | Omitted Variables and their consequences for other variables | Intuitively, the answer is that $\hat{\beta}_2$ will be biased whenever $x_2$ is correlated with $x_1$. This is because the correlation of $x_2$ with $x_1$ causes $\hat{\beta}_2$ to be correlated with $\hat{\beta}_1$; as a consequence, when $\hat{\beta}_1$ differs from $\beta_1$, $\hat{\beta}_2$ will differ from $\bet... | Omitted Variables and their consequences for other variables | Intuitively, the answer is that $\hat{\beta}_2$ will be biased whenever $x_2$ is correlated with $x_1$. This is because the correlation of $x_2$ with $x_1$ causes $\hat{\beta}_2$ to be correlated wit | Omitted Variables and their consequences for other variables
Intuitively, the answer is that $\hat{\beta}_2$ will be biased whenever $x_2$ is correlated with $x_1$. This is because the correlation of $x_2$ with $x_1$ causes $\hat{\beta}_2$ to be correlated with $\hat{\beta}_1$; as a consequence, when $\hat{\beta}_1$ d... | Omitted Variables and their consequences for other variables
Intuitively, the answer is that $\hat{\beta}_2$ will be biased whenever $x_2$ is correlated with $x_1$. This is because the correlation of $x_2$ with $x_1$ causes $\hat{\beta}_2$ to be correlated wit |
35,628 | Omitted Variables and their consequences for other variables | Here's what happens for any number of predictors and omitted variables. The answer ends up similar to what @jbowman (+1) showed for two predictors (one correlated with omitted variables and one not). Since I'd already started writing this, I'll post it in case it's useful to see another approach.
To summarize the answe... | Omitted Variables and their consequences for other variables | Here's what happens for any number of predictors and omitted variables. The answer ends up similar to what @jbowman (+1) showed for two predictors (one correlated with omitted variables and one not). | Omitted Variables and their consequences for other variables
Here's what happens for any number of predictors and omitted variables. The answer ends up similar to what @jbowman (+1) showed for two predictors (one correlated with omitted variables and one not). Since I'd already started writing this, I'll post it in cas... | Omitted Variables and their consequences for other variables
Here's what happens for any number of predictors and omitted variables. The answer ends up similar to what @jbowman (+1) showed for two predictors (one correlated with omitted variables and one not). |
35,629 | VIF for Categorical Variable with More Than 2 Categories | The "generalized variance inflation factors" (GVIF) implemented in the vif() function of the R car package were designed by Fox and Monette specifically to handle situations like this, where there are groups of predictor variables that should be considered together rather than separately. Such situations include multi-... | VIF for Categorical Variable with More Than 2 Categories | The "generalized variance inflation factors" (GVIF) implemented in the vif() function of the R car package were designed by Fox and Monette specifically to handle situations like this, where there are | VIF for Categorical Variable with More Than 2 Categories
The "generalized variance inflation factors" (GVIF) implemented in the vif() function of the R car package were designed by Fox and Monette specifically to handle situations like this, where there are groups of predictor variables that should be considered togeth... | VIF for Categorical Variable with More Than 2 Categories
The "generalized variance inflation factors" (GVIF) implemented in the vif() function of the R car package were designed by Fox and Monette specifically to handle situations like this, where there are |
35,630 | Why do we not always use the closed testing principle for multiple comparisons? | It is actually used quite often (in fact, Bonferroni and Bonferroni-Holm can be shown to be valid tests using the closed testing principle - see below). Part of the reason why some simple procedures like Bonferroni are still so popular is of course, that they are very easy to implement and it is easy to communicate wha... | Why do we not always use the closed testing principle for multiple comparisons? | It is actually used quite often (in fact, Bonferroni and Bonferroni-Holm can be shown to be valid tests using the closed testing principle - see below). Part of the reason why some simple procedures l | Why do we not always use the closed testing principle for multiple comparisons?
It is actually used quite often (in fact, Bonferroni and Bonferroni-Holm can be shown to be valid tests using the closed testing principle - see below). Part of the reason why some simple procedures like Bonferroni are still so popular is o... | Why do we not always use the closed testing principle for multiple comparisons?
It is actually used quite often (in fact, Bonferroni and Bonferroni-Holm can be shown to be valid tests using the closed testing principle - see below). Part of the reason why some simple procedures l |
35,631 | Expected min distance between N Uniform RVs | You are correct.
One route to the solution, using only well-known relationships among these variables, is the following:
Generate iid Gamma$(1)$ variates $V_i,$ $i=0,1,2,\ldots, N.$ (These are exponentially distributed variables.)
Set $Y_1=V_0$ and $Y_i=Y_{i-1}+V_{i-1}$ for $i=1,\ldots, N+1.$ These are the partial c... | Expected min distance between N Uniform RVs | You are correct.
One route to the solution, using only well-known relationships among these variables, is the following:
Generate iid Gamma$(1)$ variates $V_i,$ $i=0,1,2,\ldots, N.$ (These are expon | Expected min distance between N Uniform RVs
You are correct.
One route to the solution, using only well-known relationships among these variables, is the following:
Generate iid Gamma$(1)$ variates $V_i,$ $i=0,1,2,\ldots, N.$ (These are exponentially distributed variables.)
Set $Y_1=V_0$ and $Y_i=Y_{i-1}+V_{i-1}$ for... | Expected min distance between N Uniform RVs
You are correct.
One route to the solution, using only well-known relationships among these variables, is the following:
Generate iid Gamma$(1)$ variates $V_i,$ $i=0,1,2,\ldots, N.$ (These are expon |
35,632 | Low effective sample size but good R-hat is this a problem? | In BDA3 on page 285, they mention that $\hat{R}$ is an estimate of "the factor by which the scale of the current distribution for [a particular univariate parameter] might be reduced if the simulations were continued in the limit $n \to \infty$." Because your $\hat{R}$ is so close to $1$, this means that it's unlikely ... | Low effective sample size but good R-hat is this a problem? | In BDA3 on page 285, they mention that $\hat{R}$ is an estimate of "the factor by which the scale of the current distribution for [a particular univariate parameter] might be reduced if the simulation | Low effective sample size but good R-hat is this a problem?
In BDA3 on page 285, they mention that $\hat{R}$ is an estimate of "the factor by which the scale of the current distribution for [a particular univariate parameter] might be reduced if the simulations were continued in the limit $n \to \infty$." Because your ... | Low effective sample size but good R-hat is this a problem?
In BDA3 on page 285, they mention that $\hat{R}$ is an estimate of "the factor by which the scale of the current distribution for [a particular univariate parameter] might be reduced if the simulation |
35,633 | Low effective sample size but good R-hat is this a problem? | An old thread, but still likely relevant to many users. Instead of increasing adapt_delta, you can try decreasing it to, say, 0.8 or a value that still does not lead to divergent iterations. My understanding is this increases the mixing and, thus, may increase the ESS. | Low effective sample size but good R-hat is this a problem? | An old thread, but still likely relevant to many users. Instead of increasing adapt_delta, you can try decreasing it to, say, 0.8 or a value that still does not lead to divergent iterations. My under | Low effective sample size but good R-hat is this a problem?
An old thread, but still likely relevant to many users. Instead of increasing adapt_delta, you can try decreasing it to, say, 0.8 or a value that still does not lead to divergent iterations. My understanding is this increases the mixing and, thus, may increas... | Low effective sample size but good R-hat is this a problem?
An old thread, but still likely relevant to many users. Instead of increasing adapt_delta, you can try decreasing it to, say, 0.8 or a value that still does not lead to divergent iterations. My under |
35,634 | Mean centering interaction terms | You don't have to mean-center variables that are included in interaction terms. Back in the dark ages when people did statistical calculations by hand on mechanical (not electronic) calculators having limited precision, there might have been some practical advantages to centering first. But modern computing hardware an... | Mean centering interaction terms | You don't have to mean-center variables that are included in interaction terms. Back in the dark ages when people did statistical calculations by hand on mechanical (not electronic) calculators having | Mean centering interaction terms
You don't have to mean-center variables that are included in interaction terms. Back in the dark ages when people did statistical calculations by hand on mechanical (not electronic) calculators having limited precision, there might have been some practical advantages to centering first.... | Mean centering interaction terms
You don't have to mean-center variables that are included in interaction terms. Back in the dark ages when people did statistical calculations by hand on mechanical (not electronic) calculators having |
35,635 | Argument on Interactions in The Book of Why | You are conflating linear in parameters with linear in variables. Linearity here refers to the relationship between the variables.
Their point in the book is that, if the model is not linear in the variables, then neither the equation
$$\text{Total Effect} = \text{Direct Effect} + \text{Indirect Effect} $$
holds, nor... | Argument on Interactions in The Book of Why | You are conflating linear in parameters with linear in variables. Linearity here refers to the relationship between the variables.
Their point in the book is that, if the model is not linear in the v | Argument on Interactions in The Book of Why
You are conflating linear in parameters with linear in variables. Linearity here refers to the relationship between the variables.
Their point in the book is that, if the model is not linear in the variables, then neither the equation
$$\text{Total Effect} = \text{Direct Ef... | Argument on Interactions in The Book of Why
You are conflating linear in parameters with linear in variables. Linearity here refers to the relationship between the variables.
Their point in the book is that, if the model is not linear in the v |
35,636 | Argument on Interactions in The Book of Why | "Purely linear" models do not allow for that. If you want to model an interaction using a particular case of the General Linear Model (do not mistake this for a Generalized Linear Model), you have to introduce an artificial extra variable like the product of the two interacting ones.
This new model is still linear with... | Argument on Interactions in The Book of Why | "Purely linear" models do not allow for that. If you want to model an interaction using a particular case of the General Linear Model (do not mistake this for a Generalized Linear Model), you have to | Argument on Interactions in The Book of Why
"Purely linear" models do not allow for that. If you want to model an interaction using a particular case of the General Linear Model (do not mistake this for a Generalized Linear Model), you have to introduce an artificial extra variable like the product of the two interacti... | Argument on Interactions in The Book of Why
"Purely linear" models do not allow for that. If you want to model an interaction using a particular case of the General Linear Model (do not mistake this for a Generalized Linear Model), you have to |
35,637 | What is the relationship between minimizing prediction error versus parameter estimation error? | With the advent of statistical learning techniques, people are talking
a lot about prediction error, while in classical statistics, one is
focusing on parameter estimation error.
Exactly. This difference can be properly understood only if we realize and keep in mind that the scope of model like regression, first of al... | What is the relationship between minimizing prediction error versus parameter estimation error? | With the advent of statistical learning techniques, people are talking
a lot about prediction error, while in classical statistics, one is
focusing on parameter estimation error.
Exactly. This differ | What is the relationship between minimizing prediction error versus parameter estimation error?
With the advent of statistical learning techniques, people are talking
a lot about prediction error, while in classical statistics, one is
focusing on parameter estimation error.
Exactly. This difference can be properly und... | What is the relationship between minimizing prediction error versus parameter estimation error?
With the advent of statistical learning techniques, people are talking
a lot about prediction error, while in classical statistics, one is
focusing on parameter estimation error.
Exactly. This differ |
35,638 | What is the relationship between minimizing prediction error versus parameter estimation error? | Multicollinearity
You could have multicollinearity which can make the variance in the error of estimates of $y$ and $\beta$ a lot different (typically the error in $y$ will have lower relative variance). See for more background: https://stats.stackexchange.com/tags/multicollinearity and https://en.wikipedia.org/wiki/M... | What is the relationship between minimizing prediction error versus parameter estimation error? | Multicollinearity
You could have multicollinearity which can make the variance in the error of estimates of $y$ and $\beta$ a lot different (typically the error in $y$ will have lower relative varianc | What is the relationship between minimizing prediction error versus parameter estimation error?
Multicollinearity
You could have multicollinearity which can make the variance in the error of estimates of $y$ and $\beta$ a lot different (typically the error in $y$ will have lower relative variance). See for more backgro... | What is the relationship between minimizing prediction error versus parameter estimation error?
Multicollinearity
You could have multicollinearity which can make the variance in the error of estimates of $y$ and $\beta$ a lot different (typically the error in $y$ will have lower relative varianc |
35,639 | VAE: why we do not sample again after decoding and before reconstruction loss? | The most important point stems from the confusion that the tilde $\sim$ implies a sampling operation. But $\sim$ does not imply that something is sampled, which is an algorithmic/computational operation. It indicates that something is distributed according to some distribution.
Now, when we train a VAE, we want to get ... | VAE: why we do not sample again after decoding and before reconstruction loss? | The most important point stems from the confusion that the tilde $\sim$ implies a sampling operation. But $\sim$ does not imply that something is sampled, which is an algorithmic/computational operati | VAE: why we do not sample again after decoding and before reconstruction loss?
The most important point stems from the confusion that the tilde $\sim$ implies a sampling operation. But $\sim$ does not imply that something is sampled, which is an algorithmic/computational operation. It indicates that something is distri... | VAE: why we do not sample again after decoding and before reconstruction loss?
The most important point stems from the confusion that the tilde $\sim$ implies a sampling operation. But $\sim$ does not imply that something is sampled, which is an algorithmic/computational operati |
35,640 | VAE: why we do not sample again after decoding and before reconstruction loss? | In the case of image generation, the network will usually output a reconstructed image $\hat{\mathbf{x}}$ and you will compute the reconstruction term as
\begin{equation}
\mathcal{L}_{\text{reconstr}}(\theta) = {\lVert \mathbf{x} - f_\theta(\mathbf{x}_i,\mathbf{z}) \rVert}^2 = {\lVert \mathbf{x} - \hat{\mathbf{x}} ... | VAE: why we do not sample again after decoding and before reconstruction loss? | In the case of image generation, the network will usually output a reconstructed image $\hat{\mathbf{x}}$ and you will compute the reconstruction term as
\begin{equation}
\mathcal{L}_{\text{recons | VAE: why we do not sample again after decoding and before reconstruction loss?
In the case of image generation, the network will usually output a reconstructed image $\hat{\mathbf{x}}$ and you will compute the reconstruction term as
\begin{equation}
\mathcal{L}_{\text{reconstr}}(\theta) = {\lVert \mathbf{x} - f_\th... | VAE: why we do not sample again after decoding and before reconstruction loss?
In the case of image generation, the network will usually output a reconstructed image $\hat{\mathbf{x}}$ and you will compute the reconstruction term as
\begin{equation}
\mathcal{L}_{\text{recons |
35,641 | Is there a standard way to Determine Sample Size when doing Bayesian Inference? | There is something corresponding to power analysis for Bayesians, it is called preposterior analysis. The preposterior expectation is the prior expectation of the posterior. One example can be found here. Another post where this concept is used is Minimizing variance of an estimator under sampling cost penalty.
Som... | Is there a standard way to Determine Sample Size when doing Bayesian Inference? | There is something corresponding to power analysis for Bayesians, it is called preposterior analysis. The preposterior expectation is the prior expectation of the posterior. One example can be found h | Is there a standard way to Determine Sample Size when doing Bayesian Inference?
There is something corresponding to power analysis for Bayesians, it is called preposterior analysis. The preposterior expectation is the prior expectation of the posterior. One example can be found here. Another post where this concept i... | Is there a standard way to Determine Sample Size when doing Bayesian Inference?
There is something corresponding to power analysis for Bayesians, it is called preposterior analysis. The preposterior expectation is the prior expectation of the posterior. One example can be found h |
35,642 | Is there a standard way to Determine Sample Size when doing Bayesian Inference? | One of the more obvious approach is to simulate with model parameters drawn from your prior beliefs, then draw your Bayesian inference and assess what proportion of simulations looks they way you'd like them to look.
Alternatively, you could make frequentist style assumptions (at least with respect to the size of the q... | Is there a standard way to Determine Sample Size when doing Bayesian Inference? | One of the more obvious approach is to simulate with model parameters drawn from your prior beliefs, then draw your Bayesian inference and assess what proportion of simulations looks they way you'd li | Is there a standard way to Determine Sample Size when doing Bayesian Inference?
One of the more obvious approach is to simulate with model parameters drawn from your prior beliefs, then draw your Bayesian inference and assess what proportion of simulations looks they way you'd like them to look.
Alternatively, you coul... | Is there a standard way to Determine Sample Size when doing Bayesian Inference?
One of the more obvious approach is to simulate with model parameters drawn from your prior beliefs, then draw your Bayesian inference and assess what proportion of simulations looks they way you'd li |
35,643 | What is this "phenomenon" called? | This answer is not a direct answer to your question, because it relates to a different cause of the pattern.
But it does relate to the same graphical appearance, and therefore I post it as an answer rather than a comment (before reading your Matlab script I actually thought the pattern in your histogram was due to thi... | What is this "phenomenon" called? | This answer is not a direct answer to your question, because it relates to a different cause of the pattern.
But it does relate to the same graphical appearance, and therefore I post it as an answer | What is this "phenomenon" called?
This answer is not a direct answer to your question, because it relates to a different cause of the pattern.
But it does relate to the same graphical appearance, and therefore I post it as an answer rather than a comment (before reading your Matlab script I actually thought the patter... | What is this "phenomenon" called?
This answer is not a direct answer to your question, because it relates to a different cause of the pattern.
But it does relate to the same graphical appearance, and therefore I post it as an answer |
35,644 | What is this "phenomenon" called? | UPDATE: Re-reading this four years later, let me attempt to answer the first part too, what to call it. If it is due to two distinct distributions, as I suggest below, then the combination is called a mixture distribution; I more often hear the term mixture model, which is what we'd use to model it. If an artifact of h... | What is this "phenomenon" called? | UPDATE: Re-reading this four years later, let me attempt to answer the first part too, what to call it. If it is due to two distinct distributions, as I suggest below, then the combination is called a | What is this "phenomenon" called?
UPDATE: Re-reading this four years later, let me attempt to answer the first part too, what to call it. If it is due to two distinct distributions, as I suggest below, then the combination is called a mixture distribution; I more often hear the term mixture model, which is what we'd us... | What is this "phenomenon" called?
UPDATE: Re-reading this four years later, let me attempt to answer the first part too, what to call it. If it is due to two distinct distributions, as I suggest below, then the combination is called a |
35,645 | Why does larger perplexity tend to produce clearer clusters in t-SNE? | The larger the perplexity, the more non-local information will be retained in the dimensionality reduction result.
Yes, I believe that this is a correct intuition. The way I think about perplexity parameter in t-SNE is that it sets the effective number of neighbours that each point is attracted to. In t-SNE optimisat... | Why does larger perplexity tend to produce clearer clusters in t-SNE? | The larger the perplexity, the more non-local information will be retained in the dimensionality reduction result.
Yes, I believe that this is a correct intuition. The way I think about perplexity p | Why does larger perplexity tend to produce clearer clusters in t-SNE?
The larger the perplexity, the more non-local information will be retained in the dimensionality reduction result.
Yes, I believe that this is a correct intuition. The way I think about perplexity parameter in t-SNE is that it sets the effective nu... | Why does larger perplexity tend to produce clearer clusters in t-SNE?
The larger the perplexity, the more non-local information will be retained in the dimensionality reduction result.
Yes, I believe that this is a correct intuition. The way I think about perplexity p |
35,646 | Why does larger perplexity tend to produce clearer clusters in t-SNE? | I believe this is because of this mismatch in t-SNE between the input (Gaussian) and output (student-t) distributions. It is beneficial to make such blobs in order to separate from everything else as required by the long tail of the t-distribution. The repulsive forces dominate.
In such cases SNE may work better. Have ... | Why does larger perplexity tend to produce clearer clusters in t-SNE? | I believe this is because of this mismatch in t-SNE between the input (Gaussian) and output (student-t) distributions. It is beneficial to make such blobs in order to separate from everything else as | Why does larger perplexity tend to produce clearer clusters in t-SNE?
I believe this is because of this mismatch in t-SNE between the input (Gaussian) and output (student-t) distributions. It is beneficial to make such blobs in order to separate from everything else as required by the long tail of the t-distribution. T... | Why does larger perplexity tend to produce clearer clusters in t-SNE?
I believe this is because of this mismatch in t-SNE between the input (Gaussian) and output (student-t) distributions. It is beneficial to make such blobs in order to separate from everything else as |
35,647 | What is the $p$ in Bernoulli distribution? | In the Bayesian theory of probability, probability is our expression
of knowledge about a certain thing, not a property of that thing.
However, I always see people treat $p$ as a parameter that needs to be
estimated. They set up a prior for $p$, usually in the form of a beta
function and then update it as "real... | What is the $p$ in Bernoulli distribution? | In the Bayesian theory of probability, probability is our expression
of knowledge about a certain thing, not a property of that thing.
However, I always see people treat $p$ as a parameter that ne | What is the $p$ in Bernoulli distribution?
In the Bayesian theory of probability, probability is our expression
of knowledge about a certain thing, not a property of that thing.
However, I always see people treat $p$ as a parameter that needs to be
estimated. They set up a prior for $p$, usually in the form of a ... | What is the $p$ in Bernoulli distribution?
In the Bayesian theory of probability, probability is our expression
of knowledge about a certain thing, not a property of that thing.
However, I always see people treat $p$ as a parameter that ne |
35,648 | What is the $p$ in Bernoulli distribution? | $p$ is a parameter which specifies the "success probability", for which we have prior and posterior probability distributions.
We may for example have a coin for which we are not sure if it is fair ($p=0.5$) or not ($p\neq 0.5$). Even so, fairness, or lack thereof, is a property of the coin. We just happen to be unsur... | What is the $p$ in Bernoulli distribution? | $p$ is a parameter which specifies the "success probability", for which we have prior and posterior probability distributions.
We may for example have a coin for which we are not sure if it is fair ( | What is the $p$ in Bernoulli distribution?
$p$ is a parameter which specifies the "success probability", for which we have prior and posterior probability distributions.
We may for example have a coin for which we are not sure if it is fair ($p=0.5$) or not ($p\neq 0.5$). Even so, fairness, or lack thereof, is a prope... | What is the $p$ in Bernoulli distribution?
$p$ is a parameter which specifies the "success probability", for which we have prior and posterior probability distributions.
We may for example have a coin for which we are not sure if it is fair ( |
35,649 | What is the $p$ in Bernoulli distribution? | For a random variable $X \sim \operatorname{Bernoulli}(p)$ defined on a probability space $(\Omega, \mathcal{F}, P)$, the parameter $p$ (a number) is the probability of a certain event, namely, the event $\{X = 1\}$.
That is,
$$
p = P(X = 1).
$$
The single number $p$ completely determines the distribution of $X$ since ... | What is the $p$ in Bernoulli distribution? | For a random variable $X \sim \operatorname{Bernoulli}(p)$ defined on a probability space $(\Omega, \mathcal{F}, P)$, the parameter $p$ (a number) is the probability of a certain event, namely, the ev | What is the $p$ in Bernoulli distribution?
For a random variable $X \sim \operatorname{Bernoulli}(p)$ defined on a probability space $(\Omega, \mathcal{F}, P)$, the parameter $p$ (a number) is the probability of a certain event, namely, the event $\{X = 1\}$.
That is,
$$
p = P(X = 1).
$$
The single number $p$ completel... | What is the $p$ in Bernoulli distribution?
For a random variable $X \sim \operatorname{Bernoulli}(p)$ defined on a probability space $(\Omega, \mathcal{F}, P)$, the parameter $p$ (a number) is the probability of a certain event, namely, the ev |
35,650 | Expectation of Inverse Logit of Normal Random Variable | As mentioned already in the question comments and answer by @Martijn there doesn't appear to be an analytical solution for $E(Y)$ apart from the special case where $\mu = 0$ which gives $E(Y) = 0.5$.
Additionally by Jensen's inequality we have that $E(Y) = E(f(X)) < f(E(X))$ if $\mu > 0$ and conversely that $E(Y) = E(f... | Expectation of Inverse Logit of Normal Random Variable | As mentioned already in the question comments and answer by @Martijn there doesn't appear to be an analytical solution for $E(Y)$ apart from the special case where $\mu = 0$ which gives $E(Y) = 0.5$.
| Expectation of Inverse Logit of Normal Random Variable
As mentioned already in the question comments and answer by @Martijn there doesn't appear to be an analytical solution for $E(Y)$ apart from the special case where $\mu = 0$ which gives $E(Y) = 0.5$.
Additionally by Jensen's inequality we have that $E(Y) = E(f(X)) ... | Expectation of Inverse Logit of Normal Random Variable
As mentioned already in the question comments and answer by @Martijn there doesn't appear to be an analytical solution for $E(Y)$ apart from the special case where $\mu = 0$ which gives $E(Y) = 0.5$.
|
35,651 | Expectation of Inverse Logit of Normal Random Variable | The variable $Y$ has a logit normal or logistic normal distribution whose moments have no known analytical description. You can obtain the values computationally.
More about these distributions is described in a freely available article: Atchison, J., and Sheng M. Shen. "Logistic-normal distributions: Some properties a... | Expectation of Inverse Logit of Normal Random Variable | The variable $Y$ has a logit normal or logistic normal distribution whose moments have no known analytical description. You can obtain the values computationally.
More about these distributions is des | Expectation of Inverse Logit of Normal Random Variable
The variable $Y$ has a logit normal or logistic normal distribution whose moments have no known analytical description. You can obtain the values computationally.
More about these distributions is described in a freely available article: Atchison, J., and Sheng M. ... | Expectation of Inverse Logit of Normal Random Variable
The variable $Y$ has a logit normal or logistic normal distribution whose moments have no known analytical description. You can obtain the values computationally.
More about these distributions is des |
35,652 | What is the difference between $E[\varepsilon\mid X]=0$ and $E[\varepsilon X]=0$ in OLS regression? | I'm just looking from a probabilistic perspective, and looking forward to hear other ideas. Using law of iterated expectations, we have
$$E[\epsilon X]=E[E[\epsilon X\mid X]]=E[X E[\epsilon\mid X]]$$
If $E[\epsilon\mid X]$ is $0,$ then automatically, $E[\epsilon X]$ is $0.$ But, the other way around is not true, which ... | What is the difference between $E[\varepsilon\mid X]=0$ and $E[\varepsilon X]=0$ in OLS regression? | I'm just looking from a probabilistic perspective, and looking forward to hear other ideas. Using law of iterated expectations, we have
$$E[\epsilon X]=E[E[\epsilon X\mid X]]=E[X E[\epsilon\mid X]]$$
| What is the difference between $E[\varepsilon\mid X]=0$ and $E[\varepsilon X]=0$ in OLS regression?
I'm just looking from a probabilistic perspective, and looking forward to hear other ideas. Using law of iterated expectations, we have
$$E[\epsilon X]=E[E[\epsilon X\mid X]]=E[X E[\epsilon\mid X]]$$
If $E[\epsilon\mid X... | What is the difference between $E[\varepsilon\mid X]=0$ and $E[\varepsilon X]=0$ in OLS regression?
I'm just looking from a probabilistic perspective, and looking forward to hear other ideas. Using law of iterated expectations, we have
$$E[\epsilon X]=E[E[\epsilon X\mid X]]=E[X E[\epsilon\mid X]]$$
|
35,653 | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$ | By googling the keyword 'estimation of location parameter of Cauchy distribution', I gather that this has been a pretty well-studied problem. Similar to the answer by Robin Ryder, I suggest taking a look at the paper A Note on Estimation from a Cauchy Sample (Rothenberg et al., 1964) for an unbiased estimator of $\thet... | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$ | By googling the keyword 'estimation of location parameter of Cauchy distribution', I gather that this has been a pretty well-studied problem. Similar to the answer by Robin Ryder, I suggest taking a l | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
By googling the keyword 'estimation of location parameter of Cauchy distribution', I gather that this has been a pretty well-studied problem. Similar to the answer by Robin Ryder, I suggest taking a look at the paper A Note on Es... | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
By googling the keyword 'estimation of location parameter of Cauchy distribution', I gather that this has been a pretty well-studied problem. Similar to the answer by Robin Ryder, I suggest taking a l |
35,654 | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$ | Indeed, computing the MLE is complicated. The most straightforward choice is to use the sample median, which is consistent, but there are more efficient estimators available.
Zhang (2010) shows that a more efficient estimator is to take the mean of the central 24% of the data, i.e. use the order statistics $X_{(1)}\leq... | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$ | Indeed, computing the MLE is complicated. The most straightforward choice is to use the sample median, which is consistent, but there are more efficient estimators available.
Zhang (2010) shows that a | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
Indeed, computing the MLE is complicated. The most straightforward choice is to use the sample median, which is consistent, but there are more efficient estimators available.
Zhang (2010) shows that a more efficient estimator is ... | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
Indeed, computing the MLE is complicated. The most straightforward choice is to use the sample median, which is consistent, but there are more efficient estimators available.
Zhang (2010) shows that a |
35,655 | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$ | This is an answer I've only just come up with, so there might be a mistake somewhere. First I use a demonstration to show you how this works, and then give you a consistent estimator.
Assume for the sake of demonstraton that $EX$ exists. That is, $EX < \infty$, and take it as the mode of the distribution as with other... | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$ | This is an answer I've only just come up with, so there might be a mistake somewhere. First I use a demonstration to show you how this works, and then give you a consistent estimator.
Assume for the | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
This is an answer I've only just come up with, so there might be a mistake somewhere. First I use a demonstration to show you how this works, and then give you a consistent estimator.
Assume for the sake of demonstraton that $EX... | Consistent unbiased estimator for the location parameter of $\mathcal{Cauchy} (\theta, 1)$
This is an answer I've only just come up with, so there might be a mistake somewhere. First I use a demonstration to show you how this works, and then give you a consistent estimator.
Assume for the |
35,656 | Connection between filters and feature map in CNN | Your confusion stems from the fact that channels (feature maps) are treated somewhat differently than other dimensions.
Let's say you have a grayscale image input to the first layer and 32 kernels of shape (3,3) as per your example. But in fact, those kernels have shape (3,3,1) - 1 for the number of channels in the inp... | Connection between filters and feature map in CNN | Your confusion stems from the fact that channels (feature maps) are treated somewhat differently than other dimensions.
Let's say you have a grayscale image input to the first layer and 32 kernels of | Connection between filters and feature map in CNN
Your confusion stems from the fact that channels (feature maps) are treated somewhat differently than other dimensions.
Let's say you have a grayscale image input to the first layer and 32 kernels of shape (3,3) as per your example. But in fact, those kernels have shape... | Connection between filters and feature map in CNN
Your confusion stems from the fact that channels (feature maps) are treated somewhat differently than other dimensions.
Let's say you have a grayscale image input to the first layer and 32 kernels of |
35,657 | Does Regularized Logistic Regression Produce Calibrated Results? | The answer depends entirely on the amount of penalization used. If too little, the model will be seen to be overfitted when evaluated on an independent sample. If too much penality, it will be found to be underfitted. The goal is to solve for the penalty that gets it "just right". Two ways to do this are cross-vali... | Does Regularized Logistic Regression Produce Calibrated Results? | The answer depends entirely on the amount of penalization used. If too little, the model will be seen to be overfitted when evaluated on an independent sample. If too much penality, it will be found | Does Regularized Logistic Regression Produce Calibrated Results?
The answer depends entirely on the amount of penalization used. If too little, the model will be seen to be overfitted when evaluated on an independent sample. If too much penality, it will be found to be underfitted. The goal is to solve for the penal... | Does Regularized Logistic Regression Produce Calibrated Results?
The answer depends entirely on the amount of penalization used. If too little, the model will be seen to be overfitted when evaluated on an independent sample. If too much penality, it will be found |
35,658 | General conditional distributions for multivariate Gaussian mixtures | So, after a little while, I am able to answer my own question :) According to the definition of conditional density, we have
$$ f(\vec{x_a} | \vec{x_b}) = \frac{f(\vec{x_a}, \vec{x_b})}{f(\vec{x_b})} = \sum_{k=1}^{n} \frac{\pi_k}{f(\vec{x_b})} \mathcal{N}(\vec{x} | \mu_k, \Sigma_k). $$
where $ f(\vec{x_b}) $ is the mar... | General conditional distributions for multivariate Gaussian mixtures | So, after a little while, I am able to answer my own question :) According to the definition of conditional density, we have
$$ f(\vec{x_a} | \vec{x_b}) = \frac{f(\vec{x_a}, \vec{x_b})}{f(\vec{x_b})} | General conditional distributions for multivariate Gaussian mixtures
So, after a little while, I am able to answer my own question :) According to the definition of conditional density, we have
$$ f(\vec{x_a} | \vec{x_b}) = \frac{f(\vec{x_a}, \vec{x_b})}{f(\vec{x_b})} = \sum_{k=1}^{n} \frac{\pi_k}{f(\vec{x_b})} \mathca... | General conditional distributions for multivariate Gaussian mixtures
So, after a little while, I am able to answer my own question :) According to the definition of conditional density, we have
$$ f(\vec{x_a} | \vec{x_b}) = \frac{f(\vec{x_a}, \vec{x_b})}{f(\vec{x_b})} |
35,659 | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score to choose model? | To elaborate on some of the comments. Most models give a probability output--- say 0.78. If you convert this to a 1, then you have made a "decision". Accuracy, ROC, precision, recall, are all functions of decisions; log likelihood, brier score are functions of probabilities. Often, it's best to leave the decision t... | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score | To elaborate on some of the comments. Most models give a probability output--- say 0.78. If you convert this to a 1, then you have made a "decision". Accuracy, ROC, precision, recall, are all funct | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score to choose model?
To elaborate on some of the comments. Most models give a probability output--- say 0.78. If you convert this to a 1, then you have made a "decision". Accuracy, ROC, precision, recall, are all functio... | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score
To elaborate on some of the comments. Most models give a probability output--- say 0.78. If you convert this to a 1, then you have made a "decision". Accuracy, ROC, precision, recall, are all funct |
35,660 | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score to choose model? | Never use accuracy to evaluate a binary classifier. Here’s an extremely accurate prediction you can make at any party you attend: no one present is an astronaut. Because there are about 8 billion people in the world and only about 600 astronauts, this prediction will almost always be 100% accurate. It will also never c... | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score | Never use accuracy to evaluate a binary classifier. Here’s an extremely accurate prediction you can make at any party you attend: no one present is an astronaut. Because there are about 8 billion peop | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score to choose model?
Never use accuracy to evaluate a binary classifier. Here’s an extremely accurate prediction you can make at any party you attend: no one present is an astronaut. Because there are about 8 billion people... | In binary classification, in what specific case should I use accuracy, AUROC, logloss or Brier score
Never use accuracy to evaluate a binary classifier. Here’s an extremely accurate prediction you can make at any party you attend: no one present is an astronaut. Because there are about 8 billion peop |
35,661 | What is the "binary:logistic" objective function in XGBoost? | It appears there is an option objective: "binary:logistic"
“binary:logistic” –logistic regression for binary classification,
output probability
“binary:logitraw” –logistic regression for binary
classification, output score before logistic transformation
See http://xgboost.readthedocs.io/en/latest/parameter.html
(so l... | What is the "binary:logistic" objective function in XGBoost? | It appears there is an option objective: "binary:logistic"
“binary:logistic” –logistic regression for binary classification,
output probability
“binary:logitraw” –logistic regression for binary
class | What is the "binary:logistic" objective function in XGBoost?
It appears there is an option objective: "binary:logistic"
“binary:logistic” –logistic regression for binary classification,
output probability
“binary:logitraw” –logistic regression for binary
classification, output score before logistic transformation
See... | What is the "binary:logistic" objective function in XGBoost?
It appears there is an option objective: "binary:logistic"
“binary:logistic” –logistic regression for binary classification,
output probability
“binary:logitraw” –logistic regression for binary
class |
35,662 | What is the "binary:logistic" objective function in XGBoost? | I'm just summarizing the comments under the problem description:
$l$ is the log likelihood of the bernoulli distribution, where
$\text{logistic}(\hat{y}_i^{(t-1)} + f_t(x_i))$ is the probability.
So the formula should be
$$l = y_i \log(\text{logistic}(\hat{y}_i^{(t-1)} + f_t(x_i))) + (1-y_i)\log(1-\text{logistic}(\hat... | What is the "binary:logistic" objective function in XGBoost? | I'm just summarizing the comments under the problem description:
$l$ is the log likelihood of the bernoulli distribution, where
$\text{logistic}(\hat{y}_i^{(t-1)} + f_t(x_i))$ is the probability.
So | What is the "binary:logistic" objective function in XGBoost?
I'm just summarizing the comments under the problem description:
$l$ is the log likelihood of the bernoulli distribution, where
$\text{logistic}(\hat{y}_i^{(t-1)} + f_t(x_i))$ is the probability.
So the formula should be
$$l = y_i \log(\text{logistic}(\hat{y... | What is the "binary:logistic" objective function in XGBoost?
I'm just summarizing the comments under the problem description:
$l$ is the log likelihood of the bernoulli distribution, where
$\text{logistic}(\hat{y}_i^{(t-1)} + f_t(x_i))$ is the probability.
So |
35,663 | What are soft policies in reinforcement learning? | what are soft policies in reinforcement learning?
A "soft" policy is one that has some, usually small but finite, probability of selecting any possible action. Having a policy which has some chance of selecting any action is important theoretically when rewards and/or state transitions are stochastic - you are never 1... | What are soft policies in reinforcement learning? | what are soft policies in reinforcement learning?
A "soft" policy is one that has some, usually small but finite, probability of selecting any possible action. Having a policy which has some chance o | What are soft policies in reinforcement learning?
what are soft policies in reinforcement learning?
A "soft" policy is one that has some, usually small but finite, probability of selecting any possible action. Having a policy which has some chance of selecting any action is important theoretically when rewards and/or ... | What are soft policies in reinforcement learning?
what are soft policies in reinforcement learning?
A "soft" policy is one that has some, usually small but finite, probability of selecting any possible action. Having a policy which has some chance o |
35,664 | What values should initial weights for a ReLU network be? | There has been quite a lot of theoretical work on Neural Network initialization in the last 5 years, which apparently hasn't still propagated to the wider Deep Learning community. While it's true that there isn't still an initialization which works for all architectures and for all activation functions (and most likely... | What values should initial weights for a ReLU network be? | There has been quite a lot of theoretical work on Neural Network initialization in the last 5 years, which apparently hasn't still propagated to the wider Deep Learning community. While it's true that | What values should initial weights for a ReLU network be?
There has been quite a lot of theoretical work on Neural Network initialization in the last 5 years, which apparently hasn't still propagated to the wider Deep Learning community. While it's true that there isn't still an initialization which works for all archi... | What values should initial weights for a ReLU network be?
There has been quite a lot of theoretical work on Neural Network initialization in the last 5 years, which apparently hasn't still propagated to the wider Deep Learning community. While it's true that |
35,665 | What values should initial weights for a ReLU network be? | Initializations are a topic where only two, rather unhelpful, facts are known for certain:
Neural networks can be very sensitive to the initialization strategy that you use.
There is no universal "best choice" that is consistently good across different problem types, network architectures, activation functions, and d... | What values should initial weights for a ReLU network be? | Initializations are a topic where only two, rather unhelpful, facts are known for certain:
Neural networks can be very sensitive to the initialization strategy that you use.
There is no universal "b | What values should initial weights for a ReLU network be?
Initializations are a topic where only two, rather unhelpful, facts are known for certain:
Neural networks can be very sensitive to the initialization strategy that you use.
There is no universal "best choice" that is consistently good across different problem... | What values should initial weights for a ReLU network be?
Initializations are a topic where only two, rather unhelpful, facts are known for certain:
Neural networks can be very sensitive to the initialization strategy that you use.
There is no universal "b |
35,666 | Technical point about convergence with conditional expectation | Yes, $X_{n} \to 0$ almost surely. The argument I have is a little convoluted, so bear with me.
First, consider the events $F_{k} = \bigcup_{n \geq k} \{ C_{n} > 2 \}$. By the almost sure convergence of the $C_{n}$ it follows that $P( \bigcap_{k} F_{k} ) = 0$, and since $F_{1} \supseteq F_{2} \supseteq \cdots$ we have $... | Technical point about convergence with conditional expectation | Yes, $X_{n} \to 0$ almost surely. The argument I have is a little convoluted, so bear with me.
First, consider the events $F_{k} = \bigcup_{n \geq k} \{ C_{n} > 2 \}$. By the almost sure convergence o | Technical point about convergence with conditional expectation
Yes, $X_{n} \to 0$ almost surely. The argument I have is a little convoluted, so bear with me.
First, consider the events $F_{k} = \bigcup_{n \geq k} \{ C_{n} > 2 \}$. By the almost sure convergence of the $C_{n}$ it follows that $P( \bigcap_{k} F_{k} ) = 0... | Technical point about convergence with conditional expectation
Yes, $X_{n} \to 0$ almost surely. The argument I have is a little convoluted, so bear with me.
First, consider the events $F_{k} = \bigcup_{n \geq k} \{ C_{n} > 2 \}$. By the almost sure convergence o |
35,667 | Technical point about convergence with conditional expectation | Set $Z_n=X_n/C_n$. Then $E[Z_n]=1/n^2$ and $Z_n\ge 0$. By Markov's inequality,
$P(Z_n>\epsilon)\le E[Z_n]/\epsilon = 1/(n^2\epsilon)$ which has finite sum, so by Borel Cantelli, $P(Z_n>\epsilon \text{ infinitely often})=0$ and $Z_n\to 0$ almost surely.
If $Z_n\to0$ almost surely and $C_n\to 1$ almost surely then $X_n=Z... | Technical point about convergence with conditional expectation | Set $Z_n=X_n/C_n$. Then $E[Z_n]=1/n^2$ and $Z_n\ge 0$. By Markov's inequality,
$P(Z_n>\epsilon)\le E[Z_n]/\epsilon = 1/(n^2\epsilon)$ which has finite sum, so by Borel Cantelli, $P(Z_n>\epsilon \text{ | Technical point about convergence with conditional expectation
Set $Z_n=X_n/C_n$. Then $E[Z_n]=1/n^2$ and $Z_n\ge 0$. By Markov's inequality,
$P(Z_n>\epsilon)\le E[Z_n]/\epsilon = 1/(n^2\epsilon)$ which has finite sum, so by Borel Cantelli, $P(Z_n>\epsilon \text{ infinitely often})=0$ and $Z_n\to 0$ almost surely.
If $... | Technical point about convergence with conditional expectation
Set $Z_n=X_n/C_n$. Then $E[Z_n]=1/n^2$ and $Z_n\ge 0$. By Markov's inequality,
$P(Z_n>\epsilon)\le E[Z_n]/\epsilon = 1/(n^2\epsilon)$ which has finite sum, so by Borel Cantelli, $P(Z_n>\epsilon \text{ |
35,668 | Interpretation of "score" | This one is an interesting historical curiosity. The term "score" was initially used in Fisher (1935) in a statistical problem involving the analysis of genetic properties of families with a parent that has a genetic abnormality. Fisher noted that the children of an abnormal parent can be classified into four classes... | Interpretation of "score" | This one is an interesting historical curiosity. The term "score" was initially used in Fisher (1935) in a statistical problem involving the analysis of genetic properties of families with a parent t | Interpretation of "score"
This one is an interesting historical curiosity. The term "score" was initially used in Fisher (1935) in a statistical problem involving the analysis of genetic properties of families with a parent that has a genetic abnormality. Fisher noted that the children of an abnormal parent can be cl... | Interpretation of "score"
This one is an interesting historical curiosity. The term "score" was initially used in Fisher (1935) in a statistical problem involving the analysis of genetic properties of families with a parent t |
35,669 | Should k-means only be applied if the variables are normally distributed? | The k-means algorithm minimizes the following objective function
$\displaystyle\sum_{i=1}^n\left\|x_i-\mu_i\right\|^2=\sum_{i=1}^n\sum_{j=1}^p\left(x_i^{(j)}-\mu_i^{(j)}\right)^2=\sum_{j=1}^p\sum_{i=1}^n\left(x_i^{(j)}-\mu_i^{(j)}\right)^2$
where $\mu_i$ is the mean of the cluster associated with observation $i$ and $... | Should k-means only be applied if the variables are normally distributed? | The k-means algorithm minimizes the following objective function
$\displaystyle\sum_{i=1}^n\left\|x_i-\mu_i\right\|^2=\sum_{i=1}^n\sum_{j=1}^p\left(x_i^{(j)}-\mu_i^{(j)}\right)^2=\sum_{j=1}^p\sum_{i=1 | Should k-means only be applied if the variables are normally distributed?
The k-means algorithm minimizes the following objective function
$\displaystyle\sum_{i=1}^n\left\|x_i-\mu_i\right\|^2=\sum_{i=1}^n\sum_{j=1}^p\left(x_i^{(j)}-\mu_i^{(j)}\right)^2=\sum_{j=1}^p\sum_{i=1}^n\left(x_i^{(j)}-\mu_i^{(j)}\right)^2$
wher... | Should k-means only be applied if the variables are normally distributed?
The k-means algorithm minimizes the following objective function
$\displaystyle\sum_{i=1}^n\left\|x_i-\mu_i\right\|^2=\sum_{i=1}^n\sum_{j=1}^p\left(x_i^{(j)}-\mu_i^{(j)}\right)^2=\sum_{j=1}^p\sum_{i=1 |
35,670 | Should k-means only be applied if the variables are normally distributed? | It would help if you were to provide information regarding the magnitude of the noise or overdispersion in these features. This can be as simple as providing means, medians and std deviations in a tabular format.
My experience is that K-means, as with many traditional methods, is fairly robust to violations of normali... | Should k-means only be applied if the variables are normally distributed? | It would help if you were to provide information regarding the magnitude of the noise or overdispersion in these features. This can be as simple as providing means, medians and std deviations in a tab | Should k-means only be applied if the variables are normally distributed?
It would help if you were to provide information regarding the magnitude of the noise or overdispersion in these features. This can be as simple as providing means, medians and std deviations in a tabular format.
My experience is that K-means, a... | Should k-means only be applied if the variables are normally distributed?
It would help if you were to provide information regarding the magnitude of the noise or overdispersion in these features. This can be as simple as providing means, medians and std deviations in a tab |
35,671 | Confusion around Bellman (update) operator | The precise terminology might differ across sources, but according to the source I learned from, the first one is called the Bellman optimality equation or simply the Bellman equation. The second one is the Bellman expectation equation. The expectation equation is linear in $V$, so you can solve for $V$ using simple li... | Confusion around Bellman (update) operator | The precise terminology might differ across sources, but according to the source I learned from, the first one is called the Bellman optimality equation or simply the Bellman equation. The second one | Confusion around Bellman (update) operator
The precise terminology might differ across sources, but according to the source I learned from, the first one is called the Bellman optimality equation or simply the Bellman equation. The second one is the Bellman expectation equation. The expectation equation is linear in $V... | Confusion around Bellman (update) operator
The precise terminology might differ across sources, but according to the source I learned from, the first one is called the Bellman optimality equation or simply the Bellman equation. The second one |
35,672 | Confusion around Bellman (update) operator | A good resource will be the classical textbook on reinforcement learning Reinforcement Learning: An Introduction, mostly the part about dynamic programming. Another good resource will be Berkeley's opencourse on Artificial Intelligence on EdX.
The difference in their name (Bellman operator vs Bellman update operator) d... | Confusion around Bellman (update) operator | A good resource will be the classical textbook on reinforcement learning Reinforcement Learning: An Introduction, mostly the part about dynamic programming. Another good resource will be Berkeley's op | Confusion around Bellman (update) operator
A good resource will be the classical textbook on reinforcement learning Reinforcement Learning: An Introduction, mostly the part about dynamic programming. Another good resource will be Berkeley's opencourse on Artificial Intelligence on EdX.
The difference in their name (Bel... | Confusion around Bellman (update) operator
A good resource will be the classical textbook on reinforcement learning Reinforcement Learning: An Introduction, mostly the part about dynamic programming. Another good resource will be Berkeley's op |
35,673 | Looking for advice regarding model selection for forecasting (dynamic?) panel data | Excellent question(s). Benny_dref's suggestions are useful. The first sentence of the OPs query is concerned with "selecting an appropriate forecasting model for panel data." With several DVs, more than a few predictors and a wide variety of possible methods and models to pursue, one has two broad avenues wrt model sel... | Looking for advice regarding model selection for forecasting (dynamic?) panel data | Excellent question(s). Benny_dref's suggestions are useful. The first sentence of the OPs query is concerned with "selecting an appropriate forecasting model for panel data." With several DVs, more th | Looking for advice regarding model selection for forecasting (dynamic?) panel data
Excellent question(s). Benny_dref's suggestions are useful. The first sentence of the OPs query is concerned with "selecting an appropriate forecasting model for panel data." With several DVs, more than a few predictors and a wide variet... | Looking for advice regarding model selection for forecasting (dynamic?) panel data
Excellent question(s). Benny_dref's suggestions are useful. The first sentence of the OPs query is concerned with "selecting an appropriate forecasting model for panel data." With several DVs, more th |
35,674 | Looking for advice regarding model selection for forecasting (dynamic?) panel data | Without seeing data this is difficult to answer but because the data is quarterly, it appears you have time-series data with categorical dimensions.
If you're trying to build a regression and use the outputs for forecasting, building a multivariate model with only 30 observations is going to be difficult, and your mode... | Looking for advice regarding model selection for forecasting (dynamic?) panel data | Without seeing data this is difficult to answer but because the data is quarterly, it appears you have time-series data with categorical dimensions.
If you're trying to build a regression and use the | Looking for advice regarding model selection for forecasting (dynamic?) panel data
Without seeing data this is difficult to answer but because the data is quarterly, it appears you have time-series data with categorical dimensions.
If you're trying to build a regression and use the outputs for forecasting, building a m... | Looking for advice regarding model selection for forecasting (dynamic?) panel data
Without seeing data this is difficult to answer but because the data is quarterly, it appears you have time-series data with categorical dimensions.
If you're trying to build a regression and use the |
35,675 | Different variance-covariance matrices of random effects per fixed-effect group in lme4 | Thanks to @amoeba and using @BenBolker's brief remark here
Extensions such as allowing different residual variances or different variance-covariance matrices of random effects per (fixed-effect) group can be achieved, somewhat clunkily, by using the dummy() helper function to construct an indicator variable to multipl... | Different variance-covariance matrices of random effects per fixed-effect group in lme4 | Thanks to @amoeba and using @BenBolker's brief remark here
Extensions such as allowing different residual variances or different variance-covariance matrices of random effects per (fixed-effect) grou | Different variance-covariance matrices of random effects per fixed-effect group in lme4
Thanks to @amoeba and using @BenBolker's brief remark here
Extensions such as allowing different residual variances or different variance-covariance matrices of random effects per (fixed-effect) group can be achieved, somewhat clun... | Different variance-covariance matrices of random effects per fixed-effect group in lme4
Thanks to @amoeba and using @BenBolker's brief remark here
Extensions such as allowing different residual variances or different variance-covariance matrices of random effects per (fixed-effect) grou |
35,676 | How to test if multiple regression coefficients are not statistically different? | You can use the $F$ test to test any linear restrictions $L$ on your coefficients.
Let your null hypothesis be $H_0:L\beta = c$ and your design matrix $X$ with rank $k$. Then the $F$ statistic will be:
$$
F = \frac{(L\hat{\beta}- c)'(\hat{\sigma}^2L(X'X)^{-1}L')^{-1}(L\hat{\beta} - c)}{q}
$$
where $q$ is the number of ... | How to test if multiple regression coefficients are not statistically different? | You can use the $F$ test to test any linear restrictions $L$ on your coefficients.
Let your null hypothesis be $H_0:L\beta = c$ and your design matrix $X$ with rank $k$. Then the $F$ statistic will be | How to test if multiple regression coefficients are not statistically different?
You can use the $F$ test to test any linear restrictions $L$ on your coefficients.
Let your null hypothesis be $H_0:L\beta = c$ and your design matrix $X$ with rank $k$. Then the $F$ statistic will be:
$$
F = \frac{(L\hat{\beta}- c)'(\hat{... | How to test if multiple regression coefficients are not statistically different?
You can use the $F$ test to test any linear restrictions $L$ on your coefficients.
Let your null hypothesis be $H_0:L\beta = c$ and your design matrix $X$ with rank $k$. Then the $F$ statistic will be |
35,677 | Simulate AR(1) process in R with specified nonzero mean and AR coefficient | Notice that this is the same model as
$$
X_t - 10 = .5(X_{t-1} - 10) + Z_t .
$$
$10$ is the mean, while $5$ was the intercept. This means we can add ten to the mean zero series.
In other words, if you define $Y_t = X_t - 10$, then
$$
Y_t = .5 Y_{t-1} + Z_t
$$
and use arima.sim to simulate that. After you have $\{Y_t... | Simulate AR(1) process in R with specified nonzero mean and AR coefficient | Notice that this is the same model as
$$
X_t - 10 = .5(X_{t-1} - 10) + Z_t .
$$
$10$ is the mean, while $5$ was the intercept. This means we can add ten to the mean zero series.
In other words, if y | Simulate AR(1) process in R with specified nonzero mean and AR coefficient
Notice that this is the same model as
$$
X_t - 10 = .5(X_{t-1} - 10) + Z_t .
$$
$10$ is the mean, while $5$ was the intercept. This means we can add ten to the mean zero series.
In other words, if you define $Y_t = X_t - 10$, then
$$
Y_t = .5... | Simulate AR(1) process in R with specified nonzero mean and AR coefficient
Notice that this is the same model as
$$
X_t - 10 = .5(X_{t-1} - 10) + Z_t .
$$
$10$ is the mean, while $5$ was the intercept. This means we can add ten to the mean zero series.
In other words, if y |
35,678 | Question on Covariance for sampling without replacement | Problems in sampling from finite populations without replacement can usually be solved in terms of the sample inclusion probabilities $\pi(x)$, $\pi(x,y)$, etc.
Let $\pi(x) = \Pr(X_1 = x)$ for any $x$ in the population $\mathcal P$ (with $n=10$ elements) and let $\pi(x,y)=\Pr((X_1,X_2)=(x,y))$ for any $x$ and $y$ in $... | Question on Covariance for sampling without replacement | Problems in sampling from finite populations without replacement can usually be solved in terms of the sample inclusion probabilities $\pi(x)$, $\pi(x,y)$, etc.
Let $\pi(x) = \Pr(X_1 = x)$ for any $x | Question on Covariance for sampling without replacement
Problems in sampling from finite populations without replacement can usually be solved in terms of the sample inclusion probabilities $\pi(x)$, $\pi(x,y)$, etc.
Let $\pi(x) = \Pr(X_1 = x)$ for any $x$ in the population $\mathcal P$ (with $n=10$ elements) and let ... | Question on Covariance for sampling without replacement
Problems in sampling from finite populations without replacement can usually be solved in terms of the sample inclusion probabilities $\pi(x)$, $\pi(x,y)$, etc.
Let $\pi(x) = \Pr(X_1 = x)$ for any $x |
35,679 | Visualizing a multilevel model (HLM) in ggplot2 | When you fit an lmer model, you can use the coef() function to extract the coefficients from the model. Your code would look something like:
mod1 <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | Country), data=dat)
Then you can call coef() and extract the coefficients for each group by specifying:
coef(mod1)$Cou... | Visualizing a multilevel model (HLM) in ggplot2 | When you fit an lmer model, you can use the coef() function to extract the coefficients from the model. Your code would look something like:
mod1 <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | | Visualizing a multilevel model (HLM) in ggplot2
When you fit an lmer model, you can use the coef() function to extract the coefficients from the model. Your code would look something like:
mod1 <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | Country), data=dat)
Then you can call coef() and extract the coefficie... | Visualizing a multilevel model (HLM) in ggplot2
When you fit an lmer model, you can use the coef() function to extract the coefficients from the model. Your code would look something like:
mod1 <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | |
35,680 | Visualizing a multilevel model (HLM) in ggplot2 | When you want to plot lmer()objects, I find it easiest to use predict(). First you fit your model:
random.coef.model <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | Country),
data=dat)
Then you predict GDP values corresponding to your predictor variable (CO2.Emissions):
dat$random... | Visualizing a multilevel model (HLM) in ggplot2 | When you want to plot lmer()objects, I find it easiest to use predict(). First you fit your model:
random.coef.model <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | Country),
| Visualizing a multilevel model (HLM) in ggplot2
When you want to plot lmer()objects, I find it easiest to use predict(). First you fit your model:
random.coef.model <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | Country),
data=dat)
Then you predict GDP values corresponding to your ... | Visualizing a multilevel model (HLM) in ggplot2
When you want to plot lmer()objects, I find it easiest to use predict(). First you fit your model:
random.coef.model <- lmer(GDP ~ 1 + CO2.Emissions + (1 + CO2.Emissions | Country),
|
35,681 | Is there an accepted name for this error metric? | This is related to a coefficient of determination ($R^2$), actually, it's $1-R^2$, also called fraction of variance unexplained | Is there an accepted name for this error metric? | This is related to a coefficient of determination ($R^2$), actually, it's $1-R^2$, also called fraction of variance unexplained | Is there an accepted name for this error metric?
This is related to a coefficient of determination ($R^2$), actually, it's $1-R^2$, also called fraction of variance unexplained | Is there an accepted name for this error metric?
This is related to a coefficient of determination ($R^2$), actually, it's $1-R^2$, also called fraction of variance unexplained |
35,682 | Is it ever okay to drop missing observations? | The important distinction is in your case not the distinction between MCAR, MAR, and NMAR, but between real missing values and mechanical missing values. Real missing values are values that exist, but for some reason weren't recorded. Mechanical missing values don't exist, but the rectangular structure of a dataset for... | Is it ever okay to drop missing observations? | The important distinction is in your case not the distinction between MCAR, MAR, and NMAR, but between real missing values and mechanical missing values. Real missing values are values that exist, but | Is it ever okay to drop missing observations?
The important distinction is in your case not the distinction between MCAR, MAR, and NMAR, but between real missing values and mechanical missing values. Real missing values are values that exist, but for some reason weren't recorded. Mechanical missing values don't exist, ... | Is it ever okay to drop missing observations?
The important distinction is in your case not the distinction between MCAR, MAR, and NMAR, but between real missing values and mechanical missing values. Real missing values are values that exist, but |
35,683 | Is it ever okay to drop missing observations? | It is clear a mix of at least 2 different missingness processes.
People that die from procedure unrelated causes/abandon/etc. due to reasons other than the likely outcome of the procedure. Here some imputation under MAR makes sense (if you can clearly identify the cases).
People that give up/withdraw/drop-out due to n... | Is it ever okay to drop missing observations? | It is clear a mix of at least 2 different missingness processes.
People that die from procedure unrelated causes/abandon/etc. due to reasons other than the likely outcome of the procedure. Here some | Is it ever okay to drop missing observations?
It is clear a mix of at least 2 different missingness processes.
People that die from procedure unrelated causes/abandon/etc. due to reasons other than the likely outcome of the procedure. Here some imputation under MAR makes sense (if you can clearly identify the cases).
... | Is it ever okay to drop missing observations?
It is clear a mix of at least 2 different missingness processes.
People that die from procedure unrelated causes/abandon/etc. due to reasons other than the likely outcome of the procedure. Here some |
35,684 | Scaling/Normalization not need for tree based models | If you are scaling the outcome variable, all you are doing is multiplying everything by a constant and/or adding a constant. So, any effect that it has is irrelevant (i.e., it does not change the relativities of anything).
In the case of the predictors, the scale of the predictor variables is not a determinant of the p... | Scaling/Normalization not need for tree based models | If you are scaling the outcome variable, all you are doing is multiplying everything by a constant and/or adding a constant. So, any effect that it has is irrelevant (i.e., it does not change the rela | Scaling/Normalization not need for tree based models
If you are scaling the outcome variable, all you are doing is multiplying everything by a constant and/or adding a constant. So, any effect that it has is irrelevant (i.e., it does not change the relativities of anything).
In the case of the predictors, the scale of ... | Scaling/Normalization not need for tree based models
If you are scaling the outcome variable, all you are doing is multiplying everything by a constant and/or adding a constant. So, any effect that it has is irrelevant (i.e., it does not change the rela |
35,685 | Benefits of SVM as a tool for digit recognition | Tutorials for many different classifiers are based on digit recognition, particularly the MNIST data set. I think this is largely because this dataset is readily available, easy to understand, and requires minimal preprocessing. It's also a common benchmark dataset in the literature.
I'm not aware of arguments that any... | Benefits of SVM as a tool for digit recognition | Tutorials for many different classifiers are based on digit recognition, particularly the MNIST data set. I think this is largely because this dataset is readily available, easy to understand, and req | Benefits of SVM as a tool for digit recognition
Tutorials for many different classifiers are based on digit recognition, particularly the MNIST data set. I think this is largely because this dataset is readily available, easy to understand, and requires minimal preprocessing. It's also a common benchmark dataset in the... | Benefits of SVM as a tool for digit recognition
Tutorials for many different classifiers are based on digit recognition, particularly the MNIST data set. I think this is largely because this dataset is readily available, easy to understand, and req |
35,686 | Covariance in Gaussian Process | Noise parameter, $\sigma^2$, is the parameter of the likelihood function a.k.a noise function.
The one with $+\sigma^2$ is the variance of $y$ (observation). The one without is the variance of $f$ (latent variable = observation - noise). So they are off from each other by $\sigma^2$ which is the same for all values o... | Covariance in Gaussian Process | Noise parameter, $\sigma^2$, is the parameter of the likelihood function a.k.a noise function.
The one with $+\sigma^2$ is the variance of $y$ (observation). The one without is the variance of $f$ (l | Covariance in Gaussian Process
Noise parameter, $\sigma^2$, is the parameter of the likelihood function a.k.a noise function.
The one with $+\sigma^2$ is the variance of $y$ (observation). The one without is the variance of $f$ (latent variable = observation - noise). So they are off from each other by $\sigma^2$ whi... | Covariance in Gaussian Process
Noise parameter, $\sigma^2$, is the parameter of the likelihood function a.k.a noise function.
The one with $+\sigma^2$ is the variance of $y$ (observation). The one without is the variance of $f$ (l |
35,687 | R polynomial expansion error on "unique points" and degree | In R, the poly function generates a polynomial and then does a QR decomposition to make it orthogonal. For high degree polynomial input, you'll get a high condition number for the Vandermode matrix, and when poly tries to make the columns orthogonal, it's going to throw an error (because the input looks rank deficient ... | R polynomial expansion error on "unique points" and degree | In R, the poly function generates a polynomial and then does a QR decomposition to make it orthogonal. For high degree polynomial input, you'll get a high condition number for the Vandermode matrix, a | R polynomial expansion error on "unique points" and degree
In R, the poly function generates a polynomial and then does a QR decomposition to make it orthogonal. For high degree polynomial input, you'll get a high condition number for the Vandermode matrix, and when poly tries to make the columns orthogonal, it's going... | R polynomial expansion error on "unique points" and degree
In R, the poly function generates a polynomial and then does a QR decomposition to make it orthogonal. For high degree polynomial input, you'll get a high condition number for the Vandermode matrix, a |
35,688 | How to Estimate Population Variance from Multiple Samples | A comment clarified that the best estimator of the variance is intended to be unbiased. A standard way to find such an estimator is to restrict one's attention to linear combinations of the estimators (because almost anything else would be difficult to analyze).
Let there be $k$ independent samples indexed by $i=1,2... | How to Estimate Population Variance from Multiple Samples | A comment clarified that the best estimator of the variance is intended to be unbiased. A standard way to find such an estimator is to restrict one's attention to linear combinations of the estimator | How to Estimate Population Variance from Multiple Samples
A comment clarified that the best estimator of the variance is intended to be unbiased. A standard way to find such an estimator is to restrict one's attention to linear combinations of the estimators (because almost anything else would be difficult to analyze)... | How to Estimate Population Variance from Multiple Samples
A comment clarified that the best estimator of the variance is intended to be unbiased. A standard way to find such an estimator is to restrict one's attention to linear combinations of the estimator |
35,689 | How to Estimate Population Variance from Multiple Samples | How about putting a hat on your estimates of the variance for each of the N samples
$\hat{\sigma}_{i}^{2}\simeq s_{i}^{2}\frac{n-1}{n}$
and appealing to the Law of Propagation of Uncertainty, add them in quadrature to get the estimate of the overall variance
$\tilde{\sigma}^{2}=\sum_{i=1}^{N}\hat{\sigma}_{i}^{2}$ | How to Estimate Population Variance from Multiple Samples | How about putting a hat on your estimates of the variance for each of the N samples
$\hat{\sigma}_{i}^{2}\simeq s_{i}^{2}\frac{n-1}{n}$
and appealing to the Law of Propagation of Uncertainty, add them | How to Estimate Population Variance from Multiple Samples
How about putting a hat on your estimates of the variance for each of the N samples
$\hat{\sigma}_{i}^{2}\simeq s_{i}^{2}\frac{n-1}{n}$
and appealing to the Law of Propagation of Uncertainty, add them in quadrature to get the estimate of the overall variance
$\t... | How to Estimate Population Variance from Multiple Samples
How about putting a hat on your estimates of the variance for each of the N samples
$\hat{\sigma}_{i}^{2}\simeq s_{i}^{2}\frac{n-1}{n}$
and appealing to the Law of Propagation of Uncertainty, add them |
35,690 | How to Estimate Population Variance from Multiple Samples | I presume this is the issue where you have k samples size n of a population that are all random with a given mean m and sample unbiassed variance s^2. As you do not have the real population mean, NOR the population variance that is used in all the textbooks, you have to find a means to not only calculate this populatio... | How to Estimate Population Variance from Multiple Samples | I presume this is the issue where you have k samples size n of a population that are all random with a given mean m and sample unbiassed variance s^2. As you do not have the real population mean, NOR | How to Estimate Population Variance from Multiple Samples
I presume this is the issue where you have k samples size n of a population that are all random with a given mean m and sample unbiassed variance s^2. As you do not have the real population mean, NOR the population variance that is used in all the textbooks, you... | How to Estimate Population Variance from Multiple Samples
I presume this is the issue where you have k samples size n of a population that are all random with a given mean m and sample unbiassed variance s^2. As you do not have the real population mean, NOR |
35,691 | Formula for Newey West Standard Error | Take your model of $$y_t=\beta_0+\beta_1x_t+u_t,$$ where $t=1,...,T$. We will assume there are no other regressors and that the serial correlation only lasts up to one period (so shocks do not persist for very long).
To get the Newey-West/HAC standard error of $\beta_1$ that is robust to heteroskedasticity and autocorr... | Formula for Newey West Standard Error | Take your model of $$y_t=\beta_0+\beta_1x_t+u_t,$$ where $t=1,...,T$. We will assume there are no other regressors and that the serial correlation only lasts up to one period (so shocks do not persist | Formula for Newey West Standard Error
Take your model of $$y_t=\beta_0+\beta_1x_t+u_t,$$ where $t=1,...,T$. We will assume there are no other regressors and that the serial correlation only lasts up to one period (so shocks do not persist for very long).
To get the Newey-West/HAC standard error of $\beta_1$ that is rob... | Formula for Newey West Standard Error
Take your model of $$y_t=\beta_0+\beta_1x_t+u_t,$$ where $t=1,...,T$. We will assume there are no other regressors and that the serial correlation only lasts up to one period (so shocks do not persist |
35,692 | Probability of people not facing their partner at a round table | Analysis
Let's guess and then systematically improve the guess until it's correct.
Begin by guessing the answer is $1$. Of course that's wrong. To see how wrong, label one partner in each pair "Red" and the other "Blue". From the perspective of any Red individual, there is a $1/(2n-1)$ chance that their (Blue) partn... | Probability of people not facing their partner at a round table | Analysis
Let's guess and then systematically improve the guess until it's correct.
Begin by guessing the answer is $1$. Of course that's wrong. To see how wrong, label one partner in each pair "Red" | Probability of people not facing their partner at a round table
Analysis
Let's guess and then systematically improve the guess until it's correct.
Begin by guessing the answer is $1$. Of course that's wrong. To see how wrong, label one partner in each pair "Red" and the other "Blue". From the perspective of any Red ... | Probability of people not facing their partner at a round table
Analysis
Let's guess and then systematically improve the guess until it's correct.
Begin by guessing the answer is $1$. Of course that's wrong. To see how wrong, label one partner in each pair "Red" |
35,693 | Probability of people not facing their partner at a round table | Why would the intuitive method work?
Think of the table as a collection of pairs; that is, instead of the traditional North-West-East-South cross of a bridge table, we look at it like a table with two rows:
North-South
West-East
If we condition on North being the senior partner of one couple, then there's a 1/3rd chanc... | Probability of people not facing their partner at a round table | Why would the intuitive method work?
Think of the table as a collection of pairs; that is, instead of the traditional North-West-East-South cross of a bridge table, we look at it like a table with two | Probability of people not facing their partner at a round table
Why would the intuitive method work?
Think of the table as a collection of pairs; that is, instead of the traditional North-West-East-South cross of a bridge table, we look at it like a table with two rows:
North-South
West-East
If we condition on North be... | Probability of people not facing their partner at a round table
Why would the intuitive method work?
Think of the table as a collection of pairs; that is, instead of the traditional North-West-East-South cross of a bridge table, we look at it like a table with two |
35,694 | Which glm family to use for ordinal DV? | Given you have an ordinal response using the function MASS::polr should be more appropriate; it implements a proportional odds logistic regression routine. A very comprehensive tutorial on the analysis of ordinal response variables can be found here. It is also worth checking this thread on: How to understand output fr... | Which glm family to use for ordinal DV? | Given you have an ordinal response using the function MASS::polr should be more appropriate; it implements a proportional odds logistic regression routine. A very comprehensive tutorial on the analysi | Which glm family to use for ordinal DV?
Given you have an ordinal response using the function MASS::polr should be more appropriate; it implements a proportional odds logistic regression routine. A very comprehensive tutorial on the analysis of ordinal response variables can be found here. It is also worth checking thi... | Which glm family to use for ordinal DV?
Given you have an ordinal response using the function MASS::polr should be more appropriate; it implements a proportional odds logistic regression routine. A very comprehensive tutorial on the analysi |
35,695 | Which glm family to use for ordinal DV? | A multinomial distribution with a cumulative logit link function. You could also use a cumulative probit link function, but that's usually less desirable. | Which glm family to use for ordinal DV? | A multinomial distribution with a cumulative logit link function. You could also use a cumulative probit link function, but that's usually less desirable. | Which glm family to use for ordinal DV?
A multinomial distribution with a cumulative logit link function. You could also use a cumulative probit link function, but that's usually less desirable. | Which glm family to use for ordinal DV?
A multinomial distribution with a cumulative logit link function. You could also use a cumulative probit link function, but that's usually less desirable. |
35,696 | What is skewness of a distribution? | Skewness is related to the symmetry of a distribution.
Note that I do not write that "skewness measures symmetry" or some such. The specific relationship between symmetry and skewness is a bit complicated.
A symmetric distribution will have zero skewness, for usual definitions of skewness. (Yes, there are multiple one... | What is skewness of a distribution? | Skewness is related to the symmetry of a distribution.
Note that I do not write that "skewness measures symmetry" or some such. The specific relationship between symmetry and skewness is a bit compli | What is skewness of a distribution?
Skewness is related to the symmetry of a distribution.
Note that I do not write that "skewness measures symmetry" or some such. The specific relationship between symmetry and skewness is a bit complicated.
A symmetric distribution will have zero skewness, for usual definitions of sk... | What is skewness of a distribution?
Skewness is related to the symmetry of a distribution.
Note that I do not write that "skewness measures symmetry" or some such. The specific relationship between symmetry and skewness is a bit compli |
35,697 | Lasso and Ridge tuning parameter scope | Yes, you should taking into account the scale of the output $y$ and should also take into account the scale of the covariates in $X$.
Let $X \in \mathbb{R}^{n \times p}$ be the design matrix, whose rows are vectors with each entry being a covariate that together seek to explain the response $y \in \mathbb{R}^n$. Each e... | Lasso and Ridge tuning parameter scope | Yes, you should taking into account the scale of the output $y$ and should also take into account the scale of the covariates in $X$.
Let $X \in \mathbb{R}^{n \times p}$ be the design matrix, whose ro | Lasso and Ridge tuning parameter scope
Yes, you should taking into account the scale of the output $y$ and should also take into account the scale of the covariates in $X$.
Let $X \in \mathbb{R}^{n \times p}$ be the design matrix, whose rows are vectors with each entry being a covariate that together seek to explain th... | Lasso and Ridge tuning parameter scope
Yes, you should taking into account the scale of the output $y$ and should also take into account the scale of the covariates in $X$.
Let $X \in \mathbb{R}^{n \times p}$ be the design matrix, whose ro |
35,698 | Lasso and Ridge tuning parameter scope | In the R package glmnet, the function cv.glmnet fits a model on the whole data set to select suitable regularization path, and then does cross-validation using that path. This seems to work well in practice. | Lasso and Ridge tuning parameter scope | In the R package glmnet, the function cv.glmnet fits a model on the whole data set to select suitable regularization path, and then does cross-validation using that path. This seems to work well in pr | Lasso and Ridge tuning parameter scope
In the R package glmnet, the function cv.glmnet fits a model on the whole data set to select suitable regularization path, and then does cross-validation using that path. This seems to work well in practice. | Lasso and Ridge tuning parameter scope
In the R package glmnet, the function cv.glmnet fits a model on the whole data set to select suitable regularization path, and then does cross-validation using that path. This seems to work well in pr |
35,699 | Difference in partial dependence calculated by R and Python | Scikit-learn center the partial dependence with the mean of the target value, R does not.
Here is an example using diabetes dataset.
R
data(diabetes, package="lars")
y <- diabetes$y
x <- diabetes$x
class(x) <- "matrix"
data <- data.frame(y, as.data.frame(x))
model <- gbm::gbm(formula = y ~ . , data ... | Difference in partial dependence calculated by R and Python | Scikit-learn center the partial dependence with the mean of the target value, R does not.
Here is an example using diabetes dataset.
R
data(diabetes, package="lars")
y <- diabetes$y
x < | Difference in partial dependence calculated by R and Python
Scikit-learn center the partial dependence with the mean of the target value, R does not.
Here is an example using diabetes dataset.
R
data(diabetes, package="lars")
y <- diabetes$y
x <- diabetes$x
class(x) <- "matrix"
data <- data.frame(y, ... | Difference in partial dependence calculated by R and Python
Scikit-learn center the partial dependence with the mean of the target value, R does not.
Here is an example using diabetes dataset.
R
data(diabetes, package="lars")
y <- diabetes$y
x < |
35,700 | Can I make a decision using a Bayes factor? | This is an excellent and deep question.
While traditional textbooks (like mine) tend to promote Bayes factors as equivalent to posterior probabilities of the null and alternative hypotheses or of two models under comparison, which is formally correct as detailed in the following extract from my Bayesian Choice, I now ... | Can I make a decision using a Bayes factor? | This is an excellent and deep question.
While traditional textbooks (like mine) tend to promote Bayes factors as equivalent to posterior probabilities of the null and alternative hypotheses or of two | Can I make a decision using a Bayes factor?
This is an excellent and deep question.
While traditional textbooks (like mine) tend to promote Bayes factors as equivalent to posterior probabilities of the null and alternative hypotheses or of two models under comparison, which is formally correct as detailed in the follo... | Can I make a decision using a Bayes factor?
This is an excellent and deep question.
While traditional textbooks (like mine) tend to promote Bayes factors as equivalent to posterior probabilities of the null and alternative hypotheses or of two |
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