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 |
|---|---|---|---|---|---|---|
36,501 | How to distinguish between two biased coins | Say the odds of getting $C_1$ over $C_2$ are, in principle, $1:1$. Then, you flip the coin $n$ times and get $x$ heads. If we call the probabilities of heads $p_1=0.95$ and $p_2=0.01$, then the probability that each coin gives $x$ heads is:
$$P(x|C_1)=p_1^x(1-p_1)^{n-x}$$
$$P(x|C_2)=p_2^x(1-p_2)^{n-x}$$
If we use Bayes... | How to distinguish between two biased coins | Say the odds of getting $C_1$ over $C_2$ are, in principle, $1:1$. Then, you flip the coin $n$ times and get $x$ heads. If we call the probabilities of heads $p_1=0.95$ and $p_2=0.01$, then the probab | How to distinguish between two biased coins
Say the odds of getting $C_1$ over $C_2$ are, in principle, $1:1$. Then, you flip the coin $n$ times and get $x$ heads. If we call the probabilities of heads $p_1=0.95$ and $p_2=0.01$, then the probability that each coin gives $x$ heads is:
$$P(x|C_1)=p_1^x(1-p_1)^{n-x}$$
$$P... | How to distinguish between two biased coins
Say the odds of getting $C_1$ over $C_2$ are, in principle, $1:1$. Then, you flip the coin $n$ times and get $x$ heads. If we call the probabilities of heads $p_1=0.95$ and $p_2=0.01$, then the probab |
36,502 | How do we handle a confounder which is collinear with the exposure? | Multicollinearity will only be a problem if the correlation between X and Z is 1. In that case, X and Z can be combined into a single variable which will provide an unbiased estimate. We can see this with a simple simulation
> set.seed(1)
> N <- 100
> Z <- rnorm(N)
> X <- Z # perfect collinearity
> Y <- 4 + X + Z + r... | How do we handle a confounder which is collinear with the exposure? | Multicollinearity will only be a problem if the correlation between X and Z is 1. In that case, X and Z can be combined into a single variable which will provide an unbiased estimate. We can see this | How do we handle a confounder which is collinear with the exposure?
Multicollinearity will only be a problem if the correlation between X and Z is 1. In that case, X and Z can be combined into a single variable which will provide an unbiased estimate. We can see this with a simple simulation
> set.seed(1)
> N <- 100
> ... | How do we handle a confounder which is collinear with the exposure?
Multicollinearity will only be a problem if the correlation between X and Z is 1. In that case, X and Z can be combined into a single variable which will provide an unbiased estimate. We can see this |
36,503 | Relationship between distribution and data generating process | The probability distribution is the actual mathematical function $P({\bf x}; \theta)$ that can assign a probability to each possible vector ${\bf x}$. It is given by the parameter vector $\theta$.
The data generating process is the causal (deterministic or stochastic) mechanism from where the data originate.
The popula... | Relationship between distribution and data generating process | The probability distribution is the actual mathematical function $P({\bf x}; \theta)$ that can assign a probability to each possible vector ${\bf x}$. It is given by the parameter vector $\theta$.
The | Relationship between distribution and data generating process
The probability distribution is the actual mathematical function $P({\bf x}; \theta)$ that can assign a probability to each possible vector ${\bf x}$. It is given by the parameter vector $\theta$.
The data generating process is the causal (deterministic or s... | Relationship between distribution and data generating process
The probability distribution is the actual mathematical function $P({\bf x}; \theta)$ that can assign a probability to each possible vector ${\bf x}$. It is given by the parameter vector $\theta$.
The |
36,504 | The importance of a correct interpretation of a confidence interval | I'll be honest there: I don't think the actual distinction is all that important. Yes, saying that "the probability of the estimated parameter being included in the confidence interval is 95%" is incorrect, for the precise reason you give. However, I do not think it is a major problem. (I would be interested in any oth... | The importance of a correct interpretation of a confidence interval | I'll be honest there: I don't think the actual distinction is all that important. Yes, saying that "the probability of the estimated parameter being included in the confidence interval is 95%" is inco | The importance of a correct interpretation of a confidence interval
I'll be honest there: I don't think the actual distinction is all that important. Yes, saying that "the probability of the estimated parameter being included in the confidence interval is 95%" is incorrect, for the precise reason you give. However, I d... | The importance of a correct interpretation of a confidence interval
I'll be honest there: I don't think the actual distinction is all that important. Yes, saying that "the probability of the estimated parameter being included in the confidence interval is 95%" is inco |
36,505 | The importance of a correct interpretation of a confidence interval | Suppose you are estimating the price of a house using some given features. So when someone says to find out the $95$% confidence interval of the price, given some features. So you are basically finding an interval $[x, y]$ so that the probability of your price lying in this interval $[x,y]$ is $0.95$ (i.e $95$%).
I hop... | The importance of a correct interpretation of a confidence interval | Suppose you are estimating the price of a house using some given features. So when someone says to find out the $95$% confidence interval of the price, given some features. So you are basically findin | The importance of a correct interpretation of a confidence interval
Suppose you are estimating the price of a house using some given features. So when someone says to find out the $95$% confidence interval of the price, given some features. So you are basically finding an interval $[x, y]$ so that the probability of yo... | The importance of a correct interpretation of a confidence interval
Suppose you are estimating the price of a house using some given features. So when someone says to find out the $95$% confidence interval of the price, given some features. So you are basically findin |
36,506 | How can I calculate the probability for multiple trials with different probabilities? | Suppose you conduct the first trial $n$ times and (independently) second one $m$ times, and that the chances of success are $p$ and $q$ respectively. Let $A$ be the total number of successes in the first instance, $B$ the total in the second, and $X=A+B$ be the total number of successes. Obviously $X$ is an integer b... | How can I calculate the probability for multiple trials with different probabilities? | Suppose you conduct the first trial $n$ times and (independently) second one $m$ times, and that the chances of success are $p$ and $q$ respectively. Let $A$ be the total number of successes in the f | How can I calculate the probability for multiple trials with different probabilities?
Suppose you conduct the first trial $n$ times and (independently) second one $m$ times, and that the chances of success are $p$ and $q$ respectively. Let $A$ be the total number of successes in the first instance, $B$ the total in th... | How can I calculate the probability for multiple trials with different probabilities?
Suppose you conduct the first trial $n$ times and (independently) second one $m$ times, and that the chances of success are $p$ and $q$ respectively. Let $A$ be the total number of successes in the f |
36,507 | How can I calculate the probability for multiple trials with different probabilities? | The sum of independent non-identically distributed Bernoulli trials is called a Poisson-Binomial distribution. The probabilities can be easily calculated with an R package called poibin.
For the example in the OP's description (3 trials with 40% chance of success and 2 trials with 25% chance of success), the following... | How can I calculate the probability for multiple trials with different probabilities? | The sum of independent non-identically distributed Bernoulli trials is called a Poisson-Binomial distribution. The probabilities can be easily calculated with an R package called poibin.
For the exam | How can I calculate the probability for multiple trials with different probabilities?
The sum of independent non-identically distributed Bernoulli trials is called a Poisson-Binomial distribution. The probabilities can be easily calculated with an R package called poibin.
For the example in the OP's description (3 tri... | How can I calculate the probability for multiple trials with different probabilities?
The sum of independent non-identically distributed Bernoulli trials is called a Poisson-Binomial distribution. The probabilities can be easily calculated with an R package called poibin.
For the exam |
36,508 | How can I calculate the probability for multiple trials with different probabilities? | For at least two successes, we can calculate the probability of no success and only one success, sum them and subtract from $1$. You've already done the first. For one success, the cases are
1 success for A, 0 success for B $\rightarrow {3\choose 1}p_a(1-p_a)^2(1-p_b)^2$
0 success for A, 1 success for B $\rightarrow (... | How can I calculate the probability for multiple trials with different probabilities? | For at least two successes, we can calculate the probability of no success and only one success, sum them and subtract from $1$. You've already done the first. For one success, the cases are
1 succes | How can I calculate the probability for multiple trials with different probabilities?
For at least two successes, we can calculate the probability of no success and only one success, sum them and subtract from $1$. You've already done the first. For one success, the cases are
1 success for A, 0 success for B $\rightar... | How can I calculate the probability for multiple trials with different probabilities?
For at least two successes, we can calculate the probability of no success and only one success, sum them and subtract from $1$. You've already done the first. For one success, the cases are
1 succes |
36,509 | What is the name of a chart that visualizes the inclusion and exclusion of patients in a cohort? | This is a CONSORT 2010 Flow Diagram.
CONSORT stands for Consolidated Standards of Reporting Trials and encompasses various initiatives developed by the CONSORT Group to alleviate the problems arising from inadequate reporting of randomized controlled trials.
You can download a template for the flowchart, along with... | What is the name of a chart that visualizes the inclusion and exclusion of patients in a cohort? | This is a CONSORT 2010 Flow Diagram.
CONSORT stands for Consolidated Standards of Reporting Trials and encompasses various initiatives developed by the CONSORT Group to alleviate the problems arisin | What is the name of a chart that visualizes the inclusion and exclusion of patients in a cohort?
This is a CONSORT 2010 Flow Diagram.
CONSORT stands for Consolidated Standards of Reporting Trials and encompasses various initiatives developed by the CONSORT Group to alleviate the problems arising from inadequate repor... | What is the name of a chart that visualizes the inclusion and exclusion of patients in a cohort?
This is a CONSORT 2010 Flow Diagram.
CONSORT stands for Consolidated Standards of Reporting Trials and encompasses various initiatives developed by the CONSORT Group to alleviate the problems arisin |
36,510 | Why do we use random sample with replacement while implementing random forest? | Random forests are based on the concept of bootstrap aggregation (aka bagging). This is a theoretical foundation that shows that sampling with replacement and then building an ensemble reduces the variance of the forest without increasing the bias.
The same theoretical property is not true if you sample without replace... | Why do we use random sample with replacement while implementing random forest? | Random forests are based on the concept of bootstrap aggregation (aka bagging). This is a theoretical foundation that shows that sampling with replacement and then building an ensemble reduces the var | Why do we use random sample with replacement while implementing random forest?
Random forests are based on the concept of bootstrap aggregation (aka bagging). This is a theoretical foundation that shows that sampling with replacement and then building an ensemble reduces the variance of the forest without increasing th... | Why do we use random sample with replacement while implementing random forest?
Random forests are based on the concept of bootstrap aggregation (aka bagging). This is a theoretical foundation that shows that sampling with replacement and then building an ensemble reduces the var |
36,511 | Is initializing the weights of autoencoders still a difficult problem? | These layerwise pretraining procedures are mostly not needed anymore, for a few reasons:
Better initialization schemes, e.g. Xavier / Glorot initialization (the same thing, named after Xavier Glorot) as shimao noted. These help avoid the problems of exploding or vanishing gradients where, essentially, multiplying many... | Is initializing the weights of autoencoders still a difficult problem? | These layerwise pretraining procedures are mostly not needed anymore, for a few reasons:
Better initialization schemes, e.g. Xavier / Glorot initialization (the same thing, named after Xavier Glorot) | Is initializing the weights of autoencoders still a difficult problem?
These layerwise pretraining procedures are mostly not needed anymore, for a few reasons:
Better initialization schemes, e.g. Xavier / Glorot initialization (the same thing, named after Xavier Glorot) as shimao noted. These help avoid the problems o... | Is initializing the weights of autoencoders still a difficult problem?
These layerwise pretraining procedures are mostly not needed anymore, for a few reasons:
Better initialization schemes, e.g. Xavier / Glorot initialization (the same thing, named after Xavier Glorot) |
36,512 | Intuitive Understanding of Expected Improvement for Gaussian Process | My question is are we searching for the point in the Gaussian Process model whose expected value (determined by mean and confidence) shall be decreased the most if sampled at that point?
No. At any iteration, you've observed some inputs, and one of those inputs ($x^*$) is the current optimum with a function value $f(x... | Intuitive Understanding of Expected Improvement for Gaussian Process | My question is are we searching for the point in the Gaussian Process model whose expected value (determined by mean and confidence) shall be decreased the most if sampled at that point?
No. At any i | Intuitive Understanding of Expected Improvement for Gaussian Process
My question is are we searching for the point in the Gaussian Process model whose expected value (determined by mean and confidence) shall be decreased the most if sampled at that point?
No. At any iteration, you've observed some inputs, and one of t... | Intuitive Understanding of Expected Improvement for Gaussian Process
My question is are we searching for the point in the Gaussian Process model whose expected value (determined by mean and confidence) shall be decreased the most if sampled at that point?
No. At any i |
36,513 | Intuitive Understanding of Expected Improvement for Gaussian Process | As far as I know, in practice, the first observations in Bayesian optimization are random before the Gaussian processes take over. After the initial observations, the expected improvement can be calculated for a data point x. By doing so, we want to select the value for x that is expected to improve the results of our ... | Intuitive Understanding of Expected Improvement for Gaussian Process | As far as I know, in practice, the first observations in Bayesian optimization are random before the Gaussian processes take over. After the initial observations, the expected improvement can be calcu | Intuitive Understanding of Expected Improvement for Gaussian Process
As far as I know, in practice, the first observations in Bayesian optimization are random before the Gaussian processes take over. After the initial observations, the expected improvement can be calculated for a data point x. By doing so, we want to s... | Intuitive Understanding of Expected Improvement for Gaussian Process
As far as I know, in practice, the first observations in Bayesian optimization are random before the Gaussian processes take over. After the initial observations, the expected improvement can be calcu |
36,514 | How to understand the vertical bar (pipe) in R formulas [closed] | Assume there are only two groups: group 1 and group 2. The gls() call you specified fits two sub-models to your $y$ observations - one sub-model for the $y$ observations in the first group and another sub-model for the $y$ observations in the second group.
The sub-model for the observations $y$ in group 1 postulates ... | How to understand the vertical bar (pipe) in R formulas [closed] | Assume there are only two groups: group 1 and group 2. The gls() call you specified fits two sub-models to your $y$ observations - one sub-model for the $y$ observations in the first group and anothe | How to understand the vertical bar (pipe) in R formulas [closed]
Assume there are only two groups: group 1 and group 2. The gls() call you specified fits two sub-models to your $y$ observations - one sub-model for the $y$ observations in the first group and another sub-model for the $y$ observations in the second grou... | How to understand the vertical bar (pipe) in R formulas [closed]
Assume there are only two groups: group 1 and group 2. The gls() call you specified fits two sub-models to your $y$ observations - one sub-model for the $y$ observations in the first group and anothe |
36,515 | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence? | First order intensity and second order intensity measure different aspects of a process that can be almost independently varied. In particular, not every point process can be regarded as an inhomogeneous Poisson process.
Let's deal with that last issue first. Consider a homogeneous Poisson process on the interval $[0... | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order | First order intensity and second order intensity measure different aspects of a process that can be almost independently varied. In particular, not every point process can be regarded as an inhomogen | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence?
First order intensity and second order intensity measure different aspects of a process that can be almost independently varied. In particular, not every point process can be regarded as an inhomogeneous Poi... | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order
First order intensity and second order intensity measure different aspects of a process that can be almost independently varied. In particular, not every point process can be regarded as an inhomogen |
36,516 | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence? | In broad terms your understanding sounds right. In particular, you are right that it is basically impossible to distinguish "first order inhomogeneity" and "second order clustering due to interactions between points" based on a single point pattern. | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order | In broad terms your understanding sounds right. In particular, you are right that it is basically impossible to distinguish "first order inhomogeneity" and "second order clustering due to interactions | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order dependence?
In broad terms your understanding sounds right. In particular, you are right that it is basically impossible to distinguish "first order inhomogeneity" and "second order clustering due to interactions between... | Spatial Point Process: Does an inhomogeneous first order intensity function affect the second order
In broad terms your understanding sounds right. In particular, you are right that it is basically impossible to distinguish "first order inhomogeneity" and "second order clustering due to interactions |
36,517 | Why is random sampling a non-differentiable operation? | Gregory Gundersen wrote a blog post about this in 2018. He explictly answers the questions:
What does a “random node” mean and what does it mean for backprop to “flow” or not flow through such a node?
The following excerpt should answer your questions:
Undifferentiable expectations
Let’s say we want to take the grad... | Why is random sampling a non-differentiable operation? | Gregory Gundersen wrote a blog post about this in 2018. He explictly answers the questions:
What does a “random node” mean and what does it mean for backprop to “flow” or not flow through such a node | Why is random sampling a non-differentiable operation?
Gregory Gundersen wrote a blog post about this in 2018. He explictly answers the questions:
What does a “random node” mean and what does it mean for backprop to “flow” or not flow through such a node?
The following excerpt should answer your questions:
Undiffere... | Why is random sampling a non-differentiable operation?
Gregory Gundersen wrote a blog post about this in 2018. He explictly answers the questions:
What does a “random node” mean and what does it mean for backprop to “flow” or not flow through such a node |
36,518 | Why is random sampling a non-differentiable operation? | It'd be easier to see with sampling from a categorical distribution. Say you have categorical distribution
$$\pi_1, \pi_2,...,\pi_K, \pi_i \ge 0, \sum_{i=1}^K \pi_i = 1$$
with $p(x=i|\pi) = \pi_i$. To draw a sample $x$ from this distribution, a standard way is to do:
$$
u \sim U(0,1) \\
x = \arg\min_i \sum_{j=1}^i \pi_... | Why is random sampling a non-differentiable operation? | It'd be easier to see with sampling from a categorical distribution. Say you have categorical distribution
$$\pi_1, \pi_2,...,\pi_K, \pi_i \ge 0, \sum_{i=1}^K \pi_i = 1$$
with $p(x=i|\pi) = \pi_i$. To | Why is random sampling a non-differentiable operation?
It'd be easier to see with sampling from a categorical distribution. Say you have categorical distribution
$$\pi_1, \pi_2,...,\pi_K, \pi_i \ge 0, \sum_{i=1}^K \pi_i = 1$$
with $p(x=i|\pi) = \pi_i$. To draw a sample $x$ from this distribution, a standard way is to d... | Why is random sampling a non-differentiable operation?
It'd be easier to see with sampling from a categorical distribution. Say you have categorical distribution
$$\pi_1, \pi_2,...,\pi_K, \pi_i \ge 0, \sum_{i=1}^K \pi_i = 1$$
with $p(x=i|\pi) = \pi_i$. To |
36,519 | Why aren't auto-encoders also considered generative models? | My answer to this question would be the following:
A generative model, as defined on the Wikipedia link you provided, aims to estimate the joint distribution of your data and some latent (random) variables usually $p(\textbf{x},\textbf{z})$. Particularly, in the case of the VAE you have that the data (usually $\textbf{... | Why aren't auto-encoders also considered generative models? | My answer to this question would be the following:
A generative model, as defined on the Wikipedia link you provided, aims to estimate the joint distribution of your data and some latent (random) vari | Why aren't auto-encoders also considered generative models?
My answer to this question would be the following:
A generative model, as defined on the Wikipedia link you provided, aims to estimate the joint distribution of your data and some latent (random) variables usually $p(\textbf{x},\textbf{z})$. Particularly, in t... | Why aren't auto-encoders also considered generative models?
My answer to this question would be the following:
A generative model, as defined on the Wikipedia link you provided, aims to estimate the joint distribution of your data and some latent (random) vari |
36,520 | Can't understand why rejection sampling works | First off you need your $x$ and $y$ distributed between $-1$ and $1$ ($[0,1]$ gives you a quarter of a circle since you're only looking at $y \ge 0$ and $x \ge0$ ).
Then you realise, that you've two lists of randomly distributed variables along the $X$ and $Y$ axis - two lines with a random distribution of points. Comb... | Can't understand why rejection sampling works | First off you need your $x$ and $y$ distributed between $-1$ and $1$ ($[0,1]$ gives you a quarter of a circle since you're only looking at $y \ge 0$ and $x \ge0$ ).
Then you realise, that you've two l | Can't understand why rejection sampling works
First off you need your $x$ and $y$ distributed between $-1$ and $1$ ($[0,1]$ gives you a quarter of a circle since you're only looking at $y \ge 0$ and $x \ge0$ ).
Then you realise, that you've two lists of randomly distributed variables along the $X$ and $Y$ axis - two li... | Can't understand why rejection sampling works
First off you need your $x$ and $y$ distributed between $-1$ and $1$ ($[0,1]$ gives you a quarter of a circle since you're only looking at $y \ge 0$ and $x \ge0$ ).
Then you realise, that you've two l |
36,521 | Can't understand why rejection sampling works | In general, to sample from a distribution with density $f(x,y)$ on support $\mathcal{S}$, if using a proposal distribution with density $h(x,y)$, we need to find $M$ such that
$$\sup_{(x,y) \in\mathcal{S}} \dfrac{f(x,y)}{h(x,y)} \leq M, $$
so that we can accept a proposed value with probability
$$\alpha = \dfrac{f(x,y)... | Can't understand why rejection sampling works | In general, to sample from a distribution with density $f(x,y)$ on support $\mathcal{S}$, if using a proposal distribution with density $h(x,y)$, we need to find $M$ such that
$$\sup_{(x,y) \in\mathca | Can't understand why rejection sampling works
In general, to sample from a distribution with density $f(x,y)$ on support $\mathcal{S}$, if using a proposal distribution with density $h(x,y)$, we need to find $M$ such that
$$\sup_{(x,y) \in\mathcal{S}} \dfrac{f(x,y)}{h(x,y)} \leq M, $$
so that we can accept a proposed v... | Can't understand why rejection sampling works
In general, to sample from a distribution with density $f(x,y)$ on support $\mathcal{S}$, if using a proposal distribution with density $h(x,y)$, we need to find $M$ such that
$$\sup_{(x,y) \in\mathca |
36,522 | Can't understand why rejection sampling works | You have chosen any point in [0,1]x[0,1] with equal probability. Then you removed all of those who are outside the circle.
This does not alter the probability of being selected for each individual point inside the circle (ie: any point inside the circle has still the same probability of being chosen as any other) | Can't understand why rejection sampling works | You have chosen any point in [0,1]x[0,1] with equal probability. Then you removed all of those who are outside the circle.
This does not alter the probability of being selected for each individual poi | Can't understand why rejection sampling works
You have chosen any point in [0,1]x[0,1] with equal probability. Then you removed all of those who are outside the circle.
This does not alter the probability of being selected for each individual point inside the circle (ie: any point inside the circle has still the same p... | Can't understand why rejection sampling works
You have chosen any point in [0,1]x[0,1] with equal probability. Then you removed all of those who are outside the circle.
This does not alter the probability of being selected for each individual poi |
36,523 | verifying a posterior is proper | For $𝑘=1$, you can see directly from your equation for the likelihood that the posterior density $𝑝(𝛼,𝛽|𝑦_1,𝑥_1)$ will be constant along parallell lines at which $𝛼+𝛽𝑥_𝑖$ takes constant values. So the posterior is indeed improper and has the shape of a ridge for $𝑘=1$. Basically, any regression line fitting... | verifying a posterior is proper | For $𝑘=1$, you can see directly from your equation for the likelihood that the posterior density $𝑝(𝛼,𝛽|𝑦_1,𝑥_1)$ will be constant along parallell lines at which $𝛼+𝛽𝑥_𝑖$ takes constant values. So the | verifying a posterior is proper
For $𝑘=1$, you can see directly from your equation for the likelihood that the posterior density $𝑝(𝛼,𝛽|𝑦_1,𝑥_1)$ will be constant along parallell lines at which $𝛼+𝛽𝑥_𝑖$ takes constant values. So the posterior is indeed improper and has the shape of a ridge for $𝑘=1$. Basica... | verifying a posterior is proper
For $𝑘=1$, you can see directly from your equation for the likelihood that the posterior density $𝑝(𝛼,𝛽|𝑦_1,𝑥_1)$ will be constant along parallell lines at which $𝛼+𝛽𝑥_𝑖$ takes constant values. So the |
36,524 | verifying a posterior is proper | It's improper, I believe. I only need to prove that $$\int\limits_{\alpha \in \mathbb{R} \\ \beta>0} p(y \mid \alpha, \beta, x) = +\infty.$$
Denote function $$\sigma = \mathrm{invlogit}$$
Now that $\sigma$ is a monotonically increasing function, when $\beta > 0$, we have $$\mathrm{\sigma}(\alpha + \beta x_i) > \mathrm{... | verifying a posterior is proper | It's improper, I believe. I only need to prove that $$\int\limits_{\alpha \in \mathbb{R} \\ \beta>0} p(y \mid \alpha, \beta, x) = +\infty.$$
Denote function $$\sigma = \mathrm{invlogit}$$
Now that $\s | verifying a posterior is proper
It's improper, I believe. I only need to prove that $$\int\limits_{\alpha \in \mathbb{R} \\ \beta>0} p(y \mid \alpha, \beta, x) = +\infty.$$
Denote function $$\sigma = \mathrm{invlogit}$$
Now that $\sigma$ is a monotonically increasing function, when $\beta > 0$, we have $$\mathrm{\sigma... | verifying a posterior is proper
It's improper, I believe. I only need to prove that $$\int\limits_{\alpha \in \mathbb{R} \\ \beta>0} p(y \mid \alpha, \beta, x) = +\infty.$$
Denote function $$\sigma = \mathrm{invlogit}$$
Now that $\s |
36,525 | verifying a posterior is proper | I've already accepted an answer, but I did want to point out that the posterior isn't proper for all possible data sets. The posterior is proportional to the likelihood, which is
$$\prod_{i=1}^k [\text{invlogit}(\alpha + \beta x_i)]^{y_i}[1-\text{invlogit}(\alpha + \beta x_i)]^{n-y_i}.
$$
If $y_1 = y_2 = \cdots = y_k =... | verifying a posterior is proper | I've already accepted an answer, but I did want to point out that the posterior isn't proper for all possible data sets. The posterior is proportional to the likelihood, which is
$$\prod_{i=1}^k [\tex | verifying a posterior is proper
I've already accepted an answer, but I did want to point out that the posterior isn't proper for all possible data sets. The posterior is proportional to the likelihood, which is
$$\prod_{i=1}^k [\text{invlogit}(\alpha + \beta x_i)]^{y_i}[1-\text{invlogit}(\alpha + \beta x_i)]^{n-y_i}.
$... | verifying a posterior is proper
I've already accepted an answer, but I did want to point out that the posterior isn't proper for all possible data sets. The posterior is proportional to the likelihood, which is
$$\prod_{i=1}^k [\tex |
36,526 | Elbow Test using AIC/BIC for identifying number of clusters using GMM | Welcome to CV!
This plot shows how the AIC and BIC change as a function of the number of clusters. While the AIC continues to decrease with a larger number of clusters, you can see that the BIC stops decreasing after $k=6$ clusters. For this reason, you could choose $k = 6$.
Another way to choose the 'best' number of c... | Elbow Test using AIC/BIC for identifying number of clusters using GMM | Welcome to CV!
This plot shows how the AIC and BIC change as a function of the number of clusters. While the AIC continues to decrease with a larger number of clusters, you can see that the BIC stops | Elbow Test using AIC/BIC for identifying number of clusters using GMM
Welcome to CV!
This plot shows how the AIC and BIC change as a function of the number of clusters. While the AIC continues to decrease with a larger number of clusters, you can see that the BIC stops decreasing after $k=6$ clusters. For this reason, ... | Elbow Test using AIC/BIC for identifying number of clusters using GMM
Welcome to CV!
This plot shows how the AIC and BIC change as a function of the number of clusters. While the AIC continues to decrease with a larger number of clusters, you can see that the BIC stops |
36,527 | How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called? | It's simply a particular scaled $t$-distribution -- a $t$-distribution with a different variance to the standard $t$-distribution.
Let $\nu = \frac{2}{\alpha}-1$. Let $\sigma= \frac{\sqrt{2-\alpha}}{\alpha}$.
Then (if I did it right) $Y=X/\sigma$ is a standard $t$ with $\nu$ d.f.
Here's how my reasoning went:
$$f_Y(y... | How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called? | It's simply a particular scaled $t$-distribution -- a $t$-distribution with a different variance to the standard $t$-distribution.
Let $\nu = \frac{2}{\alpha}-1$. Let $\sigma= \frac{\sqrt{2-\alpha}}{\ | How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called?
It's simply a particular scaled $t$-distribution -- a $t$-distribution with a different variance to the standard $t$-distribution.
Let $\nu = \frac{2}{\alpha}-1$. Let $\sigma= \frac{\sqrt{2-\alpha}}{\alpha}$.
Then (if I did it right)... | How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called?
It's simply a particular scaled $t$-distribution -- a $t$-distribution with a different variance to the standard $t$-distribution.
Let $\nu = \frac{2}{\alpha}-1$. Let $\sigma= \frac{\sqrt{2-\alpha}}{\ |
36,528 | How Ridge or Lasso regression really work? | Because your "penalty" represenation of the minimization problem is just the langrange form of a constraint optimization problem:
Assume centered variables.
In both cases, lasso and ridge, your unconstrained target function is then the usual sum of squared residuals; i.e. given $p$ regressors you minimize:
$$RSS(\bol... | How Ridge or Lasso regression really work? | Because your "penalty" represenation of the minimization problem is just the langrange form of a constraint optimization problem:
Assume centered variables.
In both cases, lasso and ridge, your uncon | How Ridge or Lasso regression really work?
Because your "penalty" represenation of the minimization problem is just the langrange form of a constraint optimization problem:
Assume centered variables.
In both cases, lasso and ridge, your unconstrained target function is then the usual sum of squared residuals; i.e. giv... | How Ridge or Lasso regression really work?
Because your "penalty" represenation of the minimization problem is just the langrange form of a constraint optimization problem:
Assume centered variables.
In both cases, lasso and ridge, your uncon |
36,529 | The distribution of the initial point of an AR process | Time-series processes defined only by a recursive equation are not fully specified, since they depend on specification of a "starting distribution". Unless there is some additional restriction you are required to meet, you can use any starting distribution you like, and you will still have an AR model that conforms wi... | The distribution of the initial point of an AR process | Time-series processes defined only by a recursive equation are not fully specified, since they depend on specification of a "starting distribution". Unless there is some additional restriction you ar | The distribution of the initial point of an AR process
Time-series processes defined only by a recursive equation are not fully specified, since they depend on specification of a "starting distribution". Unless there is some additional restriction you are required to meet, you can use any starting distribution you lik... | The distribution of the initial point of an AR process
Time-series processes defined only by a recursive equation are not fully specified, since they depend on specification of a "starting distribution". Unless there is some additional restriction you ar |
36,530 | Intuition (geometric or other) of $Var(X) = Var(E[X|Y]) + E[Var(X|Y)]$ | To get some simple intuition, we will compare with a two-way analysis of variance. Let $Y_{ij} = \mu_i +\epsilon_{ij}$ where the $\epsilon_{ij}$ are iid with expectation zero and common variance $\sigma^2$, $i=1,\dotsc,k; j=1,\dotsc,n_i$.
Then we have the decomposition
$$
\sum_{i=1}^k\sum_{j=1}^{n_i} (Y_{ij}-\overl... | Intuition (geometric or other) of $Var(X) = Var(E[X|Y]) + E[Var(X|Y)]$ | To get some simple intuition, we will compare with a two-way analysis of variance. Let $Y_{ij} = \mu_i +\epsilon_{ij}$ where the $\epsilon_{ij}$ are iid with expectation zero and common variance $\sig | Intuition (geometric or other) of $Var(X) = Var(E[X|Y]) + E[Var(X|Y)]$
To get some simple intuition, we will compare with a two-way analysis of variance. Let $Y_{ij} = \mu_i +\epsilon_{ij}$ where the $\epsilon_{ij}$ are iid with expectation zero and common variance $\sigma^2$, $i=1,\dotsc,k; j=1,\dotsc,n_i$.
Then we h... | Intuition (geometric or other) of $Var(X) = Var(E[X|Y]) + E[Var(X|Y)]$
To get some simple intuition, we will compare with a two-way analysis of variance. Let $Y_{ij} = \mu_i +\epsilon_{ij}$ where the $\epsilon_{ij}$ are iid with expectation zero and common variance $\sig |
36,531 | How to do dimension reduction with sparse data | This depends on the goal of clustering.
t-SNE as well as various clustering methods (like hierarchical clustering) can work on distance matrices. And it's your job to construct a distance measure that captures what you wish to achieve. A few examples below.
Example 1
If you want to group the students based on their abi... | How to do dimension reduction with sparse data | This depends on the goal of clustering.
t-SNE as well as various clustering methods (like hierarchical clustering) can work on distance matrices. And it's your job to construct a distance measure that | How to do dimension reduction with sparse data
This depends on the goal of clustering.
t-SNE as well as various clustering methods (like hierarchical clustering) can work on distance matrices. And it's your job to construct a distance measure that captures what you wish to achieve. A few examples below.
Example 1
If yo... | How to do dimension reduction with sparse data
This depends on the goal of clustering.
t-SNE as well as various clustering methods (like hierarchical clustering) can work on distance matrices. And it's your job to construct a distance measure that |
36,532 | How to do dimension reduction with sparse data | Singular value decomposition is a very common strategy for dimension reduction applied to sparse data types. This is because you can leverage specialized sparse SVD solvers (e.g. ARPACK), and for SVD the inputs do not have to be manipulated in any special way which could disrupt sparsity. | How to do dimension reduction with sparse data | Singular value decomposition is a very common strategy for dimension reduction applied to sparse data types. This is because you can leverage specialized sparse SVD solvers (e.g. ARPACK), and for SVD | How to do dimension reduction with sparse data
Singular value decomposition is a very common strategy for dimension reduction applied to sparse data types. This is because you can leverage specialized sparse SVD solvers (e.g. ARPACK), and for SVD the inputs do not have to be manipulated in any special way which could d... | How to do dimension reduction with sparse data
Singular value decomposition is a very common strategy for dimension reduction applied to sparse data types. This is because you can leverage specialized sparse SVD solvers (e.g. ARPACK), and for SVD |
36,533 | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers | Although, you didn't include self-study tag, I first give you two hints and then full solution. You may stop reading after first or second hint and try youself.
Hint 1:
For $a \in (0,1)$ we have $$\sum_{m=0}^{\infty} ma^m = \frac{a}{(1-a)^2}$$
Hint 2:
Let $K$ be number of numbers drawn by Mr. B. And let your "target va... | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly | Although, you didn't include self-study tag, I first give you two hints and then full solution. You may stop reading after first or second hint and try youself.
Hint 1:
For $a \in (0,1)$ we have $$\su | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers
Although, you didn't include self-study tag, I first give you two hints and then full solution. You may stop reading after first or second hint and try youself.
Hint 1:
For $a \in (0,1)... | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly
Although, you didn't include self-study tag, I first give you two hints and then full solution. You may stop reading after first or second hint and try youself.
Hint 1:
For $a \in (0,1)$ we have $$\su |
36,534 | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers | Another angle of solution (summing not with P(K=k) but P(K>=k)):
$$\begin{array}\\
E(\sum Y_k) = \sum E(Y_k) & = \sum_{k=1}^\infty E(Y_k|K>=k) \cdot P(K>=k) \\
& = \sum_{k=0}^\infty \frac{1}{2} \cdot \left(\frac{x}{2} \right)^k \\
& = \frac{1}{2-x}
\end{array}$$ | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly | Another angle of solution (summing not with P(K=k) but P(K>=k)):
$$\begin{array}\\
E(\sum Y_k) = \sum E(Y_k) & = \sum_{k=1}^\infty E(Y_k|K>=k) \cdot P(K>=k) \\
& = \sum_{k=0}^\infty \frac{1}{2} \c | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers
Another angle of solution (summing not with P(K=k) but P(K>=k)):
$$\begin{array}\\
E(\sum Y_k) = \sum E(Y_k) & = \sum_{k=1}^\infty E(Y_k|K>=k) \cdot P(K>=k) \\
& = \sum_{k=0}^\infty ... | Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly
Another angle of solution (summing not with P(K=k) but P(K>=k)):
$$\begin{array}\\
E(\sum Y_k) = \sum E(Y_k) & = \sum_{k=1}^\infty E(Y_k|K>=k) \cdot P(K>=k) \\
& = \sum_{k=0}^\infty \frac{1}{2} \c |
36,535 | Alternate (?) definition of sample variance [duplicate] | Using symmetry, your double sum can be written as
$$
s^2 = \text{ave}_{i < j} f(x_i, x_j)
$$
with $f(x, y) := \frac{1}{2} (x - y)^2$ and as such is a U-statistic of degree 2 with kernel $f$ for the estimation of the parameter $\theta = E(f(X, Y)) = \ldots = \text{Var}(X)$, $X$, $Y$ iid. It is actually one of the mos... | Alternate (?) definition of sample variance [duplicate] | Using symmetry, your double sum can be written as
$$
s^2 = \text{ave}_{i < j} f(x_i, x_j)
$$
with $f(x, y) := \frac{1}{2} (x - y)^2$ and as such is a U-statistic of degree 2 with kernel $f$ for the | Alternate (?) definition of sample variance [duplicate]
Using symmetry, your double sum can be written as
$$
s^2 = \text{ave}_{i < j} f(x_i, x_j)
$$
with $f(x, y) := \frac{1}{2} (x - y)^2$ and as such is a U-statistic of degree 2 with kernel $f$ for the estimation of the parameter $\theta = E(f(X, Y)) = \ldots = \te... | Alternate (?) definition of sample variance [duplicate]
Using symmetry, your double sum can be written as
$$
s^2 = \text{ave}_{i < j} f(x_i, x_j)
$$
with $f(x, y) := \frac{1}{2} (x - y)^2$ and as such is a U-statistic of degree 2 with kernel $f$ for the |
36,536 | What is the difference between using random intercepts and slopes instead of separate regressions per subject? | There are two major differences, related to each other.
Running separate regressions for each subject takes up many more degrees of freedom, since you have an intercept and slope to estimate for every subject.
Mixed models make use of partial pooling; random effects are shrunk towards the mean. This basically means t... | What is the difference between using random intercepts and slopes instead of separate regressions pe | There are two major differences, related to each other.
Running separate regressions for each subject takes up many more degrees of freedom, since you have an intercept and slope to estimate for ever | What is the difference between using random intercepts and slopes instead of separate regressions per subject?
There are two major differences, related to each other.
Running separate regressions for each subject takes up many more degrees of freedom, since you have an intercept and slope to estimate for every subject... | What is the difference between using random intercepts and slopes instead of separate regressions pe
There are two major differences, related to each other.
Running separate regressions for each subject takes up many more degrees of freedom, since you have an intercept and slope to estimate for ever |
36,537 | Using $R^2$ to evaluate out-of-sample performance | The coefficient of determination, $R^2$, value can be calculated as
$$R^2 = 1 - SS_{res}/SS_{tot}$$ where the total sum of squares is
$$SS_{tot} = \sum_{i}(y_i - \bar{y})^2$$
and the residual sum of squares is
$$SS_{res} = \sum_{i}(y_i-\hat{y_i})^2 = \sum_{i}e_i^2$$
To apply the above equations to out-of-sample pred... | Using $R^2$ to evaluate out-of-sample performance | The coefficient of determination, $R^2$, value can be calculated as
$$R^2 = 1 - SS_{res}/SS_{tot}$$ where the total sum of squares is
$$SS_{tot} = \sum_{i}(y_i - \bar{y})^2$$
and the residual sum of | Using $R^2$ to evaluate out-of-sample performance
The coefficient of determination, $R^2$, value can be calculated as
$$R^2 = 1 - SS_{res}/SS_{tot}$$ where the total sum of squares is
$$SS_{tot} = \sum_{i}(y_i - \bar{y})^2$$
and the residual sum of squares is
$$SS_{res} = \sum_{i}(y_i-\hat{y_i})^2 = \sum_{i}e_i^2$$
... | Using $R^2$ to evaluate out-of-sample performance
The coefficient of determination, $R^2$, value can be calculated as
$$R^2 = 1 - SS_{res}/SS_{tot}$$ where the total sum of squares is
$$SS_{tot} = \sum_{i}(y_i - \bar{y})^2$$
and the residual sum of |
36,538 | Clarification on how to perform nested cross-validation | I gave a general answer to this question, and here is what applies to your question:
Train and Validation Split:
First split the input into train and validation; but I'd also take the domain knowledge into account. In your case, I would take that year parameter into account and take the last few years of the data (not ... | Clarification on how to perform nested cross-validation | I gave a general answer to this question, and here is what applies to your question:
Train and Validation Split:
First split the input into train and validation; but I'd also take the domain knowledge | Clarification on how to perform nested cross-validation
I gave a general answer to this question, and here is what applies to your question:
Train and Validation Split:
First split the input into train and validation; but I'd also take the domain knowledge into account. In your case, I would take that year parameter in... | Clarification on how to perform nested cross-validation
I gave a general answer to this question, and here is what applies to your question:
Train and Validation Split:
First split the input into train and validation; but I'd also take the domain knowledge |
36,539 | Clarification on how to perform nested cross-validation | What are you wanting to achieve by selecting knots? As a means of determining the complexity of the smooth terms in the model?
Generally there's not much to be gained from choosing knots in modern GAM theory; thin plate splines for example are the default in mgcv and have a knot at each unique data location (the subseq... | Clarification on how to perform nested cross-validation | What are you wanting to achieve by selecting knots? As a means of determining the complexity of the smooth terms in the model?
Generally there's not much to be gained from choosing knots in modern GAM | Clarification on how to perform nested cross-validation
What are you wanting to achieve by selecting knots? As a means of determining the complexity of the smooth terms in the model?
Generally there's not much to be gained from choosing knots in modern GAM theory; thin plate splines for example are the default in mgcv ... | Clarification on how to perform nested cross-validation
What are you wanting to achieve by selecting knots? As a means of determining the complexity of the smooth terms in the model?
Generally there's not much to be gained from choosing knots in modern GAM |
36,540 | Clarification on how to perform nested cross-validation | Let me add some points complementing the adready exisiting good answers:
About the nested cross validation I've come for myself to a description that I find makes it easy to get the calculations right while (by) not describing the situation as nested at all.
The basic idea is that I need two things: a way of training ... | Clarification on how to perform nested cross-validation | Let me add some points complementing the adready exisiting good answers:
About the nested cross validation I've come for myself to a description that I find makes it easy to get the calculations righ | Clarification on how to perform nested cross-validation
Let me add some points complementing the adready exisiting good answers:
About the nested cross validation I've come for myself to a description that I find makes it easy to get the calculations right while (by) not describing the situation as nested at all.
The ... | Clarification on how to perform nested cross-validation
Let me add some points complementing the adready exisiting good answers:
About the nested cross validation I've come for myself to a description that I find makes it easy to get the calculations righ |
36,541 | Is there a version of the Mahalanobis distance for matrices? | No. There are metrics that try to build on a similar concept using Wishart distribution. I have seen papers in MRI imaging that use the metrics. See p.16 in this slide deck: https://earth.esa.int/c/document_library/get_file?folderId=409343&name=DLFE-5593.pdf
There's a distance called Riemannian metric for positive defi... | Is there a version of the Mahalanobis distance for matrices? | No. There are metrics that try to build on a similar concept using Wishart distribution. I have seen papers in MRI imaging that use the metrics. See p.16 in this slide deck: https://earth.esa.int/c/do | Is there a version of the Mahalanobis distance for matrices?
No. There are metrics that try to build on a similar concept using Wishart distribution. I have seen papers in MRI imaging that use the metrics. See p.16 in this slide deck: https://earth.esa.int/c/document_library/get_file?folderId=409343&name=DLFE-5593.pdf
... | Is there a version of the Mahalanobis distance for matrices?
No. There are metrics that try to build on a similar concept using Wishart distribution. I have seen papers in MRI imaging that use the metrics. See p.16 in this slide deck: https://earth.esa.int/c/do |
36,542 | Is there a version of the Mahalanobis distance for matrices? | I'm working on a computer vision problem and I want to use the Mahalanobis distance to cluster image patches (2D matrices having the same dimensions). I haven't been able to find any generalisation up to this point and would prefer not to vectorise my patches and end-up with a huge covariance matrix.
If your "image pa... | Is there a version of the Mahalanobis distance for matrices? | I'm working on a computer vision problem and I want to use the Mahalanobis distance to cluster image patches (2D matrices having the same dimensions). I haven't been able to find any generalisation up | Is there a version of the Mahalanobis distance for matrices?
I'm working on a computer vision problem and I want to use the Mahalanobis distance to cluster image patches (2D matrices having the same dimensions). I haven't been able to find any generalisation up to this point and would prefer not to vectorise my patches... | Is there a version of the Mahalanobis distance for matrices?
I'm working on a computer vision problem and I want to use the Mahalanobis distance to cluster image patches (2D matrices having the same dimensions). I haven't been able to find any generalisation up |
36,543 | Metropolis-Within-Gibbs sampling with only marginal distribution known for a subset of variables | In an ideal world, sampling from $p_1(x_1)$ and then from $p_{1|2}(x_2|x_1)$ is a correct way to simulate from the joint. In case one of these distributions is unavailable, simulating a single step of Metropolis-Within-Gibbs targeting $p_{1|2}(\cdot|x_1^{(t-1)})$ and a single step of Metropolis-Within-Gibbs targeting ... | Metropolis-Within-Gibbs sampling with only marginal distribution known for a subset of variables | In an ideal world, sampling from $p_1(x_1)$ and then from $p_{1|2}(x_2|x_1)$ is a correct way to simulate from the joint. In case one of these distributions is unavailable, simulating a single step of | Metropolis-Within-Gibbs sampling with only marginal distribution known for a subset of variables
In an ideal world, sampling from $p_1(x_1)$ and then from $p_{1|2}(x_2|x_1)$ is a correct way to simulate from the joint. In case one of these distributions is unavailable, simulating a single step of Metropolis-Within-Gibb... | Metropolis-Within-Gibbs sampling with only marginal distribution known for a subset of variables
In an ideal world, sampling from $p_1(x_1)$ and then from $p_{1|2}(x_2|x_1)$ is a correct way to simulate from the joint. In case one of these distributions is unavailable, simulating a single step of |
36,544 | Final layer of neural network responsible for overfitting | With sample size $N=30\times10^6$ and 500 features, you already tried (most of) the usual regularization tricks, thus it doesn't look like there's much left to do at this point.
However, maybe the problem here is upstream. You haven't told us what's your dataset, exactly (what are the observations? What are the featur... | Final layer of neural network responsible for overfitting | With sample size $N=30\times10^6$ and 500 features, you already tried (most of) the usual regularization tricks, thus it doesn't look like there's much left to do at this point.
However, maybe the pr | Final layer of neural network responsible for overfitting
With sample size $N=30\times10^6$ and 500 features, you already tried (most of) the usual regularization tricks, thus it doesn't look like there's much left to do at this point.
However, maybe the problem here is upstream. You haven't told us what's your datase... | Final layer of neural network responsible for overfitting
With sample size $N=30\times10^6$ and 500 features, you already tried (most of) the usual regularization tricks, thus it doesn't look like there's much left to do at this point.
However, maybe the pr |
36,545 | Final layer of neural network responsible for overfitting | A few years late but: What optimizer are you using? Adam will usually do a pretty good job without much tuning effort. However, recent work seems to show that SGD performs better than adaptive methods in many cases, so it's also probably worth exploring. Make sure to tune both your initial learning rate as well as your... | Final layer of neural network responsible for overfitting | A few years late but: What optimizer are you using? Adam will usually do a pretty good job without much tuning effort. However, recent work seems to show that SGD performs better than adaptive methods | Final layer of neural network responsible for overfitting
A few years late but: What optimizer are you using? Adam will usually do a pretty good job without much tuning effort. However, recent work seems to show that SGD performs better than adaptive methods in many cases, so it's also probably worth exploring. Make su... | Final layer of neural network responsible for overfitting
A few years late but: What optimizer are you using? Adam will usually do a pretty good job without much tuning effort. However, recent work seems to show that SGD performs better than adaptive methods |
36,546 | Dynamic Bayesian Network library in Python [closed] | Try pgmpy.
You can also create something on your own by using more generic tools for Graphical Probabilistic Models such as PyJaggs or Edward. | Dynamic Bayesian Network library in Python [closed] | Try pgmpy.
You can also create something on your own by using more generic tools for Graphical Probabilistic Models such as PyJaggs or Edward. | Dynamic Bayesian Network library in Python [closed]
Try pgmpy.
You can also create something on your own by using more generic tools for Graphical Probabilistic Models such as PyJaggs or Edward. | Dynamic Bayesian Network library in Python [closed]
Try pgmpy.
You can also create something on your own by using more generic tools for Graphical Probabilistic Models such as PyJaggs or Edward. |
36,547 | Gap Statistic in plain English? | Since I'm looking into the gap statistics just now, I'll try to answer this old question.
$\newcommand{\Gap}{{\rm Gap}}$
Once you have partitioned your datapoints into $k$ clusters using e.g., k-means, you want some metric that describes how compact the clusters are. One way of doing this is to compute all the pairwise... | Gap Statistic in plain English? | Since I'm looking into the gap statistics just now, I'll try to answer this old question.
$\newcommand{\Gap}{{\rm Gap}}$
Once you have partitioned your datapoints into $k$ clusters using e.g., k-means | Gap Statistic in plain English?
Since I'm looking into the gap statistics just now, I'll try to answer this old question.
$\newcommand{\Gap}{{\rm Gap}}$
Once you have partitioned your datapoints into $k$ clusters using e.g., k-means, you want some metric that describes how compact the clusters are. One way of doing thi... | Gap Statistic in plain English?
Since I'm looking into the gap statistics just now, I'll try to answer this old question.
$\newcommand{\Gap}{{\rm Gap}}$
Once you have partitioned your datapoints into $k$ clusters using e.g., k-means |
36,548 | How do we incorporate new information into a Dirichlet prior distribution? | Dirichlet prior is an appropriate prior, and is the conjugate prior to a multinomial distribution. However, it seems a bit tricky to apply this to the output of a multinomial logistic regression, since such a regression has a softmax as the output, not a multinomial distribution. However, what we can do is sample from ... | How do we incorporate new information into a Dirichlet prior distribution? | Dirichlet prior is an appropriate prior, and is the conjugate prior to a multinomial distribution. However, it seems a bit tricky to apply this to the output of a multinomial logistic regression, sinc | How do we incorporate new information into a Dirichlet prior distribution?
Dirichlet prior is an appropriate prior, and is the conjugate prior to a multinomial distribution. However, it seems a bit tricky to apply this to the output of a multinomial logistic regression, since such a regression has a softmax as the outp... | How do we incorporate new information into a Dirichlet prior distribution?
Dirichlet prior is an appropriate prior, and is the conjugate prior to a multinomial distribution. However, it seems a bit tricky to apply this to the output of a multinomial logistic regression, sinc |
36,549 | Overfitting on the loss graph, but not the accuracy graph | Accuracy is not a great way to report machine learning results. (I've never found a need to report accuracy, except when explaining my results to a non-technical audience.) Accuracy only compares a predicted score $t$ to some cutoff $c$, which is not a proper scoring rule and conceals important information about model ... | Overfitting on the loss graph, but not the accuracy graph | Accuracy is not a great way to report machine learning results. (I've never found a need to report accuracy, except when explaining my results to a non-technical audience.) Accuracy only compares a pr | Overfitting on the loss graph, but not the accuracy graph
Accuracy is not a great way to report machine learning results. (I've never found a need to report accuracy, except when explaining my results to a non-technical audience.) Accuracy only compares a predicted score $t$ to some cutoff $c$, which is not a proper sc... | Overfitting on the loss graph, but not the accuracy graph
Accuracy is not a great way to report machine learning results. (I've never found a need to report accuracy, except when explaining my results to a non-technical audience.) Accuracy only compares a pr |
36,550 | Regression Using Continuous Variable with Nulls | By the way, the problem you have right now is the "missing data" problem. It can be such an issue that that's one of the reasons they do redundant checkups on you for like 5 things everytime you go to the doctor regardless of whether or not you feel it's related. No one wants to deal with the NULL monster.
It can be ... | Regression Using Continuous Variable with Nulls | By the way, the problem you have right now is the "missing data" problem. It can be such an issue that that's one of the reasons they do redundant checkups on you for like 5 things everytime you go t | Regression Using Continuous Variable with Nulls
By the way, the problem you have right now is the "missing data" problem. It can be such an issue that that's one of the reasons they do redundant checkups on you for like 5 things everytime you go to the doctor regardless of whether or not you feel it's related. No one... | Regression Using Continuous Variable with Nulls
By the way, the problem you have right now is the "missing data" problem. It can be such an issue that that's one of the reasons they do redundant checkups on you for like 5 things everytime you go t |
36,551 | Regression Using Continuous Variable with Nulls | This sounds like you have a predictor which is only relevant for part of the population. My advice is to (conceptually) split your population into two groups. Then write down equations for what your linear predictor should look like for each group. Then consider if these can merged back together into one model, or if y... | Regression Using Continuous Variable with Nulls | This sounds like you have a predictor which is only relevant for part of the population. My advice is to (conceptually) split your population into two groups. Then write down equations for what your l | Regression Using Continuous Variable with Nulls
This sounds like you have a predictor which is only relevant for part of the population. My advice is to (conceptually) split your population into two groups. Then write down equations for what your linear predictor should look like for each group. Then consider if these ... | Regression Using Continuous Variable with Nulls
This sounds like you have a predictor which is only relevant for part of the population. My advice is to (conceptually) split your population into two groups. Then write down equations for what your l |
36,552 | Why is it desirable to have linear separability in SVM? | Well, that is the whole idea behind support vector machines! svm are searching for a hyperplane that separates the classes (why the name), and that can of course be done most effectively it the points are linearly separable (that's not a deep point, it is a summary of the full idea). In the example you show, point lie... | Why is it desirable to have linear separability in SVM? | Well, that is the whole idea behind support vector machines! svm are searching for a hyperplane that separates the classes (why the name), and that can of course be done most effectively it the points | Why is it desirable to have linear separability in SVM?
Well, that is the whole idea behind support vector machines! svm are searching for a hyperplane that separates the classes (why the name), and that can of course be done most effectively it the points are linearly separable (that's not a deep point, it is a summar... | Why is it desirable to have linear separability in SVM?
Well, that is the whole idea behind support vector machines! svm are searching for a hyperplane that separates the classes (why the name), and that can of course be done most effectively it the points |
36,553 | Why is it desirable to have linear separability in SVM? | Why is it desirable to have linear separability in SVM?
SVCs are inherently a linear technique. They find linear boundaries separating (as best possible) different classes. If there is no natural linear boundary for the problem, the choices are either to use a different technique, or to use SVCs with transformed featu... | Why is it desirable to have linear separability in SVM? | Why is it desirable to have linear separability in SVM?
SVCs are inherently a linear technique. They find linear boundaries separating (as best possible) different classes. If there is no natural lin | Why is it desirable to have linear separability in SVM?
Why is it desirable to have linear separability in SVM?
SVCs are inherently a linear technique. They find linear boundaries separating (as best possible) different classes. If there is no natural linear boundary for the problem, the choices are either to use a di... | Why is it desirable to have linear separability in SVM?
Why is it desirable to have linear separability in SVM?
SVCs are inherently a linear technique. They find linear boundaries separating (as best possible) different classes. If there is no natural lin |
36,554 | Why is it desirable to have linear separability in SVM? | Not directly answer your question but,
It is important to keep in mind the difference between basis expansion and Kernel method / SVM.
We can "expand data" using basis expansion in different ways. For example, polynomial expansion, splines, Fourier series, etc. These basis expansion have little to do with SVM, kernel ... | Why is it desirable to have linear separability in SVM? | Not directly answer your question but,
It is important to keep in mind the difference between basis expansion and Kernel method / SVM.
We can "expand data" using basis expansion in different ways. Fo | Why is it desirable to have linear separability in SVM?
Not directly answer your question but,
It is important to keep in mind the difference between basis expansion and Kernel method / SVM.
We can "expand data" using basis expansion in different ways. For example, polynomial expansion, splines, Fourier series, etc. T... | Why is it desirable to have linear separability in SVM?
Not directly answer your question but,
It is important to keep in mind the difference between basis expansion and Kernel method / SVM.
We can "expand data" using basis expansion in different ways. Fo |
36,555 | Why is it desirable to have linear separability in SVM? | SVCs were originally defined to be linear classifiers which tried to find maximum margin hyperplanes. The optimization problem is derived starting with an equation of a plane and calculating the distance of points from it, and finding out its parameter matrix w. See here for details. In a 2 dimensional case(left figure... | Why is it desirable to have linear separability in SVM? | SVCs were originally defined to be linear classifiers which tried to find maximum margin hyperplanes. The optimization problem is derived starting with an equation of a plane and calculating the dista | Why is it desirable to have linear separability in SVM?
SVCs were originally defined to be linear classifiers which tried to find maximum margin hyperplanes. The optimization problem is derived starting with an equation of a plane and calculating the distance of points from it, and finding out its parameter matrix w. S... | Why is it desirable to have linear separability in SVM?
SVCs were originally defined to be linear classifiers which tried to find maximum margin hyperplanes. The optimization problem is derived starting with an equation of a plane and calculating the dista |
36,556 | Why is it desirable to have linear separability in SVM? | You are correct. When the field says "linearly separable", they mean that the data should be "differentiable": that there exists some filtering function that you can overlay onto the dataset to create two or more distinct groupings (with some small error tolerance).
That's all. But you should point out to the academi... | Why is it desirable to have linear separability in SVM? | You are correct. When the field says "linearly separable", they mean that the data should be "differentiable": that there exists some filtering function that you can overlay onto the dataset to creat | Why is it desirable to have linear separability in SVM?
You are correct. When the field says "linearly separable", they mean that the data should be "differentiable": that there exists some filtering function that you can overlay onto the dataset to create two or more distinct groupings (with some small error toleranc... | Why is it desirable to have linear separability in SVM?
You are correct. When the field says "linearly separable", they mean that the data should be "differentiable": that there exists some filtering function that you can overlay onto the dataset to creat |
36,557 | The unit of Root Mean Square Error (RMSE) | Let's say you have a model represented by the function $f(x)$ and you calculate the RMSE of the outcomes compared with the training set outcomes $y$. Let's also assume the outcome has some arbitrary unit $u$.
The RMSE is
$$RMSE(y)=\frac{1}{N^2}\sqrt{\sum_i{(f(x_i) - y_i)^2}}$$
or expressing the units explicitly
$$RMSE(... | The unit of Root Mean Square Error (RMSE) | Let's say you have a model represented by the function $f(x)$ and you calculate the RMSE of the outcomes compared with the training set outcomes $y$. Let's also assume the outcome has some arbitrary u | The unit of Root Mean Square Error (RMSE)
Let's say you have a model represented by the function $f(x)$ and you calculate the RMSE of the outcomes compared with the training set outcomes $y$. Let's also assume the outcome has some arbitrary unit $u$.
The RMSE is
$$RMSE(y)=\frac{1}{N^2}\sqrt{\sum_i{(f(x_i) - y_i)^2}}$$
... | The unit of Root Mean Square Error (RMSE)
Let's say you have a model represented by the function $f(x)$ and you calculate the RMSE of the outcomes compared with the training set outcomes $y$. Let's also assume the outcome has some arbitrary u |
36,558 | Motivation for gamma distribution with a non-integer parameter | It seems you are asking for "real-life" examples where gamma distributions are used to model some real-world observables represented by random variables. There are many such examples. Take the Erlang distribution you mention first: The integer parameter case follows from some theoretical probability model for waiting t... | Motivation for gamma distribution with a non-integer parameter | It seems you are asking for "real-life" examples where gamma distributions are used to model some real-world observables represented by random variables. There are many such examples. Take the Erlang | Motivation for gamma distribution with a non-integer parameter
It seems you are asking for "real-life" examples where gamma distributions are used to model some real-world observables represented by random variables. There are many such examples. Take the Erlang distribution you mention first: The integer parameter cas... | Motivation for gamma distribution with a non-integer parameter
It seems you are asking for "real-life" examples where gamma distributions are used to model some real-world observables represented by random variables. There are many such examples. Take the Erlang |
36,559 | Motivation for gamma distribution with a non-integer parameter | Given a radioactive sample with unknown emission rate $\lambda$, the likelihood on $\lambda$ induced by observations of emissions is gamma distributed.
Given a normal process with known mean, but unknown precision, the induced likelihood on the precision is gamma distributed.
And similarly for Pareto and gamma models. | Motivation for gamma distribution with a non-integer parameter | Given a radioactive sample with unknown emission rate $\lambda$, the likelihood on $\lambda$ induced by observations of emissions is gamma distributed.
Given a normal process with known mean, but unkn | Motivation for gamma distribution with a non-integer parameter
Given a radioactive sample with unknown emission rate $\lambda$, the likelihood on $\lambda$ induced by observations of emissions is gamma distributed.
Given a normal process with known mean, but unknown precision, the induced likelihood on the precision is... | Motivation for gamma distribution with a non-integer parameter
Given a radioactive sample with unknown emission rate $\lambda$, the likelihood on $\lambda$ induced by observations of emissions is gamma distributed.
Given a normal process with known mean, but unkn |
36,560 | Interpreting parametric and non-parametric testing | This is a welcome opportunity to discuss and clarify what statistical models mean and how we ought to think about them. Let's begin with definitions, so that the scope of this answer is in no doubt, and move on from there. To keep this post short, I will limit the examples and forgo all illustrations, trusting the re... | Interpreting parametric and non-parametric testing | This is a welcome opportunity to discuss and clarify what statistical models mean and how we ought to think about them. Let's begin with definitions, so that the scope of this answer is in no doubt, | Interpreting parametric and non-parametric testing
This is a welcome opportunity to discuss and clarify what statistical models mean and how we ought to think about them. Let's begin with definitions, so that the scope of this answer is in no doubt, and move on from there. To keep this post short, I will limit the ex... | Interpreting parametric and non-parametric testing
This is a welcome opportunity to discuss and clarify what statistical models mean and how we ought to think about them. Let's begin with definitions, so that the scope of this answer is in no doubt, |
36,561 | Most well-known set-similarity measures? | Other measures are:
Overlap Coefficient: $\frac{|A \cap B|}{min(|A|,|B|)}$
Tversky index: $|A\cap B| + \alpha|A\setminus B| + \beta|B \setminus A|$ where $\alpha$ and $\beta$ are positive numbers. | Most well-known set-similarity measures? | Other measures are:
Overlap Coefficient: $\frac{|A \cap B|}{min(|A|,|B|)}$
Tversky index: $|A\cap B| + \alpha|A\setminus B| + \beta|B \setminus A|$ where $\alpha$ and $\beta$ are positive numbers. | Most well-known set-similarity measures?
Other measures are:
Overlap Coefficient: $\frac{|A \cap B|}{min(|A|,|B|)}$
Tversky index: $|A\cap B| + \alpha|A\setminus B| + \beta|B \setminus A|$ where $\alpha$ and $\beta$ are positive numbers. | Most well-known set-similarity measures?
Other measures are:
Overlap Coefficient: $\frac{|A \cap B|}{min(|A|,|B|)}$
Tversky index: $|A\cap B| + \alpha|A\setminus B| + \beta|B \setminus A|$ where $\alpha$ and $\beta$ are positive numbers. |
36,562 | Why do we assume the exponential family in the GLM context? | When I discovered GLM I also wondered why it was always based on the exponential family. I have never answered to that question clearly. But...
I call $h$ the reciprocal of the link function. $\beta$ the parameter.
When I first learned about Generalized Linear Models I thought that
the assumption that the dependent ... | Why do we assume the exponential family in the GLM context? | When I discovered GLM I also wondered why it was always based on the exponential family. I have never answered to that question clearly. But...
I call $h$ the reciprocal of the link function. $\beta$ | Why do we assume the exponential family in the GLM context?
When I discovered GLM I also wondered why it was always based on the exponential family. I have never answered to that question clearly. But...
I call $h$ the reciprocal of the link function. $\beta$ the parameter.
When I first learned about Generalized Linea... | Why do we assume the exponential family in the GLM context?
When I discovered GLM I also wondered why it was always based on the exponential family. I have never answered to that question clearly. But...
I call $h$ the reciprocal of the link function. $\beta$ |
36,563 | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its disadvantage? | Principal Components Analysis (PCA) doesn't select among the original features unless the features are orthogonal. It lowers the dimension of the feature space by ignoring PCs (linear combinations of potentially all the predictors) that explain little of the feature-set variance, but in general it does not remove any o... | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its d | Principal Components Analysis (PCA) doesn't select among the original features unless the features are orthogonal. It lowers the dimension of the feature space by ignoring PCs (linear combinations of | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its disadvantage?
Principal Components Analysis (PCA) doesn't select among the original features unless the features are orthogonal. It lowers the dimension of the feature space by ignoring PCs (linear combinations of potenti... | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its d
Principal Components Analysis (PCA) doesn't select among the original features unless the features are orthogonal. It lowers the dimension of the feature space by ignoring PCs (linear combinations of |
36,564 | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its disadvantage? | Using PCA to do feature selection is different than using LASSO because PCA does not consider the response variable, but only check the variance in features.
See my answers here, with graphical illustration (logistic regression case / linear method without regularization).
How to decide between PCA and logistic regress... | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its d | Using PCA to do feature selection is different than using LASSO because PCA does not consider the response variable, but only check the variance in features.
See my answers here, with graphical illust | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its disadvantage?
Using PCA to do feature selection is different than using LASSO because PCA does not consider the response variable, but only check the variance in features.
See my answers here, with graphical illustration ... | Why is selection of one amongst the group of highly correlated variables by lasso is stated as its d
Using PCA to do feature selection is different than using LASSO because PCA does not consider the response variable, but only check the variance in features.
See my answers here, with graphical illust |
36,565 | Generate simulated data with predefined partial correlation structure | Why not just create data following your causal model? If the model is $X \leftarrow Z \rightarrow Y$, then just create random $Z$, and build $X$ as some function of $Z$ plus noise (same for $Y$). (I left error SD=1, but just set it higher to get more realistic significance.)
> set.seed(100)
> z = rnorm(500)
> x = z + r... | Generate simulated data with predefined partial correlation structure | Why not just create data following your causal model? If the model is $X \leftarrow Z \rightarrow Y$, then just create random $Z$, and build $X$ as some function of $Z$ plus noise (same for $Y$). (I l | Generate simulated data with predefined partial correlation structure
Why not just create data following your causal model? If the model is $X \leftarrow Z \rightarrow Y$, then just create random $Z$, and build $X$ as some function of $Z$ plus noise (same for $Y$). (I left error SD=1, but just set it higher to get more... | Generate simulated data with predefined partial correlation structure
Why not just create data following your causal model? If the model is $X \leftarrow Z \rightarrow Y$, then just create random $Z$, and build $X$ as some function of $Z$ plus noise (same for $Y$). (I l |
36,566 | Meaning of low power in neuroscience after combining results of many meta-analyses (Button et al 2013) | I'm not familiar with this particular study but am familiar with estimating the power of an area of research using a meta-analysis. Your statement that "post hoc power is always inherently associated to the achieved p-values" suggests to me that you are assuming that the post hoc power for each individual study contrib... | Meaning of low power in neuroscience after combining results of many meta-analyses (Button et al 201 | I'm not familiar with this particular study but am familiar with estimating the power of an area of research using a meta-analysis. Your statement that "post hoc power is always inherently associated | Meaning of low power in neuroscience after combining results of many meta-analyses (Button et al 2013)
I'm not familiar with this particular study but am familiar with estimating the power of an area of research using a meta-analysis. Your statement that "post hoc power is always inherently associated to the achieved p... | Meaning of low power in neuroscience after combining results of many meta-analyses (Button et al 201
I'm not familiar with this particular study but am familiar with estimating the power of an area of research using a meta-analysis. Your statement that "post hoc power is always inherently associated |
36,567 | Convergence concept for MonteCarlo(?) simulation of a tournament | Is this a Monte Carlo simulation / a naive way of drawing from a distribution?
Yes, this is a Monte Carlo simulation. There's nothing wrong with this tactic.
What is the relevant convergence concept that tells me that as the number of draws goes to infinity, my draws converge in (say) distribution to the true distribut... | Convergence concept for MonteCarlo(?) simulation of a tournament | Is this a Monte Carlo simulation / a naive way of drawing from a distribution?
Yes, this is a Monte Carlo simulation. There's nothing wrong with this tactic.
What is the relevant convergence concept t | Convergence concept for MonteCarlo(?) simulation of a tournament
Is this a Monte Carlo simulation / a naive way of drawing from a distribution?
Yes, this is a Monte Carlo simulation. There's nothing wrong with this tactic.
What is the relevant convergence concept that tells me that as the number of draws goes to infini... | Convergence concept for MonteCarlo(?) simulation of a tournament
Is this a Monte Carlo simulation / a naive way of drawing from a distribution?
Yes, this is a Monte Carlo simulation. There's nothing wrong with this tactic.
What is the relevant convergence concept t |
36,568 | Why don't we consider nonlinear estimators for the parameters of linear regression models? | In mathematics rarely things are developed in the way they're presented in textbooks. That's the real reason. Here's the explanation.
First, someone came up with a problem to fit $$y=X\beta+\varepsilon$$, i.e. find the "best" in some respect set of parameters $\beta$. Whoever did this didn't think that the solution wou... | Why don't we consider nonlinear estimators for the parameters of linear regression models? | In mathematics rarely things are developed in the way they're presented in textbooks. That's the real reason. Here's the explanation.
First, someone came up with a problem to fit $$y=X\beta+\varepsilo | Why don't we consider nonlinear estimators for the parameters of linear regression models?
In mathematics rarely things are developed in the way they're presented in textbooks. That's the real reason. Here's the explanation.
First, someone came up with a problem to fit $$y=X\beta+\varepsilon$$, i.e. find the "best" in ... | Why don't we consider nonlinear estimators for the parameters of linear regression models?
In mathematics rarely things are developed in the way they're presented in textbooks. That's the real reason. Here's the explanation.
First, someone came up with a problem to fit $$y=X\beta+\varepsilo |
36,569 | Why don't we consider nonlinear estimators for the parameters of linear regression models? | When the error term is not Gaussian, it will generally be the case that the best estimators (e.g. in terms of MSE) are not linear.
In some cases, all linear estimators may be arbitrarily bad. (It's not always so clear what all the fuss about being BLUE is, when even the best linear estimator may be terrible.)
So for ... | Why don't we consider nonlinear estimators for the parameters of linear regression models? | When the error term is not Gaussian, it will generally be the case that the best estimators (e.g. in terms of MSE) are not linear.
In some cases, all linear estimators may be arbitrarily bad. (It's | Why don't we consider nonlinear estimators for the parameters of linear regression models?
When the error term is not Gaussian, it will generally be the case that the best estimators (e.g. in terms of MSE) are not linear.
In some cases, all linear estimators may be arbitrarily bad. (It's not always so clear what all ... | Why don't we consider nonlinear estimators for the parameters of linear regression models?
When the error term is not Gaussian, it will generally be the case that the best estimators (e.g. in terms of MSE) are not linear.
In some cases, all linear estimators may be arbitrarily bad. (It's |
36,570 | Why don't we consider nonlinear estimators for the parameters of linear regression models? | The question is how you find $\beta$? How do you actually derive OLS? You differentiate according to a certain parameter, and you get a set of equations, which you then solve. If $\beta$ is not linear in the model equation, the solution will not be a linear combination of $Y$. (It will probably not be a linear combinat... | Why don't we consider nonlinear estimators for the parameters of linear regression models? | The question is how you find $\beta$? How do you actually derive OLS? You differentiate according to a certain parameter, and you get a set of equations, which you then solve. If $\beta$ is not linear | Why don't we consider nonlinear estimators for the parameters of linear regression models?
The question is how you find $\beta$? How do you actually derive OLS? You differentiate according to a certain parameter, and you get a set of equations, which you then solve. If $\beta$ is not linear in the model equation, the s... | Why don't we consider nonlinear estimators for the parameters of linear regression models?
The question is how you find $\beta$? How do you actually derive OLS? You differentiate according to a certain parameter, and you get a set of equations, which you then solve. If $\beta$ is not linear |
36,571 | Why is ReLU setting negative values to zero particularly? | The weights learned in a neural network can be both positive and negative. So in effect, either form would work. Negating the input and output weights with the $\min$ form gives the same function as with the $\max$ form. The max form is used purely by convention. | Why is ReLU setting negative values to zero particularly? | The weights learned in a neural network can be both positive and negative. So in effect, either form would work. Negating the input and output weights with the $\min$ form gives the same function as w | Why is ReLU setting negative values to zero particularly?
The weights learned in a neural network can be both positive and negative. So in effect, either form would work. Negating the input and output weights with the $\min$ form gives the same function as with the $\max$ form. The max form is used purely by conventio... | Why is ReLU setting negative values to zero particularly?
The weights learned in a neural network can be both positive and negative. So in effect, either form would work. Negating the input and output weights with the $\min$ form gives the same function as w |
36,572 | Extracting Maximum A Posteriori (MAP) estimates from MC samples | [The following is copied from earlier posts on my blog.]
I have never found MAP estimators very appealing for many reasons, one being indeed that the MAP estimator cannot correctly be expressed as the solution to a minimisation problem. I also find the point-wise nature of the estimator quite a drawback: the estimator ... | Extracting Maximum A Posteriori (MAP) estimates from MC samples | [The following is copied from earlier posts on my blog.]
I have never found MAP estimators very appealing for many reasons, one being indeed that the MAP estimator cannot correctly be expressed as the | Extracting Maximum A Posteriori (MAP) estimates from MC samples
[The following is copied from earlier posts on my blog.]
I have never found MAP estimators very appealing for many reasons, one being indeed that the MAP estimator cannot correctly be expressed as the solution to a minimisation problem. I also find the poi... | Extracting Maximum A Posteriori (MAP) estimates from MC samples
[The following is copied from earlier posts on my blog.]
I have never found MAP estimators very appealing for many reasons, one being indeed that the MAP estimator cannot correctly be expressed as the |
36,573 | Is there any difference between estimating $\sigma^2$ and $\sigma$ in a simulation study? | I find this question of interest because it highlights the artificial nature of seeking unbiasedness above everything else. A few points:
the variance $\sigma^2$ allows for an unbiased estimator, while the square root of that estimator $\hat\sigma_n$ is biased [by Jensen's inequality];
there is no generic unbiased est... | Is there any difference between estimating $\sigma^2$ and $\sigma$ in a simulation study? | I find this question of interest because it highlights the artificial nature of seeking unbiasedness above everything else. A few points:
the variance $\sigma^2$ allows for an unbiased estimator, whi | Is there any difference between estimating $\sigma^2$ and $\sigma$ in a simulation study?
I find this question of interest because it highlights the artificial nature of seeking unbiasedness above everything else. A few points:
the variance $\sigma^2$ allows for an unbiased estimator, while the square root of that est... | Is there any difference between estimating $\sigma^2$ and $\sigma$ in a simulation study?
I find this question of interest because it highlights the artificial nature of seeking unbiasedness above everything else. A few points:
the variance $\sigma^2$ allows for an unbiased estimator, whi |
36,574 | Accuracy vs Jaccard for multiclass problem | The issue has been reported on scikit-learn GitHub repository: multiclass jaccard_similarity_score should not be equal to accuracy_score #7332
scikit-learn's Jaccard score for the multiclass classification task is incorrect.
A neat overview of the most commonly used performance metrics from {1}:
The accuracy is $~\fr... | Accuracy vs Jaccard for multiclass problem | The issue has been reported on scikit-learn GitHub repository: multiclass jaccard_similarity_score should not be equal to accuracy_score #7332
scikit-learn's Jaccard score for the multiclass classific | Accuracy vs Jaccard for multiclass problem
The issue has been reported on scikit-learn GitHub repository: multiclass jaccard_similarity_score should not be equal to accuracy_score #7332
scikit-learn's Jaccard score for the multiclass classification task is incorrect.
A neat overview of the most commonly used performan... | Accuracy vs Jaccard for multiclass problem
The issue has been reported on scikit-learn GitHub repository: multiclass jaccard_similarity_score should not be equal to accuracy_score #7332
scikit-learn's Jaccard score for the multiclass classific |
36,575 | Accuracy vs Jaccard for multiclass problem | K-class multinomial classification results for n cases (tries) is a nominal variable. Therefore it can be represented as a set of k binary dummy variables.
Now, Jaccard similarity coefficient between two cases (row vectors) by a set of binary attributes is $\frac{a}{a+b+c}$; and accuracy score (I believe it is F1 score... | Accuracy vs Jaccard for multiclass problem | K-class multinomial classification results for n cases (tries) is a nominal variable. Therefore it can be represented as a set of k binary dummy variables.
Now, Jaccard similarity coefficient between | Accuracy vs Jaccard for multiclass problem
K-class multinomial classification results for n cases (tries) is a nominal variable. Therefore it can be represented as a set of k binary dummy variables.
Now, Jaccard similarity coefficient between two cases (row vectors) by a set of binary attributes is $\frac{a}{a+b+c}$; a... | Accuracy vs Jaccard for multiclass problem
K-class multinomial classification results for n cases (tries) is a nominal variable. Therefore it can be represented as a set of k binary dummy variables.
Now, Jaccard similarity coefficient between |
36,576 | Is it better to compute a ROC curve using predicted probabilities or distances from the separating hyperplane? | I am answering this from a pragmatic perspective, simply by looking at code and deducing from examples. A more theoretic answer could be a great supplement.
Generally both can be used. The difference is well explained here.
Yet most relevant, not all algorithms offer both predict_proba and decision_function. To my know... | Is it better to compute a ROC curve using predicted probabilities or distances from the separating h | I am answering this from a pragmatic perspective, simply by looking at code and deducing from examples. A more theoretic answer could be a great supplement.
Generally both can be used. The difference | Is it better to compute a ROC curve using predicted probabilities or distances from the separating hyperplane?
I am answering this from a pragmatic perspective, simply by looking at code and deducing from examples. A more theoretic answer could be a great supplement.
Generally both can be used. The difference is well e... | Is it better to compute a ROC curve using predicted probabilities or distances from the separating h
I am answering this from a pragmatic perspective, simply by looking at code and deducing from examples. A more theoretic answer could be a great supplement.
Generally both can be used. The difference |
36,577 | Variational Auto-encoders vs Restricted Boltzmann Machines | Theoretically RBMs are undirected models with no connections between any observed variables or any latent variables, VAEs are directed models with continuous latent variables.
The Deep Learning book (Chapter 20 Deep Generative Models) provides a very good summary of pros and cons of VAEs and all types of RBMs.
To quot... | Variational Auto-encoders vs Restricted Boltzmann Machines | Theoretically RBMs are undirected models with no connections between any observed variables or any latent variables, VAEs are directed models with continuous latent variables.
The Deep Learning book ( | Variational Auto-encoders vs Restricted Boltzmann Machines
Theoretically RBMs are undirected models with no connections between any observed variables or any latent variables, VAEs are directed models with continuous latent variables.
The Deep Learning book (Chapter 20 Deep Generative Models) provides a very good summa... | Variational Auto-encoders vs Restricted Boltzmann Machines
Theoretically RBMs are undirected models with no connections between any observed variables or any latent variables, VAEs are directed models with continuous latent variables.
The Deep Learning book ( |
36,578 | Quantile Regression vs OLS for homoscedasticity | Will the estimated slope coefficient $\beta_1$ always be the same for OLS and for QR for different quantiles?
No, of course not, because the empirical loss function being minimized differs in these different cases (OLS vs. QR for different quantiles).
I am well aware that in the presence of homoscedasticity all the s... | Quantile Regression vs OLS for homoscedasticity | Will the estimated slope coefficient $\beta_1$ always be the same for OLS and for QR for different quantiles?
No, of course not, because the empirical loss function being minimized differs in these d | Quantile Regression vs OLS for homoscedasticity
Will the estimated slope coefficient $\beta_1$ always be the same for OLS and for QR for different quantiles?
No, of course not, because the empirical loss function being minimized differs in these different cases (OLS vs. QR for different quantiles).
I am well aware th... | Quantile Regression vs OLS for homoscedasticity
Will the estimated slope coefficient $\beta_1$ always be the same for OLS and for QR for different quantiles?
No, of course not, because the empirical loss function being minimized differs in these d |
36,579 | Quantile Regression vs OLS for homoscedasticity | Generally the answer is yes, at least for Theil's regression, which is a special case of QR. The slope estimator for Theil's regression is an unbiased estimator of the population slope. If all the requirements for OLS are met, then it has 85% relative efficiency. There are certain circumstances where it becomes more... | Quantile Regression vs OLS for homoscedasticity | Generally the answer is yes, at least for Theil's regression, which is a special case of QR. The slope estimator for Theil's regression is an unbiased estimator of the population slope. If all the r | Quantile Regression vs OLS for homoscedasticity
Generally the answer is yes, at least for Theil's regression, which is a special case of QR. The slope estimator for Theil's regression is an unbiased estimator of the population slope. If all the requirements for OLS are met, then it has 85% relative efficiency. There... | Quantile Regression vs OLS for homoscedasticity
Generally the answer is yes, at least for Theil's regression, which is a special case of QR. The slope estimator for Theil's regression is an unbiased estimator of the population slope. If all the r |
36,580 | Using segmented linear regression as evidence for the limit of human lifespan | First of all, let's manually extract the values from their original Figure 2 and plot the data without any colors or regression lines biasing our first visual inspection of the raw data.
year <- c(1968, 1970, 1973, 1975, 1978, 1979, 1980, 1981, 1982,
1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991,
... | Using segmented linear regression as evidence for the limit of human lifespan | First of all, let's manually extract the values from their original Figure 2 and plot the data without any colors or regression lines biasing our first visual inspection of the raw data.
year <- c(19 | Using segmented linear regression as evidence for the limit of human lifespan
First of all, let's manually extract the values from their original Figure 2 and plot the data without any colors or regression lines biasing our first visual inspection of the raw data.
year <- c(1968, 1970, 1973, 1975, 1978, 1979, 1980, 19... | Using segmented linear regression as evidence for the limit of human lifespan
First of all, let's manually extract the values from their original Figure 2 and plot the data without any colors or regression lines biasing our first visual inspection of the raw data.
year <- c(19 |
36,581 | Using segmented linear regression as evidence for the limit of human lifespan | I think the nature of the conclusions are totally bunk. We see between 1950 and 2015 an increasing trend followed by a decreasing trend. It is a classic fallacy of applying data which are suggestive of a hypothesis different than the one tested and presenting them as such. With these data, a segmented regression can in... | Using segmented linear regression as evidence for the limit of human lifespan | I think the nature of the conclusions are totally bunk. We see between 1950 and 2015 an increasing trend followed by a decreasing trend. It is a classic fallacy of applying data which are suggestive o | Using segmented linear regression as evidence for the limit of human lifespan
I think the nature of the conclusions are totally bunk. We see between 1950 and 2015 an increasing trend followed by a decreasing trend. It is a classic fallacy of applying data which are suggestive of a hypothesis different than the one test... | Using segmented linear regression as evidence for the limit of human lifespan
I think the nature of the conclusions are totally bunk. We see between 1950 and 2015 an increasing trend followed by a decreasing trend. It is a classic fallacy of applying data which are suggestive o |
36,582 | Why is $B^TB+\lambda\Omega$ positive definite? | Showing that $B^TB+\lambda\Omega$ is PD amounts to showing that $\Omega$ is PD. (Thanks to Matthew Gunn for pointing that out in the comments.)
This is because $B^TB$ is, in the case that $p>n$, rank deficient and therefore PSD. This is because the quadratic form $a^TB^TBa\ge0\forall a\in\{\mathbb{R}^n\setminus0\}$ bec... | Why is $B^TB+\lambda\Omega$ positive definite? | Showing that $B^TB+\lambda\Omega$ is PD amounts to showing that $\Omega$ is PD. (Thanks to Matthew Gunn for pointing that out in the comments.)
This is because $B^TB$ is, in the case that $p>n$, rank | Why is $B^TB+\lambda\Omega$ positive definite?
Showing that $B^TB+\lambda\Omega$ is PD amounts to showing that $\Omega$ is PD. (Thanks to Matthew Gunn for pointing that out in the comments.)
This is because $B^TB$ is, in the case that $p>n$, rank deficient and therefore PSD. This is because the quadratic form $a^TB^TBa... | Why is $B^TB+\lambda\Omega$ positive definite?
Showing that $B^TB+\lambda\Omega$ is PD amounts to showing that $\Omega$ is PD. (Thanks to Matthew Gunn for pointing that out in the comments.)
This is because $B^TB$ is, in the case that $p>n$, rank |
36,583 | Is PCA invariant to orthogonal transformations? | The covariance matrix of $B$ is decomposed as:
$$ \frac{1}{n-1} B^T B = V_B L_B V^T_B $$
If we rewrite the covariance matrix of $B$ in terms of $A$ and $Q$ we have:
$$ \frac{1}{n-1} B^T B = \frac{1}{n-1} (AQ)^T AQ = \frac{1}{n-1} Q^T A^T A Q = Q^T (\frac{1}{n-1} A^T A) Q = Q^T V_A L_A V^T_A Q $$
We observe that the cov... | Is PCA invariant to orthogonal transformations? | The covariance matrix of $B$ is decomposed as:
$$ \frac{1}{n-1} B^T B = V_B L_B V^T_B $$
If we rewrite the covariance matrix of $B$ in terms of $A$ and $Q$ we have:
$$ \frac{1}{n-1} B^T B = \frac{1}{n | Is PCA invariant to orthogonal transformations?
The covariance matrix of $B$ is decomposed as:
$$ \frac{1}{n-1} B^T B = V_B L_B V^T_B $$
If we rewrite the covariance matrix of $B$ in terms of $A$ and $Q$ we have:
$$ \frac{1}{n-1} B^T B = \frac{1}{n-1} (AQ)^T AQ = \frac{1}{n-1} Q^T A^T A Q = Q^T (\frac{1}{n-1} A^T A) Q ... | Is PCA invariant to orthogonal transformations?
The covariance matrix of $B$ is decomposed as:
$$ \frac{1}{n-1} B^T B = V_B L_B V^T_B $$
If we rewrite the covariance matrix of $B$ in terms of $A$ and $Q$ we have:
$$ \frac{1}{n-1} B^T B = \frac{1}{n |
36,584 | Is PCA invariant to orthogonal transformations? | Let $R$ be a rotation (i.e. an orthogonal matrix: $R^T=R^{-1}$). $A=U\Sigma V^*$ by SVD, and correspondingly $A^TA=V\Sigma^2 V^*$.
On the other hand if $B:=RA$, then:
$$B^TB=A^TR^TRA=A^TA=V\Sigma^2V^*.$$
So PCA is "invariant" under rotations applied to $A$ from the left. Specifically, this means that the covariance mat... | Is PCA invariant to orthogonal transformations? | Let $R$ be a rotation (i.e. an orthogonal matrix: $R^T=R^{-1}$). $A=U\Sigma V^*$ by SVD, and correspondingly $A^TA=V\Sigma^2 V^*$.
On the other hand if $B:=RA$, then:
$$B^TB=A^TR^TRA=A^TA=V\Sigma^2V^* | Is PCA invariant to orthogonal transformations?
Let $R$ be a rotation (i.e. an orthogonal matrix: $R^T=R^{-1}$). $A=U\Sigma V^*$ by SVD, and correspondingly $A^TA=V\Sigma^2 V^*$.
On the other hand if $B:=RA$, then:
$$B^TB=A^TR^TRA=A^TA=V\Sigma^2V^*.$$
So PCA is "invariant" under rotations applied to $A$ from the left. ... | Is PCA invariant to orthogonal transformations?
Let $R$ be a rotation (i.e. an orthogonal matrix: $R^T=R^{-1}$). $A=U\Sigma V^*$ by SVD, and correspondingly $A^TA=V\Sigma^2 V^*$.
On the other hand if $B:=RA$, then:
$$B^TB=A^TR^TRA=A^TA=V\Sigma^2V^* |
36,585 | What is Significant about the Bayesian Approach in Machine Learning? | In any case then, do Bayesian Models offer some sort of inherent advantages or disadvantages over non-Bayesian models?
Simple answer: no. It's just a different type of thinking. Bayesian modeling just gives you a posterior density for the parameters your estimating. Usually, frequentist models like generalized linear ... | What is Significant about the Bayesian Approach in Machine Learning? | In any case then, do Bayesian Models offer some sort of inherent advantages or disadvantages over non-Bayesian models?
Simple answer: no. It's just a different type of thinking. Bayesian modeling jus | What is Significant about the Bayesian Approach in Machine Learning?
In any case then, do Bayesian Models offer some sort of inherent advantages or disadvantages over non-Bayesian models?
Simple answer: no. It's just a different type of thinking. Bayesian modeling just gives you a posterior density for the parameters ... | What is Significant about the Bayesian Approach in Machine Learning?
In any case then, do Bayesian Models offer some sort of inherent advantages or disadvantages over non-Bayesian models?
Simple answer: no. It's just a different type of thinking. Bayesian modeling jus |
36,586 | What is Significant about the Bayesian Approach in Machine Learning? | Agree with the previous answer. In fact, the idea behind almost any Bayesian model is having random variable, which can be distributed according to prior and posterior distributions. It doesn't imply that Bayesian approach works better or worse than non-Bayesian one, but usually it requires more data (for prior distrib... | What is Significant about the Bayesian Approach in Machine Learning? | Agree with the previous answer. In fact, the idea behind almost any Bayesian model is having random variable, which can be distributed according to prior and posterior distributions. It doesn't imply | What is Significant about the Bayesian Approach in Machine Learning?
Agree with the previous answer. In fact, the idea behind almost any Bayesian model is having random variable, which can be distributed according to prior and posterior distributions. It doesn't imply that Bayesian approach works better or worse than n... | What is Significant about the Bayesian Approach in Machine Learning?
Agree with the previous answer. In fact, the idea behind almost any Bayesian model is having random variable, which can be distributed according to prior and posterior distributions. It doesn't imply |
36,587 | What exactly does scipy.stats.ttest_ind test? | The quoted sentence:
This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values.
is a reference to the population means not the sample means; it is otherwise misleading and should certainly be clearer but they do t least say "(expected) values", which can only... | What exactly does scipy.stats.ttest_ind test? | The quoted sentence:
This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values.
is a reference to the population means not the sample means | What exactly does scipy.stats.ttest_ind test?
The quoted sentence:
This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values.
is a reference to the population means not the sample means; it is otherwise misleading and should certainly be clearer but they do t... | What exactly does scipy.stats.ttest_ind test?
The quoted sentence:
This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values.
is a reference to the population means not the sample means |
36,588 | What exactly does scipy.stats.ttest_ind test? | A bit more exact, see the docs you quoted
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test that
assumes equal population variances [R263]. If False, perform Welch’s
t-test, which does not assume equal population variance [R264].
So it either performs student's or Welch's t-te... | What exactly does scipy.stats.ttest_ind test? | A bit more exact, see the docs you quoted
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test that
assumes equal population variances [R263]. If False, perform | What exactly does scipy.stats.ttest_ind test?
A bit more exact, see the docs you quoted
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test that
assumes equal population variances [R263]. If False, perform Welch’s
t-test, which does not assume equal population variance [R264].
S... | What exactly does scipy.stats.ttest_ind test?
A bit more exact, see the docs you quoted
equal_var : bool, optional
If True (default), perform a standard independent 2 sample test that
assumes equal population variances [R263]. If False, perform |
36,589 | Variance of weighted mean greater than unweighted mean | Your linked question is addressing using weights as a shortcut for dealing with equally weighted per data point variance in which some data points occur more than once.
@whuber has addressed in a comment the situation in which the variances of all data points are equal. So I will address the situation in which they are... | Variance of weighted mean greater than unweighted mean | Your linked question is addressing using weights as a shortcut for dealing with equally weighted per data point variance in which some data points occur more than once.
@whuber has addressed in a comm | Variance of weighted mean greater than unweighted mean
Your linked question is addressing using weights as a shortcut for dealing with equally weighted per data point variance in which some data points occur more than once.
@whuber has addressed in a comment the situation in which the variances of all data points are e... | Variance of weighted mean greater than unweighted mean
Your linked question is addressing using weights as a shortcut for dealing with equally weighted per data point variance in which some data points occur more than once.
@whuber has addressed in a comm |
36,590 | Variance of weighted mean greater than unweighted mean | Here is a simple example using the $\frac1n\sum_i\left(x_i-\frac1n\sum_j x_j\right)^2$ and $\frac1{\sum_k w_k}\sum_i w_i\left(x_i-\frac1{\sum_k w_k}\sum_j w_j x_j\right)^2$ forms of the variance:
Suppose your population has measurements $20,30,40,50$.
Unweighted the mean is $35$ and variance is $125$
With respecti... | Variance of weighted mean greater than unweighted mean | Here is a simple example using the $\frac1n\sum_i\left(x_i-\frac1n\sum_j x_j\right)^2$ and $\frac1{\sum_k w_k}\sum_i w_i\left(x_i-\frac1{\sum_k w_k}\sum_j w_j x_j\right)^2$ forms of the variance:
Sup | Variance of weighted mean greater than unweighted mean
Here is a simple example using the $\frac1n\sum_i\left(x_i-\frac1n\sum_j x_j\right)^2$ and $\frac1{\sum_k w_k}\sum_i w_i\left(x_i-\frac1{\sum_k w_k}\sum_j w_j x_j\right)^2$ forms of the variance:
Suppose your population has measurements $20,30,40,50$.
Unweighted... | Variance of weighted mean greater than unweighted mean
Here is a simple example using the $\frac1n\sum_i\left(x_i-\frac1n\sum_j x_j\right)^2$ and $\frac1{\sum_k w_k}\sum_i w_i\left(x_i-\frac1{\sum_k w_k}\sum_j w_j x_j\right)^2$ forms of the variance:
Sup |
36,591 | Variance of weighted mean greater than unweighted mean | Searching for how to compute variance of a weighted sum I came across this question, and I don't see a satisfactory answer that applies to the situation state by @user08041991, namely "Let's say we have collected a series of heart rate measurements from a sample of people in a hospital. A weighting factor can then appl... | Variance of weighted mean greater than unweighted mean | Searching for how to compute variance of a weighted sum I came across this question, and I don't see a satisfactory answer that applies to the situation state by @user08041991, namely "Let's say we ha | Variance of weighted mean greater than unweighted mean
Searching for how to compute variance of a weighted sum I came across this question, and I don't see a satisfactory answer that applies to the situation state by @user08041991, namely "Let's say we have collected a series of heart rate measurements from a sample of... | Variance of weighted mean greater than unweighted mean
Searching for how to compute variance of a weighted sum I came across this question, and I don't see a satisfactory answer that applies to the situation state by @user08041991, namely "Let's say we ha |
36,592 | k-means++ algorithm and outliers | In K-Means++ method, the first centroid (let's call C1) can be among any of the points. In other words, all points have an equal chance of getting selected as first centroid in the initialization stage. Now, once the first centroid is selected, the second centroid is selected in such as manner that, a probability of an... | k-means++ algorithm and outliers | In K-Means++ method, the first centroid (let's call C1) can be among any of the points. In other words, all points have an equal chance of getting selected as first centroid in the initialization stag | k-means++ algorithm and outliers
In K-Means++ method, the first centroid (let's call C1) can be among any of the points. In other words, all points have an equal chance of getting selected as first centroid in the initialization stage. Now, once the first centroid is selected, the second centroid is selected in such as... | k-means++ algorithm and outliers
In K-Means++ method, the first centroid (let's call C1) can be among any of the points. In other words, all points have an equal chance of getting selected as first centroid in the initialization stag |
36,593 | k-means++ algorithm and outliers | Yes, the outlier os more likely to be picked. But there are also many more inliers, the chance of choosing one of them is also substantial. Say you have one outlier that is 10x farther, then it is 100x more likely to be picket than one inlier. If you have 100 inliers, the chances are about 50%, and if you have 1000 inl... | k-means++ algorithm and outliers | Yes, the outlier os more likely to be picked. But there are also many more inliers, the chance of choosing one of them is also substantial. Say you have one outlier that is 10x farther, then it is 100 | k-means++ algorithm and outliers
Yes, the outlier os more likely to be picked. But there are also many more inliers, the chance of choosing one of them is also substantial. Say you have one outlier that is 10x farther, then it is 100x more likely to be picket than one inlier. If you have 100 inliers, the chances are ab... | k-means++ algorithm and outliers
Yes, the outlier os more likely to be picked. But there are also many more inliers, the chance of choosing one of them is also substantial. Say you have one outlier that is 10x farther, then it is 100 |
36,594 | k-means++ algorithm and outliers | This seems to be explained on slide 27.
They propose choosing the first cluster centroid randomly, as per classic k-means. But the second is chosen differently. We look at each point x and assign it a weight equal to the distance between x and the first chosen centroid, raised to a power alpha. Alpha can take several i... | k-means++ algorithm and outliers | This seems to be explained on slide 27.
They propose choosing the first cluster centroid randomly, as per classic k-means. But the second is chosen differently. We look at each point x and assign it a | k-means++ algorithm and outliers
This seems to be explained on slide 27.
They propose choosing the first cluster centroid randomly, as per classic k-means. But the second is chosen differently. We look at each point x and assign it a weight equal to the distance between x and the first chosen centroid, raised to a powe... | k-means++ algorithm and outliers
This seems to be explained on slide 27.
They propose choosing the first cluster centroid randomly, as per classic k-means. But the second is chosen differently. We look at each point x and assign it a |
36,595 | Convergence of L-BFGS in non-convex settings | Yes, it is true that the L-BFGS-B algorithm will not converge in the true global minimum even if the learning rate is very small.
Using a Quasi-Newton method means you try to find the optimal $\theta$ using an iterative scheme similar to: $\theta_{k+1} = \theta_{k} - \alpha S_k g_k$ where $\theta$ are the parameters yo... | Convergence of L-BFGS in non-convex settings | Yes, it is true that the L-BFGS-B algorithm will not converge in the true global minimum even if the learning rate is very small.
Using a Quasi-Newton method means you try to find the optimal $\theta$ | Convergence of L-BFGS in non-convex settings
Yes, it is true that the L-BFGS-B algorithm will not converge in the true global minimum even if the learning rate is very small.
Using a Quasi-Newton method means you try to find the optimal $\theta$ using an iterative scheme similar to: $\theta_{k+1} = \theta_{k} - \alpha ... | Convergence of L-BFGS in non-convex settings
Yes, it is true that the L-BFGS-B algorithm will not converge in the true global minimum even if the learning rate is very small.
Using a Quasi-Newton method means you try to find the optimal $\theta$ |
36,596 | Should data be normalized before or after imputation of missing data? | In my opinion, since you are using kNN imputation, and kNN is based on distances you should normalize your data prior to imputation kNN. The problem is, the normalization will be affected by NA values which should be ignored.
For instance, take the e.coli, in which variables magnitude is quite homogeneous. Creating NA'... | Should data be normalized before or after imputation of missing data? | In my opinion, since you are using kNN imputation, and kNN is based on distances you should normalize your data prior to imputation kNN. The problem is, the normalization will be affected by NA values | Should data be normalized before or after imputation of missing data?
In my opinion, since you are using kNN imputation, and kNN is based on distances you should normalize your data prior to imputation kNN. The problem is, the normalization will be affected by NA values which should be ignored.
For instance, take the e... | Should data be normalized before or after imputation of missing data?
In my opinion, since you are using kNN imputation, and kNN is based on distances you should normalize your data prior to imputation kNN. The problem is, the normalization will be affected by NA values |
36,597 | Should data be normalized before or after imputation of missing data? | this paper suggests that normalization should be done before imputation
We suggest that normalization be done first followed by missing value imputations.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3489534/#:~:text=We%20suggest%20that%20normalization%20be,imputation%2C%20significance%20analysis%20and%20visualizatio... | Should data be normalized before or after imputation of missing data? | this paper suggests that normalization should be done before imputation
We suggest that normalization be done first followed by missing value imputations.
https://www.ncbi.nlm.nih.gov/pmc/articles/P | Should data be normalized before or after imputation of missing data?
this paper suggests that normalization should be done before imputation
We suggest that normalization be done first followed by missing value imputations.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3489534/#:~:text=We%20suggest%20that%20normaliza... | Should data be normalized before or after imputation of missing data?
this paper suggests that normalization should be done before imputation
We suggest that normalization be done first followed by missing value imputations.
https://www.ncbi.nlm.nih.gov/pmc/articles/P |
36,598 | Are parametric tests on rank transformed data equivalent to non-parametric test on raw data? | I think it's important to clearly distinguish between
a. using a parametric statistic on the ranks as the basis for a nonparametric test
b. using a parametric test as is, on the ranks
(we might also consider a third option -- like "b." but in some way scaling or adjusting the statistic to get a better approximation to ... | Are parametric tests on rank transformed data equivalent to non-parametric test on raw data? | I think it's important to clearly distinguish between
a. using a parametric statistic on the ranks as the basis for a nonparametric test
b. using a parametric test as is, on the ranks
(we might also c | Are parametric tests on rank transformed data equivalent to non-parametric test on raw data?
I think it's important to clearly distinguish between
a. using a parametric statistic on the ranks as the basis for a nonparametric test
b. using a parametric test as is, on the ranks
(we might also consider a third option -- l... | Are parametric tests on rank transformed data equivalent to non-parametric test on raw data?
I think it's important to clearly distinguish between
a. using a parametric statistic on the ranks as the basis for a nonparametric test
b. using a parametric test as is, on the ranks
(we might also c |
36,599 | How to measure probabilistic forecast accuracy? | Your comment sounds as if you are really looking for a density forecast rather than a point forecast, i.e., you want to forecast the full probability distribution of the future outcome(s). This is a very good idea. Density forecasting is common in financial or econometric forecasting, but unfortunately it is rarely tre... | How to measure probabilistic forecast accuracy? | Your comment sounds as if you are really looking for a density forecast rather than a point forecast, i.e., you want to forecast the full probability distribution of the future outcome(s). This is a v | How to measure probabilistic forecast accuracy?
Your comment sounds as if you are really looking for a density forecast rather than a point forecast, i.e., you want to forecast the full probability distribution of the future outcome(s). This is a very good idea. Density forecasting is common in financial or econometric... | How to measure probabilistic forecast accuracy?
Your comment sounds as if you are really looking for a density forecast rather than a point forecast, i.e., you want to forecast the full probability distribution of the future outcome(s). This is a v |
36,600 | What's the relationship between covariance, shared variance, and common variance? | Because the 1st link in your question sets against my comments in that other thread I'm just repeating here what some of the comments there said. I think that your question is chiefly about terminology, which might be used differently by different people.
I believe that "shared variance" between two variables could be ... | What's the relationship between covariance, shared variance, and common variance? | Because the 1st link in your question sets against my comments in that other thread I'm just repeating here what some of the comments there said. I think that your question is chiefly about terminolog | What's the relationship between covariance, shared variance, and common variance?
Because the 1st link in your question sets against my comments in that other thread I'm just repeating here what some of the comments there said. I think that your question is chiefly about terminology, which might be used differently by ... | What's the relationship between covariance, shared variance, and common variance?
Because the 1st link in your question sets against my comments in that other thread I'm just repeating here what some of the comments there said. I think that your question is chiefly about terminolog |
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