Questions tagged [multivariate-distribution]

Probability distribution over vectors (as opposed to univariate distributions that are over numbers).

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Generate a multivariate normal vector

I'm working in R and I was wondering, let's say I want to generate a random vector $X \in \mathbb{R^p}$, with $X \sim N(0,I)$ where $I$ is the identity matriz in $\mathbb{R}^{p \times p}$. The ...
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17 views

Combining marginal bivariate probability distributions into a single multivariate one

Consider a set of 3 correlated random variables $X$, $Y$ and $Z$. I have calculated bivariate marginal distributions over any pairs of these random variables. That is, I know probability density ...
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10 views

How to determine if a multi-dimensional vector comes from a multi-variate distribution or not?

I have a complex multivariate distribution given by a multivariate random number generator (black box). In other words, whenever I call the "generator" I get a random vector containing ...
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27 views

Simplifying assumption for regular vine distributions

I am trying to understand vine copulas and specifically, conditional bivariate copulas. Let $c_{X_1,X_3|X_2}(\cdot,\cdot | \cdot)=:c$ be the probability density function of a bivariate conditional ...
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2answers
74 views

Can a random variable be uncorrelated with its product with a correlated random variable?

I have a random variable $X.$ I want to find a random variable $Y$ such that $Y$ is correlated with $X,$ but $Y$ is not correlated with the product of $X$ and $Y.$ Is it always possible?
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25 views

Conditional Expectation of a normal distribution [duplicate]

say we have a multivariate normal distribution with ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$ The conditional expection is $\overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\...
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21 views

Test goodness of fit of a multivariate distribution via testing GOF of linear combinations of components

I am modeling a continuous bivariate distribution of a random vector $(X_1,X_2)$ using a copula. I would like to assess how well I am doing. Given a data sample, I could probably do a bivariate ...
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1answer
168 views

KL divergence for joint probability distributions?

I have a pair of joint probability distributions. I want to measure their similarity/dissimilarity. If they were single-dimensional probability distributions, then I could measure the Kullback–Leibler ...
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1answer
73 views

What is a generalisation of t-test to the case of multivariate distribution?

When we have a sample of numbers, one of the most basic tests is the t-test, in which we check the null hypothesis that the population mean is equal to zero. I am interested in a generalisation of ...
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23 views

Mutual information relationship to copula entropy is borked?

I have posted a related Question based on a paper, Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-54. In the paper, they ...
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38 views

Copula entropy: calculation is borked?

I came across a pretty cool paper whose idea makes a lot of sense to me. Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
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1answer
153 views

What is the best way to sample points from an arbitrary 2D distribution?

My problem is the following: I want to randomly sample points $(x,y)$ according to the Himmelblau function $$f(x,y) = (x^2 + y - 11)^2 + (x + y^2 - 7)^2$$ which I treat as a kind of multivariate ...
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14 views

Covariance matrix in multivariate is Wishart random variable

I'm working on sample variance distribution, on page 111 of Methods of Multivariate Analysis, it says "The joint distribution of these $p(p + 1)/2$ distinct variables in $W =(n−1)S = \sum_i(y_i − ...
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33 views

Is it reasonable to look at the output of simulating from a multivariate distribution as univariate distribution? If yes, what is this called?

Suppose I have $X_{n} \sim MVN(\underline{\mu},\Sigma)$ where $n$ is large (several thousands). However, the $\mu_i's$ and the elements of $\Sigma$ are such that almost every simulation from $X_n$ ...
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28 views

Measuring entropy/concentration in multivariate datasets

I am interested in measuring how concentrated or widely dispersed a certain binary property is among a population, which is defined by a number of categorical variables. For instance, let's say I have ...
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23 views

Multivariate Bernoulli Logistic Regression on OpenBUGS

I am new to WinBUGS/OpenBUGS, and am trying to solve this Multivariate Bernoulli Logistic Regression $$P(θ1,θ2|X,Z) ∝ P(X|θ1) P(θ1) P(Z|X,θ2) P(θ2)$$ Is this valid on OpenBUGS? This is my OpenBUGS ...
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Does the t-copula capture serial nonlinear dependence?

Say $\mathbf{Y}=[y_1,\cdots,y_n]'$ with elements that are uncorrelated, yet serially dependent. As a result, in the literature, the means of allowing for nonlinear serial dependence for processes ...
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25 views

PDF for Bayesian Decision

I have the discrete-time sequence $X(n) = 1$ with probability 3/4 and $X(n) = 0$ with probability 1/4 for $n = 1,...,10$. The observed signal is $Y(n) = X(n) + N(n)$ where $N(n)$ is an AWGN with ...
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126 views

Interpretation of multivariate conditional gaussian function form?

I've been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate ...
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29 views

Kth Order statistic of a multivariate distribution

The Kth order statistic for a univariate is equal to its kth-smallest value. For instance, given $\{6,9,3,8\}$, the 2nd-smallest value would be the 2nd order statistic. How does this concept ...
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36 views

Multivariate Gamma parameter estimation

Consider $X$ a d-dimensional random variable with positive values, mean $\mu\in\mathbb{R}_+^d$, and covariance matrix $\Sigma\in\mathbb{R}^{d\times d}$. If I have $n$ samples $\{ x_1, ..., x_n \}$ ...
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Derive expectation of the log determinant of a precision matrix from a Wishart distribution

I'm reading through section 21.6 of Murphy's Machine Learning: A probabilistic perspective where they derive the variational bayes algorithm for fitting a mixture of gaussians. One of the steps ...
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15 views

plotting bivariate normal distribution samples

I have generated two samples $\underline{X_i}$ and $\underline{Z_i}$ $\underline{X_1}$ $\underline{X_2}\dots \underline{X_{5000}}$ , while $\underline{X_i} \sim N_2[(1,2)^T,\begin{pmatrix}2&1.5\\ ...
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36 views

Why is there covariance instead of simply variance in multidimensional normal distributions?

Maybe it is just because I'm only experimenting with 2 dimensional normal distributions, but multi-dimensional normal distributions for me seem like just multiple one dimensional normal distributions. ...
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48 views

Probability distribution associated with nuclear norm?

The $\ell_1$ norm of model parameters is often added to loss functions because it induces sparsity in the solution of the overall cost function: $$ c(\theta) = \log L(x|\theta) + \lambda ||\theta||_1$$...
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Help in calculating diagonal covariance matrix for generative model for binary classification

I am given this data. I want to fit a generative model $\cal{N}(\mu_0, \sigma_0^2 I_2)$, $\cal{N}(\mu_1, \sigma_1^2 I_2)$ for the $0$ and $1$ classes respectively using $\textbf{MLE}$ and plot ...
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Condition Random Variable on Range of Another Random Variable [duplicate]

Assume that $v_s \sim N(\mu_s,\sigma_s^2)$ and $v_b \sim N(\mu_b,\sigma_b^2)$, denote their correlation by $\rho$, and assume they are jointly normally distributed. How would I assess $E[v_b|v_s\leq c]...
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113 views

Multivariate Normal distribution Cramer-Rao bound

How to compute the Cramer-Rao lower bound for $\mu$ of a multivariate normal distribution $\mathcal{N}_{p}(2\mu,\Sigma_0)$. Also I would like to know the minimum variance unbiased estimators for $\mu$....
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Bivariate random effect problems in selection models (Mixture Cure model)

I am currently working on a mixed effects selection model. The selection model is a logistic model with a Gaussian random effect. The principal model is a survival model with a Gaussian random effect (...
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1answer
122 views

Find expectation of conditional normal distribution

I am struggling with some finding expectation value question . the question is to find $E[Y|X]$ from the result $P(Y|X)$ with given mean and covariance $$\mu=[\mu_x, \mu_y]^T$$ $$\Sigma=\begin{bmatrix}...
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28 views

What is the entropy of multivariate data multiplied by a vector?

It is a general rule that for multivariate data $\boldsymbol{X}$ and a matrix $\boldsymbol{A}$, their entropy is $$h(\boldsymbol{A} \boldsymbol{X}) = h(\boldsymbol{X}) + \ln |\det \boldsymbol{A}|$$ (...
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113 views

What is the distribution of the CDF of a sample drawn from a multivariate normal?

Introduction: Lets say we have a random variable $X$ that follows a normal distribution, $X \sim N(\mu, \sigma^2)$ , with a CDF function $F_X(x) = P(X \leq x)$. Then we draw some random samples $S_1$, ...
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1answer
96 views

What is the fisher information matrix of the multivariate t distribution?

$\newcommand{\bx}{\mathbf{x}}$ $\newcommand{\bSigma}{\boldsymbol{\Sigma}}$ $\newcommand{\bE}{\mathbf{E}}$ $\newcommand{\bD}{\mathbf{D}}$ Consider the multivariate central t distribution with p.d.f. \...
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Is this causation?

Consider the following joint distribution for the random variables $A$ and $B$: $$ \begin{array} {|r|r|}\hline & B=1 & B=2 \\ \hline A=1 & 49\% & 1\% \\ \hline A=2 & 49\% & 1\...
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Is the mvt (multivariate t) adjustment for multiplicity appropriate for repeated measures and longitudinal trials?

As in the title. If I have a longitudinal experiment with t0...tn timepoints, and want to verify the Dunnett contrast (all versus baseline), can I use the mvt adjustment for multiplicity? I think it ...
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1answer
139 views

Correlation Coefficient for Hypergeometric-type Distribution

Problem Statement: A box contains $N_1$ white balls, $N_2$ black balls, and $N_3$ red balls $(N_1+N_2+N_3=N).$ A random sample of $n$ balls is selected from the box (without replacement). Let $Y_1,Y_2,...
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1answer
455 views

Probability distribution of the product of two dependent random variables

It is well known that being $X$ and $Y$ two independent random variables with distributions $f_X(x)$ and $f_Y(y)$, respectively, then the probability distribution of the multiplicative function $z = ...
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2answers
182 views

What does i.i.d. mean for multivariate case?

When we say a random variable is i.i.d., it's often used to describe the dependency between the observations of that random variable, which I call the row dimension, indexed by time if it's a time ...
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27 views

How does the maximum entropy principle affect joint entropy, mutual information, and other info measures?

The maximum entropy principle says that we should use the probability distribution of a univariate dataset that has the highest level of entropy because it offers the lowest information. How does the ...
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81 views

Rescaling multivariable normal pdf and normalizing constant

I am trying to understand change of variables for a random variable and how it changes the pdf and the normalizing constant. Let $\mathbf{x}$ be $N$-dimensional normal variable and let $\mathbf{\...
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27 views

What are the best known techniques to verify that a GAN samples correctly from a given distribution?

I would like to know what are the best known techniques to check that a generative adversarial network (GAN) samples from the correct distribution. Naively, I would say it all boils down to a ...
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0answers
18 views

Multivariate rare events

My data is something like this: I have U urns, and I have taken a bag of $n$ objects from each urn. Each urn has $N$ objects, and I have sampled $n$ with replacement. $n$ is comprised of coloured ...
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45 views

Algorithm to generate non normal (discrete and continuous) correlated variables using a correlation matrix

What I am trying to do here is, given a dataset (let's say n observations of N variables), and thus a correlation matrix M that result from said dataset, I would like to write a Python algorithm that ...
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41 views

Empirical conditional density of continuous variables

I have a dataframe, with data of several continuous variables. The variables are not independent. My goal is to sample from the distribution that generated this data. What's a relatively easy and ...
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1answer
57 views

Multivariate distributions

Let $x$ and $y$ be random variables with the following joint density function: $f(x,y) = e^{-x}$ for $0<x< \infty$, and $0<y<1$ If $z= x+2y$, what is the joint density function of $x$ and ...
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1answer
16 views

How to choose what feature vector to plot in multivariate regression analysis?

I'm new to the field of machine learning and I have been having this doubt for a long time now. If we want to plot a scatter plot, we plot it as, x as a function of y where x is a 1-D array. But in ...
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1answer
1k views

Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?

If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$. But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
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0answers
46 views

How to draw huge high-dimensional normally distributed vectors in a memory-efficient way

I would like to simulate a few hundred multivariate normally distributed vectors $H$ of dimension $10^6$ with covariance matrix $\Sigma$, preferrably in R. The individual entries of $\Sigma$ are ...
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1answer
282 views

What's the best function to test multivariate normality when sample size more than 3000?

Before MANOVA, I need to test multivariate normality. Then I tried MVN::mvn() function in R, output as below: ...
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14 views

bivariate conditional joint sensor model

I am struggling to find $P\left( V_t | z \right)$ from $P\left( V_t | z , V_p \right)$. Here $z$ and $V_p$ are independent variables. ...