Questions tagged [multivariate-distribution]

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

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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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6 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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7 views

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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107 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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21 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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15 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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13 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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25 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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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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10 views

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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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
79 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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24 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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48 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
51 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
79 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
192 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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104 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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20 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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54 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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26 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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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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36 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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32 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
53 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
12 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
413 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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43 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
120 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. ...
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34 views

Calculate fifth and sixth polynomials for Headrick (2002) method for non-normal multivariate distribution

I am trying to perform a 3-variable correlated multivariate Monte Carlo simulation. As the asset class returns are non-normal, I found the following function ...
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22 views

Likelihood of three dimensional data

I'm having a lot of trouble finding the likelihood and log-likelihood of $\tau$ that corresponds to the following equation: $a_i = y(t_i, \tau) + \epsilon_i$ where $\epsilon_i \sim \mathcal{N}(0, \...
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1answer
25 views

Normal random matrix as linear transformation of standard normal random matrix?

Wikipedia has the following definition of a normal random vector: A real random vector $\mathbf{X} = (X_1,\ldots,X_k)^{\mathrm T}$ is called a normal random vector if there exists a random $\ell$-...
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31 views

How to learn dependency of variables from data?

I have a data set $X$ that consist of $m$ vectors $\vec{x}$ of $n$ real valued components. Each vector component lies within a corresponding predefined interval of valid values, which is the same for ...
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93 views

Probability density of conditional multivariate distribution [duplicate]

We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning ${\boldsymbol Y}$ into $${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\ {...
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2answers
45 views

How to prove a multivariate r.v. does not follow the nonparanormal distribution?

Background You may find the definition of the non-paranormal distribution at the 2nd paragraph in p.2296 of this paper. In short, $(X_1, \ldots, X_p)$ is non-paranormal if there exists a set of ...
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11 views

Modelling probability distribution of multivariate dataset

my background is not in stats so apologies if this is an obvious question. I have a dataset with 60,000 observations, and 128 features. Each feature follows a hurdle distribution with either a gamma, ...
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41 views

Distribution of quadratic forms in mixed model

I have a question related to the distribution or asymptotic distribution of quadratic forms that arise in the linear mixed model. Suppose, $$Y=X\beta + H\delta + \epsilon$$ where $\epsilon \sim N_n(0,...
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1answer
48 views

Conditional Distribution Multivariate Normal Distribution [closed]

Let $X_1, X_2, X_3$, be jointly distributed according to a multivariate normal distribution. $[X_1, X_2, X_3]^T\sim N(\mu=[0,0,0]^T , \Sigma = [[5,0,0],[0,2,1], 0,1,3]])$ $U = X_1 + 2X_2$ and $V = ...
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12 views

Calculating the credible interval of each variable in a multivariate distribution

I have a k-dimensional Dirichlet posterior with parameters $\alpha_1,...,\alpha_k$ and variables $\theta_1,...,\theta_k$. This posterior comes from a Dirichlet prior and a multinomial likelihood. ...
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1answer
265 views

Multivariate normal distribution transformation

Suppose that $X $ has a multivariate normal distribution $X\sim MVN (\mu, \Sigma) $, How can I transform $X$ into $Z$ so that $Z\sim MVN(\mu, I) $ where $I$ is the identity matrix? For instance, ...
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1answer
23 views

DeGroot P.155 integration problem for multivariate distributions

I am stuck with the integral for equation 3.7.4 and do not see how it was done. Could someone provide me with some hints or resources to read around?
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1answer
102 views

Confusion over multinomial and multivariate- hypergeometric distributions

You, your parents, your sister, go to visit grandma for her birthday. Grandma made a cake for the party. If she puts $20$ raisins in the cake at random in the cake, and she divides the cake into $5$ ...
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45 views

Simulations using correlated random numbers from Multivariate Normal and fat-tailed distributions

This question makes use of the LaplacesDemon package in R, but it is not a coding question, so I believe this is the most appropriate forum. First, the unsurprising results. I generate k correlated ...