Questions tagged [multivariate-normal-distribution]

The multivariate normal distribution, is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. (Also called, multivariate Gaussian)

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derive multivariate pdf from matrix variate pdf

I am working on the proof part in Definition section of https://en.wikipedia.org/wiki/Matrix_normal_distribution. I can understand what s going on, except for the last part how inv(V(kron)U) becomes ...
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Jointly complete and sufficient statistics for multivariate normal distribution [duplicate]

Consider the random sample X from the multivariate normal distribution where xi are i.i.d as N(µ,Σ). *Show that the sample mean x̄ and Sample covariance matrix S are jointly complete and sufficient ...
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Partition of a Multivariate Normal Distribution with Inverse-Wishart Covariance Prior

I'm trying to partition a multivariate normal with the following structure: $$\bf{X} = \left. \begin{bmatrix} \bf{X}_1 \\ \bf{X}_2 \end{bmatrix}\right|\Sigma \sim N_p \left(\begin{bmatrix} \mu_1 \\ \...
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How is this equation deduced?

These two equations are from the book Gaussian Process for Machine Learning. First we already have equation (2.8). $p(\mathbf{w}|X, \mathbf{y}) ∼ N (\frac1{\sigma_n^2}A^{−1}X\mathbf{y}, A^{−1})$ (2.8) ...
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Intuition for correlation of N≥3 dimensional Normal distribution

What is an intuitive way to think about the covariance matrix in an N≥3 dimensional Normal distribution? In two dimensions the covariance matrix can be visualized by plotting a region of constant ...
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How to simulate samples using MH from multi-variate Gaussian proposal distribution?

I have the task of obtaining samples {$\theta ,. . .,\theta_N$} using the Metropolis-Hastings algorithm, where the proposal distribution $q(\theta)$ is a multi-variate distribution The proposal ...
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Conjugate prior bayesian inference on multivariate GMM

I am trying to understand how the posterior looks like when running Bayesian inference on a multivariate Gaussian-mixture model. $p(\mathbf{x}) \propto \sum_{i=1}^M w_iN(\mathbf{x}|\mu_i,\Sigma_i)$. ...
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Sampling from multivariate normal conditional on a negative minimum

Let $X\sim \mathcal{N}(\mu,\Sigma)$, where $\mu\in\mathbb{R}^n$ and $\Sigma\in\mathbb{R}^{n\times n}$. How can I efficiently sample from $X | {\min{X}\le 0}$? (I.e. from the distribution of $X$ ...
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Power Against Specific Alternative when testing the mean of a MVN distribution

Consider testing a null hypothesis about the mean of a p-dimensional MVN distribution (with known covariance matrix $\Sigma$) where the form of the null is that the mean is equal to a certain vector, ...
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60 views

Normality of sum of normal random variables

If $(X,Y)$ and $(X+Y,Z)$ both follow nondegenerate bivariate Gaussian distributions, is it possible that $(X,Y,Z)$ follow a nondegenerate trivariate distribution that is not Gaussian? I want to make a ...
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What is the definition and upper bound on the variable "m" in the definition of the multivariate normal Fisher Information?

Multivariate normal distribution [edit] The FIM for a $N$-variate multivariate normal distribution, $X \sim N(\mu(\theta), \Sigma(\theta))$ has a special form. Let the $K$-dimensional vector of ...
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Proving Equivalence between Multivariate distributions and Gaussian Bayesian Networks

I am studying Probabilistic Graphical Models by Daphne Koller. In Chap 7, the authors say the following. I can't convince myself of the highlighted part. Induction typically has a statement for n, ...
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293 views

Multivariate Gaussian fitting

When trying to approximate a distribution of random vectors $D$ by using multivariate Gaussian, what properties must we ensure that $D$ has? I.e., what distributions can be estimated by multivariate ...
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How can I construct a desired variance-covariance matrix for simulating multivariate Gaussian distribution samples using MATLAB?

I want to simulate multivariate normal distribution samples to help understand PCA, biplot, etc. For example, I want to see how the correlation structure affects the appearance of 2-D biplot. Two ...
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Correlation matrix as maximum likelihood estimator under constraint

The Problem Although it seems to be straight forward I am struggling to prove the following statement. Assume, we have $p$-variate Gaussian observations $\left\{x_1, \ldots, x_N \right\} \subset \...
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Mean and covariance estimators for indirect (transformed) observations of Gaussian

Suppose we have a multivariate Gaussian random variable $\mathbf{x}\sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, and $n$ measurements $\mathbf{y}_1 = H_1 \mathbf{x}_1, \mathbf{y}_2 = H_2 \mathbf{x}...
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What is the physical significance of the height of the the bell shaped curve in normal distribution?

Hello enthusiasts and explorers!! Going through the concepts of normal distribution, one can understand that the height and the width of the bell shaped curve in a normal distribution graph tells ...
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Multivariate normal, conditioned on inequalities

$X_1,\ldots,X_n$ are jointly normal random variables with some mean vector $\mu$ and covariance matrix $S$. How do I evaluate $P \left (X_1 \le x \mid \sum_{i=1}^nX_i \ge y\right )$? Is there a quick ...
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If normally distributed random vector $X$ has a PD covariance matrix, then any conditional distribution induced by $X$ has a PD covariance matrix?

Suppose I have a random vector $X$ whose distribution is joint normal. I know that the covariance matrix of $X$ is positive definite. I wonder if I partition $X$ in any way: e.g., $X=(X_1, X_2)$, then ...
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Density of a degenerate complex normal distribution

I'm looking for the properties of a degenerate complex normal distribution. Wikipedia has a section on degeneracy in the real multivariate normal case here, stating that the density (in a subspace of $...
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Multivariate Normal Underflow

Good day everyone At the moment I am attempting to write code in R to calculate the following. $$ \tau_{k j}^{(m)}=\frac{\pi_{k}^{(m)} f_{k}\left(x_{j} ; \theta_{k}^{(m)}\right)}{f\left(x_{j} ; \Theta^...
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678 views

Generating samples from high-dimensional multivariate Gaussian with few training samples

Say I have a $n\times d$ dataset $D$ where $n\ll d$ ($n$ number of observations, $d$ number of dimensions). Currently, if I want $m$ samples from $D$ assuming it is multivariate Gaussian, I can do ...
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Distribution of residuals of LME models

I'm asking if you can help me please. I have a random slope model with longitudinal data, where I consider two times (time 1 and time 2). As a result I have residuals for each subject at both time 1 ...
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Multinormal assessment of residuals (Multilevel models)

I need some help. my_model<-lme(OUTCOME ~ VISIT + TREATMENT+ VISIT*TREATMENT, random= ~ 1+VISIT | ID, my_data) I built this multilevel random slope model (above), where we have two visits for each ...
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High dimensional multivariate normal calculations in R

Good day everyone At the moment I am attempting to write code in R to calculate the following. $$ \tau_{k j}^{(m)}=\frac{\pi_{k}^{(m)} f_{k}\left(x_{j} ; \theta_{k}^{(m)}\right)}{f\left(x_{j} ; \Theta^...
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Maximum likelihood estimate for multivariate sum of normal distributions

For each $j = 1,\dots,N$, let $\mu_j \in \mathbb{R}^N$ denote a known column vector, $\Sigma_j \in \mathbb{R}^{N\times N}$ a known covariance matrix, and $\theta_j \in \mathbb{R}$ an unknown parameter,...
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How to find mean of multivariate normal distribution when holding a variable constant? [duplicate]

I wanted to know if there is a way to calculate the mean of a multivariate normal distribution when a certain variable is held constant. For example, if I had a continuous bivariate normal ...
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32 views

estimate multivariate normal parameters

Suppose that I have "a realization" of random vector $x=(x_1,\cdots,x_N)$ where $N$ is sufficiently large $N>100$. I know that random vector is joint normally distributed $$x \sim N(\mu,\...
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335 views

Numerical computation of the means and covariance in a truncated bivariate normal distribution

How can I compute the means and covariance of a truncated bivariate normal distribution? I am particularly worried about the case when the truncation occurs very far from the mean. Is there a robust ...
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DFT of Multivariate Normal Random Vector

I have a real zero-mean multivariate rv $X \sim \mathcal{N}(0, \Sigma)$, with $N^d$ entries. $X \in \mathbb{R}^{N^d}$ is the flattened representation of a $d$-dimensional signal, of "side length&...
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Assumptions of Path Analysis - Multivariate Normality inspite Univariate Non Normality

I am currently checking if my data meets the assumptions for path analysis: mainly multivariate normality of the three endogenous variables $m_1,m_2,y$ (As recommended by e.g. Streiner, 2005). I ...
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1answer
28 views

Conditional distribution of multivariate cauchy distribution

In the example of multivariate normal distribution, $$ \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \end{bmatrix} \sim \mathcal{N}\left(\begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}, \begin{...
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Sum of two different freedom degree multivariate t dsitributions

there are two multivariate t distributions whose freedom degree $\nu$ are different to each other. $$ \mathbf{x} \sim \mathcal{T}(\nu_x, \mathbf{0}, \Sigma_x) $$ $$ \mathbf{y} \sim \mathcal{T}(\nu_y, \...
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How to calculate univariate conditional distribution of a trivariate gaussian [duplicate]

I am trying to find the conditional distribution of a trivariate gaussian. So here is a hypothetical trivariate gaussian: $$\mathcal{N}(\mu_{ABC},\Sigma_{ABC}),\;\mu_{ABC}=\begin{bmatrix}\mu_A \\ \...
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EM algorithm for multivariate gaussian with diagonal covariance matrix

Ok so quick question. Say I need to use the EM-algorithm to estimate the parameters of a multivariate gaussian $$ f_{k}\left(x ; \theta_{k}\right)=\frac{1}{(2 \pi)^{P / 2}|V|} \exp \left(-\frac{1}{2}\...
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Standard Gaussianity test for high dimensional data

I'm using a Gaussianity assumption over 500-dimensional data in my work and I wanted to check the validity of my assumption. I developed a transformation that relies on this assumption and I have good ...
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1answer
32 views

Derivative of a Bivariate normal CDF with respect to its variables

Following up on the question (and answers) here, I'm trying to derive $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\theta}})}{\partial x_1}$ and $\frac{\partial \Phi(x_1, x_2|\mathbf{\underline{\...
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984 views

Implication of relationship between multivariate normal distribution and chi-square distribution

I am wondering what is the implication of the above relation/theorem. I know how to prove this using "sphering $Y$" but I am failing to get intuitive understanding of the theorem. What does it mean ...
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Find the variance-covariance matrix for a linear combination of multiple bivariate normal distribution?

I have an arbitrary number of independnet bivariate normal distributions with $\mu_i = [x_i,z_i]$ & $ \Sigma_i= \left(\begin{array}{cc} \sigma^2_{x_i} & \sigma^2_{x_i,z_i}\\\ \sigma^2_{x_i, ...
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1answer
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How to implement a mixture model for Dirac Delta and Normal distributions?

How could I fit data with observations from one Dirac delta component and $n$ normal distributed components? Where $n$ usually is between 1 and 5. My prior knowledge is that one component really is a ...
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32 views

Expectation of a constant matrix multiplied with a random vector and its transpose?

It is given that a random vector $\mathbf{y} \sim N(\mathbf{0}, \boldsymbol{\Theta})$, and $\mathbf{A}$ is a constant matrix. Assuming that the dimensions are compatible, what would $$E(\mathbf{A} \...
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What is the covariance matrix of a fixed matrix multiplied by a vector of random variables? [duplicate]

Suppose we have a random variable $\epsilon\sim N(0,\sigma^2)$, then we know that if a constant, say $a$, is multiplied by $\epsilon$, then \begin{equation} a\epsilon\sim N\left(0,a^2\sigma^2\right) \...
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Alternative way of solving Mahalanobis distance

Mahalanobis distance is given by: $(x-\mu)^T \Sigma^{-1}(x-\mu)$ Apparently the solution to the above formula is equivalent to: $Trace(\Sigma^{-1} (x-\mu)(x-\mu)^T)$ How can you prove that the two are ...
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Normalizing a custom Weight Shifted or Spiked Gaussian distribution

I have a custom weight shifted bivariate gaussian distribution that I wish to normalize. W is the weighted symmetric matrix that shifts the entire distribution and the λ below is the diagonal matrix ...
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66 views

Prior for covariance matrix?

Given a set of data $\{(x_i\pm e_{x,i},\,y_i\pm e_{y,i})\}_i$ (with uncorrelated uncertainties), I want to model it as a multivariate Gaussian function with an unknown mean $\boldsymbol{\mu} $ and a ...
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Distribution of the dot product of a multivariate Laplace random variable and a fixed vector

This question is basically a follow-up: Distribution of the dot product of a multivariate gaussian random variable and a fixed vector But instead of a multivariate Gaussian random variable, what about ...
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Best measure of similarity between multivariate Gaussian distributions?

I am working to describe differences in covariances between a "baseline" multivariate Gaussian random process $\mathbf{x}_0$ and other processes $\mathbf{x}_i$. All are discrete and ...
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101 views

truncation of bivariate normal under quadratic condition

Consider a complex normal variable $Z \sim \mathcal{CN}(\mu,2\sigma^2)$ with real component $X \sim \mathcal{N}(\mu,\sigma^2)$ and imaginary component $Y \sim \mathcal{N}(0,\sigma^2)$. We can write ...
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285 views

Inverse of cumulative density function for Multivariate Normal Distribution

How do I calculate the inverse of the cumulative distribution function (CDF) of a multivariate normal distribution? Does it even exist for the multivariate case? I know this is possible for a ...
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1answer
47 views

Calculate mean of $X$ when $X$ and $Y$ are jointly normal and $Y$ is truncated above [duplicate]

Suppose I have two random variable $X$ and $Y$ and they are distributed joint normally and $Y$ is truncated above by constant $c$ $$\begin{pmatrix} X \\ Y \end{pmatrix} = TN\left(\underbrace{\begin{...

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