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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Sampling distribution parameters of Normal Inverse-Wishart marginal wrt x

Assume $\mu$ is a known vector and $\lambda$ is a known scalar, then for $$ x|\Sigma \sim MVN_n \left(\mu,\frac{1}{\lambda}\Sigma\right) $$ $$ \Sigma \sim W^{-1}\left(\Psi, \nu\right) $$ we know that $...
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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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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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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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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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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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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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30 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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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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25 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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1answer
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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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1answer
30 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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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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1answer
30 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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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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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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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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1answer
208 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
45 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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1answer
64 views

Distribution of $\mathbf{v}^{\top} \Sigma^{- 1} \mathbf{v}$, when $\mathbf{v}$ is a multivariate normal with covariance $\Sigma$? [duplicate]

What is the distribution of the quadratic form $\mathbf{v}^{\top} \Sigma^{-1} \mathbf{v}$, when $\mathbf{v}$ is a multivariate normal with covariance $\Sigma$ and zero means? I suspect this is related ...
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On the distribution of sample intraclass correlation for normal data

Consider i.i.d random vectors $\boldsymbol X_1,\boldsymbol X_2,\ldots,\boldsymbol X_n$ having a $p$-variate normal distribution $N_p(\boldsymbol \mu,\Sigma)$ where $\Sigma$ has compound symmetry of ...
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Computationally tractable (pseudo-)likelihood methods for Gaussian data with missing values (MCAR)

I want to calculate a Gaussian (pseudo-)likelihood for some data where each data point has random dimensions missing. Calculating the marginal log-likelihood is computationally intractable because for ...
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how can I generate GEE or paired data with multivarite normal distrubition?

I want to generate paired data and all variables must have multivaritare normal distrubition. help me please , I have not find any codes for paired data from multivaritare normal distrubition . I ...
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Intergrating product of multivariate normal and univariate normal to find marginal density [duplicate]

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $U\sim MN(0,I_p)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and ...
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Integrating product of multivariate normal and univariate normal to find marginal

Suppose, $y_i|u_i\sim MN(X_i(t_i), \sigma_e^2I_{m_i})$ and $u_i\sim N(0,1)$. Now how to find the marginal distribution of $f(y_i)=\int f(y_i|u_i)f(u_i)du_i$? Since one is multivariate normal and other ...
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1answer
186 views

Understanding the pdf of a truncated normal distribution

Suppose $\boldsymbol{x} = (x_1, \ldots, x_m)^T$ follows a multivariate normal distribution with 2-sided truncation $a_i \leq x_i \leq b_i$. This is a truncated multivariate normal defined by $TN(\mu, \...
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Integrating Monte Carlo Error Out of Likelihood Function

I am calculating the likelihood for a multivariate normal process in which the conditional mean in computed with Monte Carlo integration. I'm trying to account for the Monte Carlo error within the ...
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How to generate a multivariate normal distribution by conditioning on the normally distributed mean of each sample?

I would like to generate a joint distribution of random variables $X$ and $Y$ from a multivariate normal distribution with mean zero and symmetric covariance matrix. However, each generated $x_i$ and $...
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1answer
109 views

Self-normalized weighted sum of random variables

Given $n$ i.i.d. random variables $X_1, \dots, X_n$ with $X_i \sim \mathcal{N}(0,1)$ and weights $a_1, \dots, a_n \in [-1, 1]$ such that $$ Y = \frac{P}{Q} = \frac{\sum_{i = 1}^{n} a_i X_i }{\sum_{i =...
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Posterior conditionals for bivariate normal

I'm new to bayesian statistics and I'm studying Bayesian models. I'm having trouble writing the Gibbs sampler for a particular case of bivariate normals. Assume now that $d_i = (x_i, y_i)$ for $i=1,2\...
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1answer
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How to scale (in the range 0 to 1) and mathematically explain two mutually exclusive probabilities of a data point belonging to a normal distribution?

I have two set of n-dimensional multivariate data, with the assumption that both set of data is normally distributed. When I get a new data point, my goal is to classify it into one of the two sets. ...
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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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Conditional distributions of correlated normal random variables

Suppose that $X$ and $Y$ are normally distributed with mean zero and nonzero covariance. I want to know the distributions of $X | X - Y > c$ and $Y | X - Y > c$, which I believe should be ...
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Self Study: Trivariate Normal Expectation with Inequality Condition

I'm reading a paper and found an interesting expectation. I know the result the author found but I can't figure out the intermediary steps because the author provided none. My attempt is getting ...
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28 views

Conditional Normal Distribution. Intuition based on projection?

It is known that, when $(X,Y)^T\sim\mathcal{N}_{p+q}(\mu,\Sigma)$, $$X\mid Y\sim\mathcal{N}_p(\mu_{X\mid Y}, \Sigma_{X\mid Y}),$$ where $\mu_{X\mid Y}=\mu_X+\Sigma_{XY}\Sigma_{YY}^{-1}(Y-\mu_Y)$ and $\...
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428 views

How to determine if a sample is within a standard deviation of a multivariate normal distribution

In the case of a sample from a one dimensional normal distribution $x \sim \mathcal{N}(\mu, \sigma)$, I can calculate whether or not a sample is within some multiple $\eta$ of $\sigma$ by measuring if ...

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