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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Conditional expectation in the multivariate normal distribution

Suppose $(X_1, X_2, X_3)^T$ is multivariate normal. What is the conditional expectation $E(X_1 \mid X_2, I(X_3 > 0))$? Here, $I(X_3>0)$ is the random variable that takes the value one when $...
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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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Can we apply a constraint on the distribution of the layer output?

As far I understood, the hidden layer outputs can be anything based on the learning algorithm or optimization rules. I was wondering if it possible to some constraints on the layer output. For ...
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Faster computation of high-dimensional multivariate normal probabilities

My goal is to find a faster way to calculate something like mvtnorm::pmvnorm(upper = rep(1,100)) that is, the tail probability of multivariate normal distribution ...
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Covariance as a function of explanatory variable

Suppose we have a collection of bivariate random variables $X_{1i}$ and $X_{2i}$ indexed by a continuous variable $t$ such that, for the vector ${\bf{X}} = (X_{1i} \ X_{2i})^T$ we can assume \begin{...
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Bayesian Multivariate Normal-Normal Model when covariance depends on the mean

Let $y$ denote a D-dimensional multivariate normal random variable with density $$y \sim N(\mu, \Sigma(\mu)) $$ such that the covariance $\Sigma$ is a deterministic non-linear function of $\mu$. ...
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Resulting univariate marginal distributions are not $t$ distributed - why? (S Wood Core Statistics)

In the Core Statistics by Simon Wood it says: "If we replace the random variables $Z_i\sim_\text{i.i.d.} N(0,1)$ with random variables $T_i \sim_\text{i.i.d.} t_k$ in the definition of a ...
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What is the expected value of $X_i/\|X\|^2$ when $X \sim \mathcal{N}(\mu, \sigma^2I)$

Let $X$ be an N-dimensional normal random vector with non-zero mean $\mu$ and diagonal covariance matrix $\sigma^2I$. I would like to understand if it is possible to derive the expected value of the ...
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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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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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How to build a probability distribution from locations with accuracies?

I have a set of $n$ GPS locations $l_i$ with latitude, longitude in degrees and accuracy in meters, corresponding to $3 σ$, i.e. the probability ≈ 0.997) $(lat_i, lng_i, acc_i)$ or $(lat_i, lng_i, σ_i ...
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Mean and covariance estimators for IDD normal

Suppose $X_1,\cdots, X_n$ are sampled iid from a multivariate Gaussian $\mathcal N(\mu, \Sigma)$. We denote the sample mean and covariance as follows \begin{align} \bar X &= \frac{1}{n} \sum_{i=1}^...
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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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Swiss Cheese Distributions

I am curious about a normal distribution with no probability mass in certain regions, sort of like the complement of the truncated normal. In particular, it will have zero mass in a circular region. ...
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Prove that the joint density of independent multivariate normal variables is a matrix-normal

Let $X_1,...,X_n \sim N_p(\mu_i,\Sigma_i)$ be Multivariate Normal a.v. independent. Show that $W = (X_1,...,X_n) \sim MN(M,\mathbb{I},\Sigma)$ where $M = [\mu_1 \mu_2...\mu_n]$ and $\mathbb{I}$ ...
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Select ellipsoid data cloud "equivalent" to a spherical data cloud

What will be your considerations when you've got to realize the intuitively plain, but meditatively manifold idea to create a data cloud which "the same as" that data cloud, only that it is not ...
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How does the multivariate normal distribution work with multiple conditions?

The following multivariate distribution is given: $\left(\begin{array}{c}X_1 \\ X_2 \\ X_3\end{array}\right) \sim N_3\left(\left(\begin{array}{c}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{ccc}1 ...
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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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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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Inequality of Probability

This problem is from Matrix Analysis for Statistics by James R. Schott. Problem. Show that if $x$ and $y$ are two independently distributed random vectors with $x\sim{\rm Normal}(0,\Omega_1)$ and $y\...
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Efficient computation of marginalized multivariate normal posterior distribution

In general,if we know that the marginal Gaussian distribution for some variable $\textbf{x}$ and a conditional Gaussian distribution for some $\textbf{y}|\textbf{x}$ of the forms: $$p(\textbf{x}) = \...
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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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Covariance matrix of integral of multivariate normal distribution

If $t = [t_0, t_1, \dots, t_{N-1}] \in \mathbb{R}^N$ with $t_i \sim N(\mu_i, \sigma_i^2)$ and its covariance matrix $C \in \mathbb{R}^{N \times N}$ where $C_{ij} = Cov(t_i, t_j)$ is given If I define ...
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Joint distribution of log hazard ratio estimates for two outcomes

I have a study where individuals are randomized to a treatment or control group, and there are two time-to-event outcomes $S_i$ and $T_i$ measured for each individual $i$. The two outcomes have some ...
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Problem with two correlated random normals

Imagine you have a two-dimensional multivariate normal random variable with $\mu = [0, 0]$ and $\Sigma\ = \begin{bmatrix}1 & r\\r & 1\end{bmatrix}$. (Conceptually, you have two random normal ...
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How to characterize the effect of $(\textrm{Diag}(\Sigma^{-1}))^{-1}$ badly approximating $\textrm{Diag}(\Sigma)$

I have an almost singular covariance matrix $\Sigma\in\mathbb{R}^{n\times n}$ that has a few large eigenvalues, followed by many many comparatively very small ev's. If I were to try to approximate ...
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Equality of two multivariate normal CDF's

Let $\pmb{X} \sim N_d(\pmb{\mu}, \pmb{\Sigma})$ and $\pmb{Y} \sim N_d(\pmb{\nu}, \pmb{\Omega})$; $\pmb{\mu} \neq \pmb{\nu}, \pmb{\mu} \neq \pmb{0}, \pmb{\nu} \neq \pmb{0}$, and $\pmb{\Sigma}\neq\pmb{\...
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Distribution of quadratic form of multivariate normal with linear term

Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of $$X^tAX+b^tX \quad ?$$ The distribution without ...
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Truncated mulitvariate normal: first two moments

Let $X\in \mathbb{R}$ be a univariate random varible for which it holds that $$ X \sim N(\mu,\sigma^2).$$ where $\mu\in \mathbb{R}$ gives the expected value and $\sigma^2>0$ is the variance. If ...
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Argmax order statistics for multivariate normal

The Problem Given a gaussian random variable $\mathbf{x} = (x_1, ..., x_N)^T \sim \mathcal{N}(0, \Sigma)$ what is the probability that $i = \underset{i\in\{1, \dots, N\}}{argmax}\{|x_1|, \dots, |x_N|\...
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transformation with mvtnorm

I'm struggling to understand why the following R outputs don't yield the same value. ...
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Creating multi-variate normal with constraint:

Is it possible to create multivariate normal from the same normal variable? That is i have a normal variable $x_1 \sim N(0,1)$, and $p$ normal variable $y_1$ which their marginal distribution is also ...
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Marginalizing multivariate-normal distribution canonical form

Regarding the problem of margenalization of canonical forms of multivariate gaussian distribution it was mentioned in probabilistic graphical models text book that $$\int{C(X,Y;k,h,g)}dY$$ is ...
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Does a "pruned" i.i.d. multivariate sample behave as the i.i.d. sample?

Let $z_1,\cdots,z_n$ be $n$ points drawn i.i.d. from $\mathbb{C}N_p(0,\Sigma_n)$. The distribution of the covariance $S_n=\frac1n\sum_{i=1}^n z_i z_i^*$ is well known in the limit as $n,p(n)\to\infty$ ...
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A question concerning distribution of $\mathbf{Y}/\|\mathbf{Y}\|_2$ where $\mathbf{Y}\sim \mathcal{N}(\boldsymbol{\mu},\mathbf{I})$

I know that when $\mathbf{Y}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, $\mathbf{Y}/\|\mathbf{Y}\|_2$ is distributed uniformly on the unit sphere. But to my surprise, I failed to find a simple closed ...
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Multivariate distribution for products of random variables

Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would ...
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Sampling distribution of variances of multivariate normal RVs

Is there an analytical expression for the distribution of variances of MVN RVs? I mean if $X=[x_1, \dots, x_D]\sim \mathcal{N}(0, \Sigma)$ where $\Sigma$ is a $D$-dimensional covariance matrix, is ...
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Composition of Multivariate Gaussians

This question is teasing my intuition for a moment : $X \sim N(0, S_1)$ $Y|X \sim N(X, S_2)$ Does $Y \sim N(0, S_3)$ with some $S_3 = f(S_1, S_2)$ like (for instance) $S_1 + S_2$ ? What I found is ...
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Clarification on LDA and the multivariate Gaussian

From my understanding, to calculate the posterior probability of a sample $x$ belonging to a class $k$ using Linear Discriminant Analysis you would first calculate the eigenvector matrix $W$ required ...
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Notation for distributions on subspaces

Question What notation should I use for a normal distribution on a subspace $U \subset \mathbb{R}^n$? Motivation I have a geometric intuition for Bessel's correction which goes something like this: A ...
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Probability of N-dim Gaussian marginalized over a plane, and a likelihood of a point on a plane

Context: I'd like to fit a series of coordinates, each with their uncertainties, to a (hyper)plane. I am trying to derive the likelihood for a point on a plane observed at a point. I need a ...
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Relationship between univariate and multivariate normal distribution

I have a portfolio consisting of N assets with known average historical returns ($r_1$, $r_2$, ...$r_N$) and a known set of weights ($a_1$, $a_2$, ...$a_N$) subject to $\sum_{i=1}^N a_i$ = 1 I am ...
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Conjugate Prior for Multivariate Normal Variances and Correlations

Is there a way to separately specify conjugate priors for the variance and correlations of a multivariate normal? The inverse Wishart is conjugate if you want to specify the covariance, but covariance ...
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linear transform of a random variable follows multivariate normal, what is the distribution before the transform?

$x$ is a $n\times 1$ random vector,$A$ is a $m\times n$ matrix. Given that $x$'s linear transform $z = Ax$ follows a multivariate normal distribution: $$ z = Ax \sim N(\mu_z,\Sigma_z) $$ The question ...
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Joint distribution of sample correlations of variables taken from a multivariate normal distribution

Let us assume multivariate normal vector $(X_1, \cdots, X_n)$ with mean vector $\mu$ and variance-covariance matrix $\Sigma$. A sample correlation will not exactly equal its population parameter, but ...
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Henze-Zirkler Multivariate Normality on R and Python

I am trying to run some ANCOVA analysis but beforehand, I would like to show that data follows multivariate normal distribution, which is one of the assumptions. On R, there is ...
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Using multivariate normal likelihood when determinant of covariance matrix is zero

Estimating a multivariate normal (MVN) model requires minimising the negative log-likelihood of MVN (constant term dropped): $$ \ell(X|\mu,\Sigma)=\frac{n}{2}\log|\Sigma|+\frac{1}{2}(X-\mu)^T\Sigma^{-...
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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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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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