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Questions tagged [wishart-distribution]

The Wishart distribution is a common matrix distribution on square symmetric semi-definite matrices.

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Distribution of the sample covariance of a multivariate exponential family

I am wondering if there is a known form for the distribution of the sample covariance matrix of a random variable that follows a multivariate exponential family distribution. I guess it would be a ...
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Bounds of integration for the Wishart density

I once took a course that included zillions of exercises concerning the Wishart distribution, but as far as I recall, never mentioned the Wishart density. I asked something about that in this question,...
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Limiting distribution of the Wishart process

Consider the Wishart process: $$ dS_t = \sqrt{S_t} \, dB_t Q + Q^\top \, dB^\top_t \sqrt{S_t} + (S_t K + K^\top S_t + \Omega \Omega^\top) \, dt $$ Or the restricted version where $\Omega = \sqrt{\...
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Posterior of Inverse Wishart distribution with a subset of data observed

Suppose: \begin{equation} x_1\in \mathbb{R}^{p_1}\\ x_2\in \mathbb{R}^{p_2} \end{equation} such that \begin{equation} x \sim \mathcal{N}( \begin{bmatrix} x_1\\ x_2 \end{bmatrix}; \begin{bmatrix} \...
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$E[W\otimes W]$ for Wishart R.V. $W$

What is the value of $E[W\otimes W]$ for Wishart R.V. $W$? $\otimes$ refers to Kronecker product I found related formula for $E[WAW]$ on page 467 of Seber's Matrix handbook, wondering if $E[W\otimes W]...
Yaroslav Bulatov's user avatar
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Distribution of the sum of Wishart distributed random matrices

Suppose \begin{equation} A_1 \sim \mathcal{W}(A_1; \Psi_1, v_1)\\ A_2 \sim \mathcal{W}(A_2; \Psi_2, v_2), \end{equation} where $\mathcal{W}$ is the Wishart distribution and $A_i$, and $\Psi_i$ are PSD ...
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Expectation of constant matrix sandwiched between two random matrices

I have a $N\times P$ random matrix $X$ with i.i.d. coefficients from a standard normalized Gaussian $\mathcal{N}(0, 1)$. The corresponding Wishart matrix is $$W = \frac{1}{P}X X^{\top}$$ Calculating ...
SphericalApproximator's user avatar
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Is Inverse-Wishart a conjugate prior for Wishart likelihood?

Suppose I have a noisy observation $Z$ of a covariance matrix $F$, given a prior on $F: p(F)$, I would like to find the posterior of $p(F|Z)$, does the following specification forms conjugacy?: $$ F \...
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Wishart conditionned by the determinant

I am interested in the density of the Wishart distribution under the constraint that the determinant of the outcome is 1. It suffice do divide the density of the Wishart by the marginal density of the ...
Chevallier's user avatar
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Questions on the Wishart distribution

If $X$ is an $n\times p$ matrix where each row is iid multivariate normal, then $X^TX$ has a Wishart distribution. What is known about the limiting distribution of $X^TX$ for large $n$ when the rows ...
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What is the distribution of the Frobenius distance between two covariance matrices?

I am computing the Frobenius norm of the difference between two covariance matrices, \begin{align} |\mathbf{C}-\mathbf{C}'|_F=\sqrt{\sum_{i,j}\left(c_{ij}-c'_{ij}\right)^2}. \end{align} Each of these ...
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Random variate of a singular Wishart distribution with non-integral degrees of freedom

Let's say that I have a random variable $X$ that follows a singular Wishart distribution with $\nu$ degrees of freedom and a shape matrix $\Sigma$ that is $p\times p$. We can write "symbolically&...
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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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What is the expectation of the Cholesky factor of a Wishart distributed random matrix?

Let a $d-\text{dimensional}$ Wishart random variable with $\nu$ degrees of freedom $\Sigma$ be distributed according to $\mathcal{W}(\Sigma|\Sigma_0, \nu) \propto |\Sigma|^\frac{\nu-d-1}2\exp{(-\...
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On the Wishart distribution

Definitions Consider the following bivariate model \begin{equation*}y\triangleq S(p)h+v\end{equation*} where $p\triangleq \begin{bmatrix}\theta & \ell_1 & \ell_2\end{bmatrix}'$ and $\theta \...
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Clustering with gaussian mixtures: choice of hyperparameters

Question: I am interested in general in understanding how to choose the hyperparameters if we are interested in clustering bivariate vectors assuming a mixture of Gaussian mixture with conjugate ...
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Is there a closed form solution for the Hellinger Distance between two Wishart distributions?

Given two Wishart distributions $X_0 \sim W_{p_0}(V_0,n_0)$, $X_1 \sim W_{p_1}(V_1,n_1)$, what is the Hellinger Distance between them? Can it be obtained in closed form assuming $p_0 = p_1$? For ...
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Marginal likelihood of bivariate Gaussian model

I assume the following model for a sample $y_1 \in \mathbb{R}^2$ of size $1$ with bivariate Gaussian likelihood and independent bivariate Gaussian and inverse-Wishart prior for the mean and variance ...
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Variance of $\operatorname{tr}(W^2)$ with $W \sim \text{Wishart}(n, \Sigma)$

Suppose $W \sim \text{Wishart}(n, \Sigma)$, where $\Sigma \in \mathbb R^{p\times p}$, the expectation of $\operatorname{tr}(W^2)$ is $$E[\operatorname{tr}(W^2)] =n(n+1)\operatorname{tr}(\Sigma^2) + n\...
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Finite integral vs Finite probability Density

I was reading the Bayesian Data Analysis book from Gelman et al. I was going through Appendix A which describes probability distributions, and realize the book talks about something I don't really ...
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How to impose restrictions on a random matrix via its prior distribution?

I am reading the paper Factor analysis and outliers: A Bayesian approach. The author starts with a factor analysis model given by $${\bf y}_i = {\bf \Lambda} {\bf z}_i + {\bf e}_i, \quad i = 1, \ldots,...
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Kullback–Leibler divergence between two inverse Wishart distributions

Is there a closed form formula for the KL divergence between two inverse Wishart distributions?
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Marginal distribution for the off-diagonal elements of a Matrix following an Inverse-Wishart

I've found on CV the marginal distribution for the main-diagonal elements of a matrix which follows an Inverse-Wishart distribution. However, what will happen to the off-diagonal elements? what ...
An old man in the sea.'s user avatar
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How to Estimate Covariance Matrix using Wishart Distribution?

I learned that Wishart distribution can help estimate covariance matrix of MVN without sampling directly from MVN. But there are some points that are still unclear for me: Why we bother to estimate ...
user273192's user avatar
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How to calculate the Jacobian of the transformation ( for covariance matrix)

I'm reading this Paper about a separation strategy for modeling covariance matrices with focus on Bayesian analysis. Direct decomposition of covariance matrix is as follows: $\Sigma = \text{diag}(S)\,...
Dedula33's user avatar
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Kullback Leibler Divergence between two Normal Whishart Distributions

I'm having trouble to compute the KL Divergence between two normal-Wishart distributions. KL divergence from $Q$ to $P$ is defined as: $$D_{\mathrm{KL}}(P \Vert Q) = \int p(x) \log \frac{p(x)}{q(x)}...
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Sum of Log Chi-Squared Asymptotic Distribution

I'd like to find the asymptotic distribution of $$\sqrt{n}\left(\log|\mathbf{S}| - \log|\boldsymbol{\Sigma}|\right), ~~~~~n \rightarrow \infty$$ where $\mathbf{S} \sim W_j\left(n, \frac{\boldsymbol{\...
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Sampling prior covariance matrices - nested sampling

I am trying to fit a multivariate Gaussian with a non-diagonal covariance matrix $\Sigma$ using nested sampling. Usually, in other Bayesian analyses, we would use a Inverse Wishart or LKJ prior on ...
Lucidnonsense's user avatar
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Semi-conjugate inverse Wishart posterior, can we obtain the marginal?

In Hoff's text (A First Course in Bayesian Statistical Methods), he uses a semi-conjugate inverse-Wishart prior for the covariance matrix of a multivariate normal process. In equation 7.9, he has the ...
bayes's user avatar
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Showing a useful result for Wisharts and Multivariate Beta random matrices

Let $\mathbf{A} \sim \text{Wishart}_m\left(k_a,\mathbf{V} \right)$ and $\mathbf{B} \sim \text{Wishart}_m\left(k_b,\mathbf{V} \right)$ be two full rank Wishart random matrices. Define $$ \mathbf{S} = \...
Taylor's user avatar
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Inverse Wishart Prior for linear model

I know some bayesian methods employ an inverse wishart distribution for the prior distribution of the covariance matrix in a linear regression. I.e. for the model: $$Y=X\beta+\epsilon$$ Where $\...
JDoe2's user avatar
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Estimate parameters of inv-Wishart distribution using Bayesian

$\Sigma \sim Inv-Wishart_{v}(\Lambda^{-1})$ Suppose that I have a set of observations of $\Sigma$s, I wonder if there is a conjugate way to estimate the value of $v$ and $\Lambda$ (especially $\...
Ding Li's user avatar
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Integrate out (covariance) matrix in Normal-Wishart distribution

In Gelman's Bayesian Data Analysis Chapter 3.6, he introduces the multivariate normal with unknown mean and variance, with the priors $\Sigma\sim \text{Inv-Wishart}_{\nu_0}(\Lambda_0^{-1})$ $\mu\...
bayes's user avatar
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Is assigning an inverse-Wishart distribution to a diagonal matrix problematic?

I'm reading the paper Bayesian Vector Autoregressions by Thomas Wozniak. He considers the model $$y_t = \mu + A_1 y_{t-1} + \cdots A_k y_{t-k} + u_t$$ where each $y_i$ is a $N$-vector, each $A_j$ is a ...
user avatar
7 votes
1 answer
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Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
Michael Hardy's user avatar
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What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions?

What's the role of the scale matrix for the Inverse-Wishart and Wishart distributions? The purpose of finding this information is to enlighten me on how should one decide on a prior for a positive-...
An old man in the sea.'s user avatar
3 votes
2 answers
753 views

Deriving the sampling distribution of MLE for Normal distribution

Let $X_1,\ldots,X_n$ be an observed random sample from $N_p(\mu, \Sigma)$. I know that the MLE of $\Sigma$ is $\frac{1}{n} \sum_i^n(X_i -\bar X)(X_i -\bar X)^T$, which is biased. We define $S = \...
Jason's user avatar
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Marginal Distribution of Matrix Normal with Two Inverse Wisharts

Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions $$X \sim MN_{p\times n}(0, \Sigma, \Omega)$$ $$ \Sigma \sim IW(a, A) $$ $$ \Omega \sim IW(b, B)$$ where $a$ and $b$ are ...
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Appropriate Distribution for Diagonal Covariance Matrices

Let's say I have a model like: \begin{align} X\mid\mu,\Sigma_X &\sim \mathcal{N}(\mu,\Sigma_X)\\ \mu\mid m, \Sigma_\mu &\sim \mathcal{N}(m,\Sigma_\mu) \\ \Sigma_X\mid \Psi, c &\sim \...
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Intuitive explanation of Inverse Wishart prior for covariance estimation

I am trying to understand what is going on in the use of an Inverse Wishart prior for (Gaussian) covariance, and what is the motivation for it. I am seeing this posed as a solution for when the ...
adamconkey's user avatar
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How do I find the "elliptical confidence region" from columns of a matrix that follows the Wishart distribution?

The subject is about the sample mean and the sample covariance estimators and their respective confidence regions for the estimated parameters. Suppose that $n$ samples are taken from a $p$-variate ...
Marcos Vinicius's user avatar
5 votes
1 answer
592 views

Transformation of Inverse Wishart

Let $\Sigma$ be an $p\times p$ dimensional covariance matrix that is distributed Inverse Wishart with degrees of freedom $\nu$ and Prior scale matrix $\Psi$ such that we write $\Sigma \sim W^{-1}(\nu, ...
jds's user avatar
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Expected eigenvalues of a Wishart Matrix

I consider a $n\times n$ Wishart Matrix with expected value $p \cdot I_n$, i.e. a matrix of the form $$W = XX'$$ with $X$ a $n\times p$ matrix with independent standard normal entries. It is easy ...
Elvis's user avatar
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What is the relationship between the normalization constants of the normal distribution and the (Inverse-)Wishart distribution?

I was looking at the probability density function of the multivariate normal distribution and that of the inverse-Wishart distribution: $$ \begin{array}{rccll} p_{\mathcal{N}}\left(\mathbf{x}; \...
Xiubo Zhang's user avatar
5 votes
1 answer
701 views

Scaling for Wishart distribution

If $X$ has a Wishart distribution $W_p(n,\Sigma)$ , what's the distribution for $cX$ where $c>0$ ? I know that for a $\chi^2_n (x)$ distribution with $n$ degrees of freedom, $c\chi^2 $ follows $\...
omega's user avatar
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Expectation of the inverse of $\textbf{z} \textbf{z}^{H}$ where $\textbf{z}$ is a complex Gaussian vector

Considering the vector $\textbf{z} \sim \mathcal{CN}(\textbf{0}_{M},\Theta_{M \times M})$, what would be the expectation of $\frac{1}{\textbf{z} \textbf{z}^{H}}$, i.e., $\mathbb{E} \left\lbrace \frac{...
Felipe Augusto de Figueiredo's user avatar
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Property of Wishart distribution

I am solving the next exercise about a property of a Wishart Distribution: $$M_1\sim W_p(\Sigma,n_1)$$ $$M_2\sim W_p(\Sigma,n_2)$$ are independent, then $M_1+M_2\sim W_p(\Sigma,n_1+n_2)$ I have ...
Boris's user avatar
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5 votes
1 answer
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Result for covariance between elements of the sample covariance matrix

Given data matrix $X_{n,p}$ where $n$ is the sample size, $p$ is the dimension, the sample covariance is $$\hat{\Sigma}=\frac{1}{n-1}X^\top X=\{\hat{\sigma}_{ij}\}_{1\le i,j\le p}$$ Is there any ...
breezeintopl's user avatar
4 votes
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1k views

Maximum Likelihood of Wishart parameters

I'm having some difficulty in deriving the ML estimation of the parameters of a Wishart distribution. Given a set of matrices $\{W_1, W_2,\dots,W_N\} \in \mathbb{C}^{k\times k}$ for which $W_i \sim ...
aepound's user avatar
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3 votes
1 answer
519 views

Is Wishart distance always positive?

From what is told on page 268 of the Polarimetric Radar Imaging: From Basics To Applications: I have written the following code for wishart distance calculation in Matlab: ...
Sepideh Abadpour's user avatar