Questions tagged [wishart]

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

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1answer
2k views

Expected value of the log-determinant of a Wishart matrix

Let $\Lambda \sim \mathcal W_D(\nu, \Psi)$, i.e. distributed according to a $D \times D$ dimensional Wishart distribution with mean $\nu \Psi$ and degrees of freedom $\nu$. I would like an expression ...
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548 views

Distribution of inverse Wishart to a power?

In a related question, I had asked about the norm induced by an inverse Wishart matrix. I am interested in generalizing that result somewhat. Let $A\sim\mathcal{W}_p\left(I,n\right)$, a Wishart matrix ...
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2answers
1k views

What are the parameters of a Wishart-Wishart posterior?

When infering the precision matrix $\boldsymbol{\Lambda}$ of a normal distribution used to generate $N$ D-dimensional vectors $\mathbf{x_1},..,\mathbf{x_N}$ \begin{align} \mathbf{x_i} &\sim \...
11
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1answer
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Covariance matrix for Gaussian Process and Wishart distribution

I'm reading through this paper on Generalised Wishart Processes (GWP). The paper calculates the covariances between different random variables (following Gaussian Process) using squared exponential ...
8
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1answer
2k views

Hyperprior distributions for the parameters (scale matrix and degrees of freedom) of a wishart prior to an inverse covariance matrix

I'm estimating several inverse covariance matrices of a set of measurements across different subpopulations using an wishart prior in jags/rjags/R. Instead of specifying a scale matrix and degrees ...
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1answer
347 views

Generate covariance matrix with fixed values in certain cells

I want to be able to generate a covariance matrix of dimensions $D$ x $D$, such that certain specified cells of this matrix contain a fixed predetermined values (at least approximately). For e.g. For ...
6
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1answer
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Downsides of inverse Wishart prior in hierarchical models

I am working with a Bayesian hierarchical model that has a number of parameters for each experimental unit (6 parameters). I really do not know all that much about them a-priori, but it is quite ...
6
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1answer
4k views

How to specify the Wishart distribution scale matrix

I'm running the below Bayesian mixing model in R using the rjags package, but I am having difficultly in specifying the scale matrix for the Wishart distribution. Essentially, I want Sigma.inv to be a ...
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265 views

Entropy of Inverse-Wishart distribution

What is the entropy of the Inverse-Wishart distribution? I need just a reference, but derivation (e.g. using inverse property) would be interesting too.
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Sampling distribution of average of some covariance matrices

I have $K$ datasets, each with $N$ variables and $M$ samples (they are in fact EEG time series, but I discard time and treat them as $K$ iid multivariate samples) and assume they are coming from the ...
5
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1answer
596 views

Distribution of Trace of non-centered Wishart matrix

I am looking for the distribution of trace of the non-central Wishart matrix with different variations along different axes. Is there a general formula for such distribution? If not, is there a ...
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175 views

Wishart Conditional

I am looking for the conditional probability of a Wishart distribution, i.e., if I have a Wishart distributed variable $ S \sim W(\Sigma,n), $ where $$ S = \begin{bmatrix} S_1 \quad S_{12} \\ S_{21} ...
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256 views

Intuitive explanation for Marchenko-Pastur law

I am looking for an intuitive reasoning behind the Marchenko Pastur law, which is described as a law of large numbers analog for random matrices. I know the law gives the probability density function ...
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626 views

Sum of independent Wishart with same degrees of freedom but different scale matrices

Is there any result showing that a sum of independent Wishart with same degrees of freedom but different scale matrices is a Wishart? For example, if I have two random variables: $$ Y \sim W_p(n,\...
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184 views

Joint distribution of two distances

Suppose there are three points in 3D space, each with coordinates $A_i=(X_i,Y_i,Z_i)\leadsto \mathcal{N}(\mu_i,\tau^2\mathbb{I}_3)$. We compute the distance between the three points, e.g. $D_{ij} = \|...
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1answer
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How do I use Stan to fit a covariance matrix? [closed]

I'm new to Stan (and bayesian methods in general), so this is likely very simple. I'm trying to model some multivariate normal data. All I want to know is the covariance matrix generating the data, ...
4
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1answer
181 views

Distribution of the product of a Wishart matrix

If $\mathbf{M} \sim W_2(\Sigma, 3)$ is a Wishart matrix and $\Sigma =\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$ then what is the distribution of $(3, 1) \mathbf{M}^{-1}(3,1)^T$ ? Thank ...
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2answers
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Posterior covariance of Normal-Inverse-Wishart not converging properly

I am trying to implement a simple normal-inverse-Wishart conjugate prior distribution for a multivariate normal with unknown mean and covariance in numpy/scipy such that it can take a data vector and ...
4
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1answer
704 views

Marginal distributions of off-diagonal terms in a Wishart-distributed random variable

I am interested in finding expressions for the marginal distributions of the off-diagonal terms in a Wishart-distributed random variable. More specifically, suppose $X$ is an $n \times p$ matrix, ...
4
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1answer
121 views

How do you translate a density from Cholesky factor to density of the matrix?

Suppose $L$ is a random $p\times p$ lower triangular matrix, with known density, $f(L)$. To compute the density of $C=L L^{\top}$, one needs to use the change of density formula. This is a little bit ...
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385 views

Sampling from Wishart distributions with scalar degrees of freedom $(\upsilon>p-1)$

Let $\upsilon$ be the degrees of freedom of a Wishart distribution and $p$ the dimensions of its scale matrix. If the degrees of freedom is a scalar, then its range is: $$ \upsilon > p−1 $$ and ...
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1answer
188 views

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 ...
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1answer
47 views

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 ...
3
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1answer
93 views

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}; \...
3
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1answer
162 views

What is the Fisher's information matrix for the Wishart distribution?

I have been struggling computing the Fisher's information of the Wishart distribution. I'll write what I have gone through. Let's $\Omega$ denote a $p\times p$ Wishart random variate denoted by $\...
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1answer
166 views

Can Wishart be close to Normal distribution?

I'm trying to figure out if Wishart can be close to Normal for a number of degrees of freedom enough large. About chi-squared distribution, Wikipedia states: ...
3
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1answer
86 views

Simulation under Wishart-like constraint in $\mathbb{R}^{k\times p}$

Given a $(p,p)$ symmetric positive semi-definite matrix $\mathbf{H}$ of rank $k\le p$, I am looking for a (possibly efficient) way of generating a set of $k$ vectors $\alpha_i\in\mathbb{R}^p$ ...
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72 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, ...
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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 ...
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141 views

Distribution for sum of Wishart Matrices with scale matrices proportional to identity

Consider the independent $n\times p$ dimensional matrices $\mathbf{X}_i, i=1,...,G$, from the matrix variate normal distribution, $N_{n,p}\left(0,\mathbf{I}_n, \mathbf{\Sigma}_i\right)$: $$P_i\left(\...
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Intuition behind random matrices [closed]

I am looking for an intuition for random matrices. Say, Gaussian or Binary squares matrices to begin with. I am considering three possible viewpoints: As a $n$ random points in $\mathbb{R}^n$. As a ...
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374 views

Distribution of a normalized inverse Wishart times Gaussian

Suppose $z\sim\mathcal{N}\left(\lambda^2 e_1,I_n\right)$ where $e_1$ is the first column of the $n$-dimensional identity matrix, denoted here as $I_n$. Suppose $S\sim\mathcal{W}\left(m,I_n\right)$ is ...
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3answers
822 views

Prior distribution importance in Bayesian inference

I am performing a Bayesian multivariate regression, and therefore I have to construct the prior and the subsequent posterior. But the paper that I am using as a reference, uses a "Uninformative prior"...
2
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1answer
3k views

How to sample from a Wishart distribution?

From Wikipedia, we know that $n$, the degrees of freedom, should be larger than $p-1$ where $p$ is the dimension of the scale matrix. Also, from the bottom part of the same article, we see "Bartlett ...
2
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2answers
207 views

Understanding Wishart Definition

I'm trying to understand the definition of the wishart distribution. In wiki, $X_{(i)}{=}(x_i^1,\dots,x_i^p)\sim N_p(0,V).$ What do they mean by this? Each component is drawn from a univariate $N(0,...
2
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1answer
910 views

Normal-inverse-Wishart distribution

The Normal-inverse-Wishart distribution is a conjugate prior for the multivariate normal distribution when the mean and covariance are unknown. I understand that conjugate priors are mathematically ...
2
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2answers
550 views

What is the distribution of norm induced by an inverse Wishart?

Suppose $S$ is distributed as a Wishart matrix with $n$ degrees of freedom and scale matrix $\Sigma$, and let $\vec{a}$ be a fixed vector. It is well known that $\vec{a}^{\top}S\vec{a}$ is equal to $\...
2
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1answer
63 views

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\...
2
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1answer
186 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: ...
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35 views

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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186 views

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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88 views

expectation associated with Wishart distribution

Suppose that $W$ follows the Wishart distribution with parameter $\Sigma$ and d.f. $n$. Then, I would like to know the result of the following expectations. $E[tr(WAWB)tr(WC)]$ $E[tr(WAWB)tr(WCWD)]$...
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329 views

Informative prior for Normal-Inverse Wishart distribution

In Bayesian analysis we use the Normal-Inverse Wishart distribution for the parameters of multivariate models these prior distributions have some hyperparameters. So how do we find the values of ...
2
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1answer
365 views

Covariance Matrix Eigenvalue Distribution Relation to Size [closed]

I'm trying to run PCA on sample covariance matrices of various sizes (ranging between 20 x 20 to 4000 x 4000). Assume the data follows a joint multivariate normal distribution. While derivations are ...
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59 views

Show that distribution of $\small(n-1)\overline{X}'(S^{-1}-\frac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0})\overline{X}$ is $\small T^2(p-1,n-1)$

Show that the distribution of $(n-1)\overline{X}'\left(S^{-1}-\dfrac{S^{-1}\mu_0\mu_0'S^{-1}}{\mu_0'S^{-1}\mu_0}\right)\overline{X}$ is $T^2(p-1,n-1)$ where $X_i\sim N_p(\mu,\Sigma)$, where $\Sigma$ ...
2
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0answers
560 views

Proving a property of the Wishart distribution

I would like to prove that, if $X \sim W(V,n)$, then $CXC^T \sim W(CVC^T,n)$ where $W$ is a Wishart distribution. A point in the right direction would be welcome - happy to admit that I may have ...
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0answers
58 views

Derive the distribution of the fraction $\frac{[M^+ (M^+)^h]_{i}}{ || M^+||^2}$

Let $M^+$ represents the pseudo-inverse of matrix $M$; $M^+=(M^hM)^{-1}M^h$, where $h$ denotes the conjugate transpose. We assume that the elements of $M$ are complex Gaussian with zero mean and unit ...
2
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1answer
257 views

Mode of inverse-Wishart distribution (sampled vs. calculated)

I recently investigated the sampling behavior of covariance matrices through simulation. I noticed that the mode of simulated inverse-Wishart distributed matrices somehow differs from the "theoretical"...
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714 views

Eigenvectors of a Wishart matrix

I have been trying to find a good source (or clarifications) to help me understand this point. I am very new to random matrix theory so any pointers will be appreciated. Here is what I think I have ...
2
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187 views

rWishart: should be $dof>p-1$ or $dof \ge p$?

The degrees of freedom $n$ of a Wishart distribution parametrized like in wikipedia (and like most people do) are restricted to: $$ n>p-1 $$ where $p$ are the dimensions of the data, to ensure ...