Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [wishart]

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

0
votes
0answers
36 views

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 ...
1
vote
0answers
24 views

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-...
1
vote
1answer
104 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 = \...
1
vote
0answers
29 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 ...
2
votes
0answers
173 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 \...
1
vote
1answer
390 views

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 ...
1
vote
0answers
120 views

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 ...
3
votes
0answers
59 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, ...
2
votes
0answers
87 views

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 ...
3
votes
1answer
92 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}; \...
0
votes
1answer
71 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 $\...
0
votes
0answers
183 views

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{...
0
votes
0answers
125 views

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 ...
1
vote
0answers
37 views

How to generate an inverse Wishart matrice when $\Sigma$ is not definitive positive? [closed]

In MATLAB, how to generate an inverse Wishart matrice when $\Sigma$ is not definitive positive? Any ideas?
3
votes
1answer
157 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 ...
0
votes
0answers
333 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 ...
2
votes
1answer
170 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: ...
2
votes
0answers
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. $E[tr(WAWB)tr(WC)]$ $E[tr(WAWB)tr(WCWD)]$...
1
vote
3answers
305 views

Unsolvable Integral?

Is the following integral solvable? $$P(X) = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} P(X|\mu,K)P(\mu|K)P(K) d\mu dK$$ with $$P(K) = \frac{|K| ^{(v-d-1)/2}}{2^{vd/2}|V|^{v/2}\Gamma_d|\frac{...
0
votes
1answer
153 views

use inverse Wishart for variance in MCMC

When you have a posterior that looks like as one step in Gibber Sampler $P(\xi | \Sigma_\xi, \theta) ∝ exp\{-1/2 \xi\Sigma_\xi^{-1}\xi\}P(data | \xi, \theta)$ Do you always assume inverse Wishart ...
3
votes
0answers
137 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(\...
1
vote
1answer
282 views

What is the correct form of Metropolis Hasting step in scaled Inverse Wishart prior for covariance matrix?

I was going through the paper of O'Malley and Zaslavsky (2008) for the scaled inverse Wishart priors for a covariance matrix, in order to write an R-code for hierarchical Bayesian estimation of mixed ...
0
votes
0answers
23 views

When are $X^tC_1 X$ and $X^tC_2X$ independent?

Let $X$ be a $n\times p$ data matrix from $N_p(0,\Sigma)$. Let $C_1$ and $C_2$ be two symmmetric idempotent matrices. When are $X^tC_1 X$ and $X^tC_2X$ independent?
1
vote
1answer
199 views

Reconciling Wishart mean and Inverse-Wishart mean

The Inverse-Wishart wikipedia article states that $ X \sim \operatorname{Wish}^{-1}(\Psi,\nu)$ if $X^{-1} \sim \operatorname{Wish}(\Psi^{-1},\nu),$ where $\Psi$ and $X$ are $p\times p$. The mean ...
5
votes
0answers
167 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} ...
1
vote
1answer
153 views

The multivariate normal distribution has the same relationship with the Wishart distribution as the multivariate t-distribution with the …?

Is there a name for the distribution resulting from the sum of outer products of t-distributed random vectors? Alternatively, is there a matrix-valued distribution with the support of positive ...
2
votes
0answers
319 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
votes
3answers
781 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"...
3
votes
1answer
161 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
votes
1answer
151 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 $\...
1
vote
0answers
35 views

Eigenvalue order of magnitude for Wishart random matrix

If we have a $P\times N$ matrix $\mathbf{A}$ whose elements $A_i$ are samples (in this case, P samples) from a multivariate gaussian distribution in an $N$ dimensional space, we can define the Wishart ...
1
vote
0answers
189 views

Simulating from an Inverse Wishart with constraints

Let $\Sigma$ be a $p\times p$ positive definite matrix and $\nu>p-1$. I would like to simulate a realization of the Inverse-Wishart distribution $X\sim\mathcal W^{-1}(\Sigma, \nu)$ with the ...
0
votes
0answers
34 views

How to find the distribution of $B*=BSB^T$, where $S$ is the sample covariance from normal observations?

Let $X_1, X_2, \dots, X_{30}$ be a random sample of size $n=30$ from a $N_5(μ, Σ)$ population. If B is a $2\times 5$ matrix $$ B = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & ...
1
vote
0answers
189 views

Log Expectation of Inflated Determinant of Wishart Distribution

Let $\Lambda \sim \mathcal W(\nu, \Psi)$, i.e., following a $n \times n$ dimensional Wishart distribution with mean $\nu \Psi$ and degrees of freedom $\nu$. The expectation of the log determinant of $\...
3
votes
1answer
84 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 $\alpha_i\in\mathbb{R}^p$ ...
2
votes
1answer
361 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 ...
5
votes
0answers
215 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 ...
1
vote
0answers
121 views

Posterior pointwise uncertainty of multivariate normal-Wishart (variational GMM)

Given a variational mixture of Gaussians (as per, e.g., Chapter 10 of Bishop, 2006), we can compute the posterior predictive pdf: $$ \left\langle p(x|\alpha,\beta,\nu,\mu,V) \right\rangle $$ where $\...
4
votes
1answer
173 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 ...
1
vote
2answers
197 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
votes
0answers
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 $\Sigma$ ...
5
votes
1answer
2k views

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 ...
1
vote
0answers
235 views

Expected value of $E[S_{ki}S_{kj}]$ in wishart dist

Some details: We know that the Wishart distribution with $\nu$ degrees of freedom and positive definite $p \times p$ scale matrix $V$, $\mathcal{W}_p(\nu,V)$, has the pdf $p(S|V,\nu) = \frac{|S|^{(\...
1
vote
0answers
210 views

Infinite Mixture of Infinite Gaussian Mixture Model

I'm struggling to implement the following model in R: When I generate H, I get a matrix. For the DP, I am using the Chinese Restaurant Process (CRP). However, I'm not sure how to incorporate a matrix,...
2
votes
0answers
552 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 ...
2
votes
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
votes
1answer
234 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"...
1
vote
0answers
78 views

Deriving expectation involving Wishart distributions $E[\bf{A(A'WA)^-A'W}]=\bf{A(A'\Sigma A)^-A'\Sigma}$

I have a problem deriving two expectation involving Wishart distributions with mean zero. Let $\bf{W} \sim {W_p}({\bf{\Sigma }},$$n\bf{)}$ and $\bf{A}$$: p\times q$. Prove that $E[\bf{A(A'WA)^-A'W}]...
4
votes
1answer
5k views

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, ...
2
votes
0answers
685 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 ...