Questions tagged [wishart]

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

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1answer
423 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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443 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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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 $\...
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1answer
217 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
278 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,...
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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 $\Sigma$ ...
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1answer
3k 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 ...
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266 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|^{(\...
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223 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,...
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592 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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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 ...
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1answer
348 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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95 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}]...
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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, ...
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864 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 ...
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409 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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93 views

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 ...
3
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1answer
217 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 ...
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1answer
1k 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 ...
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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 \...
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159 views

Question on Inverse-Wishart Distribution when reading Peter Hoff's book

I have a couple of questions when reading the chapter 7 The Multivariate Normal Model of Peter Hoff's "A First Course in Bayesian Statistical Methods". First, could anyone give me any resource about "...
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1answer
702 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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223 views

Calculation the Expectation of an Inverse Wishart matrix

I have $\boldsymbol{A} = \boldsymbol{G}^H \boldsymbol{G}$ is a Wishart matrix, i.e, $\boldsymbol{G}^H \boldsymbol{G} \sim \mathcal{W}_K (M, \boldsymbol{\Lambda})$ with $\boldsymbol{\Lambda} = \mathrm{...
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151 views

Wilks's lambda distribution [closed]

Please help me, I don´t have any idea how to solve this problem. If $\textbf{A} \sim W_p(\mathbf{\Sigma}, m)$ and $\textbf{B} \sim W_p(\mathbf{\Sigma}, m)$ are independent Wishart matrices, show that $...
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1answer
361 views

Bayesian estimation of multivariate Gaussian from noisy observations with known error variances

I have a dataset $\mathbf{D} = \{ (\tau_i, \Gamma_i) : 1 \le i \le n \}$ of observations $\tau_i = X_i + \epsilon_i$ from a $p$-dimensional Gaussian $X_i \sim \mathcal{N}(\mu, \Sigma)$ contaminated by ...
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1answer
869 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, ...
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662 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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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 ...
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779 views

Sample from Wishart distribution with inverse Scale matrix

I tried to model precision matrix in a hierarchical Bayesian setup with Wishart prior given d.f. and inverse scale matrix, and matrix normal likelihood, since it's a conjugate prior, my posterior on ...
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1answer
320 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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1answer
787 views

Matching X'X with Wishart Samples in R

$X'X \sim Wishart(\Sigma,n)$, however I'm having a tough time producing this in R. Example: ...
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1answer
5k 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 ...
4
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1answer
129 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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1answer
369 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 ...
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1answer
614 views

Degrees of freedom for Gaussian Process

I am reading this paper on Generalised Wishart Process (GWP). It is about modelling covariance matrix of D - dimensional gaussian processes (GP) as GWP. I fail to understand interpretation of "degrees ...
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1answer
2k views

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 ...
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186 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} = \|...
9
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1answer
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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
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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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379 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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0answers
613 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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0answers
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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 ...
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1answer
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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 ...
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675 views

Generating correlation matrices using Wishart distribution

I have problem on generating correlation matrices using Wishart distribution. I read some articles about Wishart distribution, and it turns out that Wishart distribution is commonly used to generate ...
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2answers
580 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 $\...

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