Questions tagged [wishart]

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

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1answer
41 views

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)\,...
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10 views

Kullback Leibler Divergence between two Normal Whishart Distributions

I'm having trouble to compute the KL Divergence between two normal wishart distributions. DKL from $Q$ to $P$ is defined as $$DKL(P||Q) = \int p(x) \log \frac{p(x)}{q(x)}dx$$ and a normal wishart ...
2
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1answer
285 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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0answers
67 views

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{\...
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7 views

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 ...
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1answer
153 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 = \...
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245 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 \...
3
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1answer
80 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 ...
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29 views

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} = \...
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1answer
77 views

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 $\...
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1answer
123 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\...
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22 views

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 $\...
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53 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 ...
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1answer
370 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 $\Sigma$ ...
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1answer
456 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 ...
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59 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-...
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1k views

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
178 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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45 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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217 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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331 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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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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731 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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170 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 ...
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94 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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98 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 ...
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1answer
98 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}; \...
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1answer
90 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 $\...
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209 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{...
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145 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 ...
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1answer
264 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
337 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 ...
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441 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
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1answer
210 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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130 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 $\...
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89 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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351 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 ...
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1answer
176 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: ...
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3answers
308 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{...
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1answer
241 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 ...
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1answer
171 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 ...
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0answers
153 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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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?
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1answer
164 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 ...
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185 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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0answers
144 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 $...
2
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3answers
939 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"...
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0answers
43 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 ...
2
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0answers
227 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 ...