Questions tagged [wishart]

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

Filter by
Sorted by
Tagged with
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
706 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 ...
4
votes
0answers
382 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 ...
3
votes
0answers
81 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 ...
2
votes
0answers
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 ...
2
votes
1answer
878 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 ...
11
votes
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 \...
1
vote
0answers
126 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 "...
5
votes
1answer
587 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 ...
2
votes
0answers
212 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{...
1
vote
0answers
140 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 $...
1
vote
1answer
329 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 ...
4
votes
1answer
698 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, ...
5
votes
0answers
624 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,\...
4
votes
2answers
2k views

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 ...
1
vote
0answers
714 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 ...
6
votes
0answers
261 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.
0
votes
1answer
679 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: ...
6
votes
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 ...
4
votes
1answer
120 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 ...
7
votes
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 ...
0
votes
1answer
472 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 ...
10
votes
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 ...
5
votes
0answers
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} = \|...
8
votes
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 ...
16
votes
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 ...
3
votes
0answers
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 ...
16
votes
0answers
536 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 ...
6
votes
0answers
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 ...
2
votes
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
votes
0answers
638 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 ...
2
votes
2answers
547 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 $\...