# Questions tagged [wishart-distribution]

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

112 questions
Filter by
Sorted by
Tagged with
31 views

### Distribution of the sample covariance of a multivariate exponential family

I am wondering if there is a known form for the distribution of the sample covariance matrix of a random variable that follows a multivariate exponential family distribution. I guess it would be a ...
1 vote
43 views

### Bounds of integration for the Wishart density

I once took a course that included zillions of exercises concerning the Wishart distribution, but as far as I recall, never mentioned the Wishart density. I asked something about that in this question,...
• 10.3k
119 views

• 6,297
204 views

### Distribution of the sum of Wishart distributed random matrices

Suppose $$A_1 \sim \mathcal{W}(A_1; \Psi_1, v_1)\\ A_2 \sim \mathcal{W}(A_2; \Psi_2, v_2),$$ where $\mathcal{W}$ is the Wishart distribution and $A_i$, and $\Psi_i$ are PSD ...
• 331
15 views

### Expectation of constant matrix sandwiched between two random matrices

I have a $N\times P$ random matrix $X$ with i.i.d. coefficients from a standard normalized Gaussian $\mathcal{N}(0, 1)$. The corresponding Wishart matrix is $$W = \frac{1}{P}X X^{\top}$$ Calculating ...
215 views

• 111
217 views

• 227
677 views

### Clustering with gaussian mixtures: choice of hyperparameters

Question: I am interested in general in understanding how to choose the hyperparameters if we are interested in clustering bivariate vectors assuming a mixture of Gaussian mixture with conjugate ...
• 511
1 vote
70 views

### Is there a closed form solution for the Hellinger Distance between two Wishart distributions?

Given two Wishart distributions $X_0 \sim W_{p_0}(V_0,n_0)$, $X_1 \sim W_{p_1}(V_1,n_1)$, what is the Hellinger Distance between them? Can it be obtained in closed form assuming $p_0 = p_1$? For ...
• 19.4k
1 vote
149 views

### Marginal likelihood of bivariate Gaussian model

I assume the following model for a sample $y_1 \in \mathbb{R}^2$ of size $1$ with bivariate Gaussian likelihood and independent bivariate Gaussian and inverse-Wishart prior for the mean and variance ...
• 511
1 vote
69 views

• 655
1 vote
231 views

### Kullback–Leibler divergence between two inverse Wishart distributions

Is there a closed form formula for the KL divergence between two inverse Wishart distributions?
• 954
192 views

### Marginal distribution for the off-diagonal elements of a Matrix following an Inverse-Wishart

I've found on CV the marginal distribution for the main-diagonal elements of a matrix which follows an Inverse-Wishart distribution. However, what will happen to the off-diagonal elements? what ...
292 views

### How to Estimate Covariance Matrix using Wishart Distribution?

I learned that Wishart distribution can help estimate covariance matrix of MVN without sampling directly from MVN. But there are some points that are still unclear for me: Why we bother to estimate ...
2k views

• 680
1 vote
34 views

• 165
1 vote
257 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 ...
243 views

### Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $n$ degrees of freedom on $p\times p$ positive-definite ...
• 10.3k
187 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-...
753 views

• 978
298 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 ...
• 12.9k
273 views

I was looking at the probability density function of the multivariate normal distribution and that of the inverse-Wishart distribution: $$\begin{array}{rccll} p_{\mathcal{N}}\left(\mathbf{x}; \... • 398 5 votes 1 answer 701 views ### Scaling for Wishart distribution If X has a Wishart distribution W_p(n,\Sigma) , what's the distribution for cX where c>0 ? I know that for a \chi^2_n (x) distribution with n degrees of freedom, c\chi^2  follows \... • 427 0 votes 0 answers 573 views ### Expectation of the inverse of \textbf{z} \textbf{z}^{H} where \textbf{z} is a complex Gaussian vector Considering the vector \textbf{z} \sim \mathcal{CN}(\textbf{0}_{M},\Theta_{M \times M}), what would be the expectation of \frac{1}{\textbf{z} \textbf{z}^{H}}, i.e., \mathbb{E} \left\lbrace \frac{... 0 votes 0 answers 497 views ### Property of Wishart distribution I am solving the next exercise about a property of a Wishart Distribution:$$M_1\sim W_p(\Sigma,n_1)M_2\sim W_p(\Sigma,n_2)$$are independent, then M_1+M_2\sim W_p(\Sigma,n_1+n_2) I have ... • 1,003 5 votes 1 answer 1k views ### Result for covariance between elements of the sample covariance matrix Given data matrix X_{n,p} where n is the sample size, p is the dimension, the sample covariance is$$\hat{\Sigma}=\frac{1}{n-1}X^\top X=\{\hat{\sigma}_{ij}\}_{1\le i,j\le p} Is there any ...
• 488
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 ...