The Wishart distribution models random d × d symmetric, non-negative definite matrices, the same structural class as covariance and precision matrices. It is the multivariate counterpart of the gamma distribution: instead of distributing a scalar precision, it distributes a matrix precision. Only d(d + 1)/2 independent entries vary because symmetry identifies entries across the diagonal. section 13.10 section 13.10.1 section 13.11
A Wishart distribution is parameterized by:
- ν, a scalar degree-of-freedom parameter satisfying ν > d − 1. section 13.10.1
- W, a d × d symmetric non-negative definite matrix. section 13.10.1
- Its density uses matrix determinants, the trace operator, and the multivariate gamma function. section 13.10.1
- Its expected random precision is E(Λ) = νW, while its mode is Λ = (ν − d − 1)W when ν ≥ d + 1. section 13.10.1
For a Gaussian likelihood with known mean, the precision matrix Λ = Σ⁻¹ receives a Wishart prior. Conjugacy makes the posterior another Wishart distribution, with updated parameters obtained by combining the prior and training-data terms through determinant and trace identities. The posterior mode supplies a precision estimate when its degrees-of-freedom condition is satisfied. section 13.10.2 section 13.5
Precision estimation for a multivariate Gaussian
Given training data X and a known Gaussian mean, place a Wishart prior on Λ, derive the Wishart posterior, and estimate precision from its mode: Λ = (νₙ − d − 1)Wₙ for νₙ ≥ d + 1. section 13.10.2
| 𝔼(Λ) = νW |
Use the expectation table to connect the Wishart parameters with the average random precision matrix. section 13.10.1
Use the mode table to interpret the Wishart distribution as a source of a maximum-density precision estimate. section 13.10.1
