Budan's theorem

Template:Probability distribution In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and covariance matrix (the inverse of the precision matrix).

Definition

Suppose

$\mathit{\mu }|{\mathit{\mu }}_0,\lambda ,\mathrm{\Sigma }\sim \mathcal{𝒩}\left(\mathit{\mu }\right|{\mathit{\mu }}_0,\frac{1}{\lambda }\mathrm{\Sigma })$

has a multivariate normal distribution with mean ${\mathit{\mu }}_0$ and covariance matrix $\frac{1}{\lambda }\mathrm{\Sigma }$, where

$\mathrm{\Sigma }|\mathrm{\Psi },\nu \sim {\mathcal{𝒲}}^{-1}(\mathrm{\Sigma }|\mathrm{\Psi },\nu )$

has an inverse Wishart distribution. Then $(\mathit{\mu },\mathrm{\Sigma })$ has a normal-inverse-Wishart distribution, denoted as

$(\mathit{\mu },\mathrm{\Sigma })\sim \mathrm{N}\mathrm{I}\mathrm{W}({\mathit{\mu }}_0,\lambda ,\mathrm{\Psi },\nu ).$

Characterization

Probability density function

$f(\mathit{\mu },\mathrm{\Sigma }|{\mathit{\mu }}_0,\lambda ,\mathrm{\Psi },\nu )=\mathcal{𝒩}\left(\mathit{\mu }\right|{\mathit{\mu }}_0,\frac{1}{\lambda }\mathrm{\Sigma }){\mathcal{𝒲}}^{-1}(\mathrm{\Sigma }|\mathrm{\Psi },\nu )$

Properties

Marginal distributions

By construction, the marginal distribution over $\mathrm{\Sigma }$ is an inverse Wishart distribution, and the conditional distribution over $\mathit{\mu }$ given $\mathrm{\Sigma }$ is a multivariate normal distribution. The marginal distribution over $\mathit{\mu }$ is a multivariate t-distribution.

Generating normal-inverse-Wishart random variates

Generation of random variates is straightforward:

1. Sample $\mathrm{\Sigma }$ from an inverse Wishart distribution with parameters $\mathrm{\Psi }$ and $\nu$
2. Sample $\mathit{\mu }$ from a multivariate normal distribution with mean ${\mathit{\mu }}_0$ and variance $\frac{1}{\lambda }\mathrm{\Sigma }$

Related distributions

• The normal-Wishart distribution is essentially the same distribution parameterized by precision rather than variance. If $(\mathit{\mu },\mathrm{\Sigma })\sim \mathrm{N}\mathrm{I}\mathrm{W}({\mathit{\mu }}_0,\lambda ,\mathrm{\Psi },\nu )$ then $(\mathit{\mu },{\mathrm{\Sigma }}^{-1})\sim \mathrm{N}\mathrm{W}({\mathit{\mu }}_0,\lambda ,{\mathrm{\Psi }}^{-1},\nu )$ .
• The normal-inverse-gamma distribution is the one-dimensional equivalent.
• The multivariate normal distribution and inverse Wishart distribution are the component distributions out of which this distribution is made.

Notes

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References

• Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
• Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [1]

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