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

${\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Sigma }}\sim {\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right)$

has a multivariate normal distribution with mean ${\boldsymbol {\mu }}_{0}$ and covariance matrix ${\tfrac {1}{\lambda }}{\boldsymbol {\Sigma }}$, where

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

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

$({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu ).$

Characterization

Probability density function

$f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right){\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )$

Properties

Marginal distributions

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

Generating normal-inverse-Wishart random variates

Generation of random variates is straightforward:

1. Sample ${\boldsymbol {\Sigma }}$ from an inverse Wishart distribution with parameters ${\boldsymbol {\Psi }}$ and $\nu$
2. Sample ${\boldsymbol {\mu }}$ from a multivariate normal distribution with mean ${\boldsymbol {\mu }}_{0}$ and variance ${\boldsymbol {\tfrac {1}{\lambda }}}{\boldsymbol {\Sigma }}$

Related distributions

• The normal-Wishart distribution is essentially the same distribution parameterized by precision rather than variance. If $({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )$ then $({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}^{-1})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }}^{-1},\nu )$ .
• 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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