Normal-inverse-Wishart 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).^[1]

Suppose

${\displaystyle {\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Sigma }}\sim {\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right)}${\displaystyle {\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 ${\displaystyle {\boldsymbol {\mu }}_{0}}${\displaystyle {\boldsymbol {\mu }}_{0}} and covariance matrix ${\displaystyle {\tfrac {1}{\lambda }}{\boldsymbol {\Sigma }}}${\displaystyle {\tfrac {1}{\lambda }}{\boldsymbol {\Sigma }}}, where

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

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

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

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

The full version of the PDF is as follows:^[2]

${\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\frac {\lambda ^{D/2}|{\boldsymbol {\Psi }}|^{\nu /2}|{\boldsymbol {\Sigma }}|^{-{\frac {\nu +D+2}{2}}}}{(2\pi )^{D/2}2^{\frac {\nu D}{2}}\Gamma _{D}({\frac {\nu }{2}})}}{\text{exp}}\left\{-{\frac {1}{2}}Tr({\boldsymbol {\Psi \Sigma }}^{-1})-{\frac {\lambda }{2}}({\boldsymbol {\mu }}-{\boldsymbol {\mu }}_{0})^{T}{\boldsymbol {\Sigma }}^{-1}({\boldsymbol {\mu }}-{\boldsymbol {\mu }}_{0})\right\}}${\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\frac {\lambda ^{D/2}|{\boldsymbol {\Psi }}|^{\nu /2}|{\boldsymbol {\Sigma }}|^{-{\frac {\nu +D+2}{2}}}}{(2\pi )^{D/2}2^{\frac {\nu D}{2}}\Gamma _{D}({\frac {\nu }{2}})}}{\text{exp}}\left\{-{\frac {1}{2}}Tr({\boldsymbol {\Psi \Sigma }}^{-1})-{\frac {\lambda }{2}}({\boldsymbol {\mu }}-{\boldsymbol {\mu }}_{0})^{T}{\boldsymbol {\Sigma }}^{-1}({\boldsymbol {\mu }}-{\boldsymbol {\mu }}_{0})\right\}}

Here ${\displaystyle \Gamma _{D}[\cdot ]}${\displaystyle \Gamma _{D}[\cdot ]} is the multivariate gamma function and ${\displaystyle Tr({\boldsymbol {\Psi }})}${\displaystyle Tr({\boldsymbol {\Psi }})} is the Trace of the given matrix.

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

Suppose the sampling density is a multivariate normal distribution

${\displaystyle {\boldsymbol {y_{i}}}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}${\displaystyle {\boldsymbol {y_{i}}}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}

where ${\displaystyle {\boldsymbol {y}}}${\displaystyle {\boldsymbol {y}}} is an ${\displaystyle n\times p}${\displaystyle n\times p} matrix and ${\displaystyle {\boldsymbol {y_{i}}}}${\displaystyle {\boldsymbol {y_{i}}}} (of length ${\displaystyle p}${\displaystyle p}) is row ${\displaystyle i}${\displaystyle i} of the matrix .

With the mean and covariance matrix of the sampling distribution is unknown, we can place a Normal-Inverse-Wishart prior on the mean and covariance parameters jointly

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

The resulting posterior distribution for the mean and covariance matrix will also be a Normal-Inverse-Wishart

${\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|y)\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{n},\lambda _{n},{\boldsymbol {\Psi }}_{n},\nu _{n}),}${\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|y)\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{n},\lambda _{n},{\boldsymbol {\Psi }}_{n},\nu _{n}),}

where

${\displaystyle {\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\bar {\boldsymbol {y}}}}{\lambda +n}}}${\displaystyle {\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\bar {\boldsymbol {y}}}}{\lambda +n}}}

${\displaystyle \lambda _{n}=\lambda +n}${\displaystyle \lambda _{n}=\lambda +n}

${\displaystyle \nu _{n}=\nu +n}${\displaystyle \nu _{n}=\nu +n}

${\displaystyle {\boldsymbol {\Psi }}_{n}={\boldsymbol {\Psi +S}}+{\frac {\lambda n}{\lambda +n}}({\boldsymbol {{\bar {y}}-\mu _{0}}})({\boldsymbol {{\bar {y}}-\mu _{0}}})^{T}~~~\mathrm {with} ~~{\boldsymbol {S}}=\sum _{i=1}^{n}({\boldsymbol {y_{i}-{\bar {y}}}})({\boldsymbol {y_{i}-{\bar {y}}}})^{T}}${\displaystyle {\boldsymbol {\Psi }}_{n}={\boldsymbol {\Psi +S}}+{\frac {\lambda n}{\lambda +n}}({\boldsymbol {{\bar {y}}-\mu _{0}}})({\boldsymbol {{\bar {y}}-\mu _{0}}})^{T}~~~\mathrm {with} ~~{\boldsymbol {S}}=\sum _{i=1}^{n}({\boldsymbol {y_{i}-{\bar {y}}}})({\boldsymbol {y_{i}-{\bar {y}}}})^{T}}.

To sample from the joint posterior of ${\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}${\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}, one simply draws samples from ${\displaystyle {\boldsymbol {\Sigma }}|{\boldsymbol {y}}\sim {\mathcal {W}}^{-1}({\boldsymbol {\Psi }}_{n},\nu _{n})}${\displaystyle {\boldsymbol {\Sigma }}|{\boldsymbol {y}}\sim {\mathcal {W}}^{-1}({\boldsymbol {\Psi }}_{n},\nu _{n})}, then draw ${\displaystyle {\boldsymbol {\mu }}|{\boldsymbol {\Sigma ,y}}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }}_{n},{\boldsymbol {\Sigma }}/\lambda _{n})}${\displaystyle {\boldsymbol {\mu }}|{\boldsymbol {\Sigma ,y}}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }}_{n},{\boldsymbol {\Sigma }}/\lambda _{n})}. To draw from the posterior predictive of a new observation, draw ${\displaystyle {\boldsymbol {\tilde {y}}}|{\boldsymbol {\mu ,\Sigma ,y}}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}${\displaystyle {\boldsymbol {\tilde {y}}}|{\boldsymbol {\mu ,\Sigma ,y}}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})} , given the already drawn values of ${\displaystyle {\boldsymbol {\mu }}}${\displaystyle {\boldsymbol {\mu }}} and ${\displaystyle {\boldsymbol {\Sigma }}}${\displaystyle {\boldsymbol {\Sigma }}}.^[3]

Generation of random variates is straightforward:

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

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

• Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
• Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [2]

• Multivariate continuous distributions
• Conjugate prior distributions
• Normal distribution

• Articles with short description
• Short description matches Wikidata