Normal-inverse-Wishart distribution

normal-inverse-Wishart

Notation	${\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )}$
Parameters	${\displaystyle {\boldsymbol {\mu }}_{0}\in \mathbb {R} ^{D}\,}$ location (vector of real) ${\displaystyle \lambda >0\,}$ (real) ${\displaystyle {\boldsymbol {\Psi }}\in \mathbb {R} ^{D\times D}}$ inverse scale matrix (pos. def.) ${\displaystyle \nu >D-1\,}$ (real)
Support	${\displaystyle {\boldsymbol {\mu }}\in \mathbb {R} ^{D};{\boldsymbol {\Sigma }}\in \mathbb {R} ^{D\times D}}$ covariance matrix (pos. def.)
PDF	${\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\mathcal {N}}({\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},{\tfrac {1}{\lambda }}{\boldsymbol {\Sigma }})\ {\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )}$

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]

Definition

Suppose

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

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

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

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

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

Characterization

Probability density function

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

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

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

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

Properties

Marginal distributions

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

Posterior distribution of the parameters

Suppose the sampling density is a multivariate normal distribution

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

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

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

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

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

where

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

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

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

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

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

Generating normal-inverse-Wishart random variates

Generation of random variates is straightforward:

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

Related distributions

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

Notes

1. Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution."
2. Simon J.D. Prince(June 2012). Computer Vision: Models, Learning, and Inference. Cambridge University Press. 3.8: "Normal inverse Wishart distribution".
3. Gelman, Andrew, et al. Bayesian data analysis. Vol. 2, p.73. Boca Raton, FL, USA: Chapman & Hall/CRC, 2014.

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."