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

## 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

$f({\boldsymbol {\mu )),{\boldsymbol {\Sigma ))|{\boldsymbol {\mu ))_{0},\lambda ,{\boldsymbol {\Psi )),\nu )={\mathcal {N))\left({\boldsymbol {\mu )){\Big |}{\boldsymbol {\mu ))_{0},{\frac {1}{\lambda )){\boldsymbol {\Sigma ))\right){\mathcal {W))^{-1}({\boldsymbol {\Sigma ))|{\boldsymbol {\Psi )),\nu )$ The full version of the PDF is as follows:

$f({\boldsymbol {\mu )),{\boldsymbol {\Sigma ))|{\boldsymbol {\delta )),\gamma ,{\boldsymbol {\Psi )),\alpha )={\frac {\gamma ^{D/2}|{\boldsymbol {\Psi ))|^{\alpha /2}|{\boldsymbol {\Sigma ))|^{-{\frac {\alpha +D+2}{2)))){(2\pi )^{D/2}2^{\frac {\alpha D}{2))\Gamma _{D}({\frac {\alpha }{2))))){\text{exp))\left\{-{\frac {1}{2))Tr({\boldsymbol {\Psi \Sigma ))^{-1})-{\frac {\gamma }{2))({\boldsymbol {\mu ))-{\boldsymbol {\delta )))^{T}{\boldsymbol {\Sigma ))^{-1}({\boldsymbol {\mu ))-{\boldsymbol {\delta )))\right\)$ Here $\Gamma _{D}[\cdot ]$ is the multivariate gamma function and $Tr({\boldsymbol {\Psi )))$ is the Trace of the given matrix.

## 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.

## Posterior distribution of the parameters

Suppose the sampling density is a multivariate normal distribution

${\boldsymbol {y_{i))}|{\boldsymbol {\mu )),{\boldsymbol {\Sigma ))\sim {\mathcal {N))_{p}({\boldsymbol {\mu )),{\boldsymbol {\Sigma )))$ where ${\boldsymbol {y))$ is an $n\times p$ matrix and ${\boldsymbol {y_{i)))$ (of length $p$ ) is row $i$ 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

$({\boldsymbol {\mu )),{\boldsymbol {\Sigma )))\sim \mathrm {NIW} ({\boldsymbol {\mu ))_{0},\lambda ,{\boldsymbol {\Psi )),\nu ).$ The resulting posterior distribution for the mean and covariance matrix will also be a Normal-Inverse-Wishart

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

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

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

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