# Normal-inverse-Wishart distribution

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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).[1]

## 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

1. ^ Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [1]

## 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." [2]