MultivariateNormalInverseGammaDistribution

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final class MultivariateNormalInverseGammaDistribution<Arg1, Arg2, Arg3, Arg4>(ν:Arg1, Λ:Arg2, k:Arg3, γ:Arg4) < Distribution<Real[_]>

Multivariate normal-inverse-gamma distribution.

This represents the joint distribution:

$$\begin{align*} \sigma^2 & \sim \mathrm{Inverse-Gamma}(\alpha, \beta) \\ x \mid \sigma^2 & \sim \mathrm{N}(\mu, \Sigma\sigma^2), \end{align*}$$

which may be denoted:

$$(x, \sigma^2) \sim \mathrm{Normal-Inverse-Gamma}(\mu, \Sigma, \alpha, \beta),$$

and is a conjugate prior of a Gaussian distribution with both unknown mean and variance. The variance scaling is independent and identical in the sense that all components of $x$ share the same $\sigma^2$.

In model code, it is not usual to use this class directlyDistribution. Instead, establish a conjugate relationship via code such as the following:

σ2 ~ InverseGamma(α, γ);
x ~ Gaussian(μ, Σ*σ2);
y ~ Gaussian(x, σ2);

where the last argument in the distribution of y must appear in the last argument of the distribution of x. The operation of Σ on σ2 may be multiplication on the left (as above) or the right, or division on the right.

Note

This class does not inherit from MultivariateTDistribution, as it is typically used for a random variable that is marginalized out.

Member Variables

Name	Description
ν:Arg1	Precision times mean.
Λ:Arg2	Precision.
k:Arg3	Degrees of freedom.
γ:Arg4	Variance scale.