MultivariateNormalInverseGamma

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final class MultivariateNormalInverseGamma(μ:Expression<Real[_]>, Σ:Expression<LLT>, σ2:InverseGamma) < 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 directly. Instead, establish a conjugate relationship via code such as the following:

σ2 ~ InverseGamma(α, β);
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.

Factory Functions

Name	Description
MultivariateNormalInverseGamma

Member Variables

Name	Description
Λ:Expression<LLT>	Precision.
ν:Expression<Real[_]>	Precision times mean.
α:Expression<Real>	Variance shape.
γ:Expression<Real>	Variance scale accumulator.
σ2:InverseGamma	Variance scale.

Factory Function Details

function MultivariateNormalInverseGamma(μ:Expression<Real[_]>, Σ:Expression<LLT>, σ2:InverseGamma) -> MultivariateNormalInverseGamma