# NormalInverseGamma

final class NormalInverseGamma(μ:Expression<Real>, a2:Expression<Real>, σ2:InverseGamma) < Distribution<Real>

Normal-inverse-gamma distribution.

This represents the joint distribution:

\sigma^2 \sim \mathrm{Inverse-Gamma}(\alpha, \beta)x \mid \sigma^2 \sim \mathrm{N}(\mu, a^2\sigma^2),

which may be denoted:

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

and is the conjugate prior of a Gaussian distribution with both unknown mean and unknown variance.

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

σ2 ~ InverseGamma(α, β);
x ~ Gaussian(μ, a^2*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 a2 on σ2 may be multiplication on the left (as above) or the right, or division on the right.

### Factory Functions

Name Description
NormalInverseGamma

### Member Variables

Name Description
μ:Expression<Real> Mean.
λ:Expression<Real> Precision scale.
σ2:InverseGamma Variance.

### Factory Function Details

function NormalInverseGamma(μ:Expression<Real>, a2:Expression<Real>, σ2:InverseGamma) -> NormalInverseGamma