# Normal-normal-inverse-gamma conjugate model¶

## Posterior predictive distribution¶

If $$X| \mu, \sigma^2 \sim \mathcal{N}(\mu, \sigma^2)$$ with $$(\mu, \sigma) \sim \mathcal{N}\Gamma^{-1}(\mu_0, \lambda, \alpha, \beta)$$, then the posterior predictive probability density function, the expected value and variance of $$X$$ are

$f(x; \mu_0, \lambda, \alpha, \beta) = \frac{\alpha}{\beta(1 + \lambda^{-1})} \frac{\left(1 + \frac{1}{2\alpha} \left(\frac{\alpha(x - \mu_0)}{\beta(1+\lambda^{-1})} \right)^2 \right)^{-\alpha - 1/2}} {\sqrt{2\alpha}B(\alpha, 1/2)},$
$\mathrm{E}[X] = \mu_0, \quad \mathrm{Var}[X] = \frac{\left(\beta(1 + \lambda^{-1})\right)^2}{\alpha(\alpha - 1)},$

where $$\mathrm{E}[X]$$ is defined for $$\alpha > 1/2$$ and $$\mathrm{Var}[X]$$ is defined for $$\alpha > 1$$.

## Proofs¶

Posterior predictive probability density function

Note that this is the probability density function of the Student’s t-distribution, thus

$X \sim t_{2 \alpha}\left(\mu_0, \frac{\beta (1 + \lambda^{-1})}{\alpha}\right),$

see [Mur07].

Posterior predictive expected value

Apply properties of the Student’s t-distribution.

$\mathrm{E}[X] = \mu_0.$

Posterior predictive variance

Apply properties of the Student’s t-distribution.

$\mathrm{Var}[X] = \frac{\left(\beta(1 + \lambda^{-1})\right)^2}{\alpha(\alpha - 1)}.$