ParameterMixtureDistribution

A distribution whose own parameter $\theta$ is itself random, drawn from $prior$ — the marginal of $dist(\theta)$ averaged over $\theta \sim prior$ .

ParameterMixtureDistribution(dist(theta), Distributed(theta, prior))dist(theta) with theta itself drawn from prior.

Typed signatures

Type	Package	Applies
`(any, expression<Distributed>) -> distribution`	`statistics`	always

Domain: Statistics

Details

Two named conjugate closed forms: PoissonDistribution(theta) mixed over a GammaDistribution(shape, scale) prior on theta is exactly NegativeBinomialDistribution(shape, 1/(1+scale)) (the standard Gamma-Poisson identity — PDF/Mean/Variance all delegate to that equivalent NegativeBinomial). BinomialDistribution(n, theta) mixed over a BetaDistribution(a, b) prior is the Beta-Binomial: $PDF(x) = \binom{n}{x} B(x+a,n-x+b)/B(a,b)$ , $Mean = na/(a+b)$ , $Variance = nab(a+b+n)/\left((a+b)^2(a+b+1)\right)$ .
Anything else falls back to the LAW OF TOTAL EXPECTATION/VARIANCE — $E[X] = E_\theta[Mean(dist(\theta))]$ , $Var(X) = E_\theta[Variance(dist(\theta))] + Var_\theta[Mean(dist(\theta))]$ — built entirely out of the (already fully extended) Mean/Variance/Expectation operators, so it composes with any inner distribution/prior pair those already answer. Resolves exactly when Mean(dist(theta)) (and, for Variance, Variance(dist(theta)) too) is linear or quadratic in theta — Expectation's own recognized shapes; anything past that stays unevaluated.
PDF has no generic fallback (only the two named closed forms above) — an unrecognized inner/prior pair's PDF stays unevaluated.