Index: The Book of Statistical Proofs ▷ General Theorems ▷ Bayesian statistics ▷ Probabilistic modeling ▷ Prior predictive distribution is marginal of joint likelihood

Theorem: The prior predictive distribution is the marginal distribution of the joint likelihood of the new data $y_{\mathrm{new}}$, unconditional on the measured data $y$:

\[\label{eq:priorpred-jl} p(y_{\mathrm{new}}) = \int p(y_{\mathrm{new}}, \theta) \, \mathrm{d}\theta\]

Proof: The prior predictive distribution is defined as the marginal distribution of new data $y_{\mathrm{new}}$, predicted based on the prior distribution before seeing the measured data $y$:

\[\label{eq:prior-pred} p(y_{\mathrm{new}}) = \int p(y_{\mathrm{new}} \vert \theta) \, p(\theta) \, \mathrm{d}\theta \; .\]

By using the law of conditional probability, we can write the integrand as

\[\label{eq:jl-post} p(y_{\text{new}} \vert \theta) \, p(\theta) = p(y_{\text{new}}, \theta)\]

which is the joint likelihood. Thus, expression \eqref{eq:prior-pred} can be written as:

\[\label{eq:priorpred-jl-qed} p(y_{\mathrm{new}}) = \int p(y_{\text{new}}, \theta) \, \mathrm{d}\theta \; .\]
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Metadata: ID: P510 | shortcut: priorpred-jl | author: JoramSoch | date: 2025-07-04, 13:33.