In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values.

Given a set of N i.i.d. observations ${\displaystyle \mathbf {X} =\{x_{1},\dots ,x_{N}\}}$, a new value ${\displaystyle {\tilde {x}}}$ will be drawn from a distribution that depends on a parameter ${\displaystyle \theta \in \Theta }$, where ${\displaystyle \Theta }$ is the parameter space.

It may seem tempting to plug in a single best estimate ${\displaystyle {\hat {\theta }}}$ for ${\displaystyle \theta }$, but this ignores uncertainty about ${\displaystyle \theta }$, and because a source of uncertainty is ignored, the predictive distribution will be too narrow. Put another way, predictions of extreme values of ${\displaystyle {\tilde {x}}}$ will have a lower probability than if the uncertainty in the parameters as given by their posterior distribution is accounted for.

A posterior predictive distribution accounts for uncertainty about ${\displaystyle \theta }$. The posterior distribution of possible ${\displaystyle \theta }$ values depends on ${\displaystyle \mathbf {X} }$:

And the posterior predictive distribution of ${\displaystyle {\tilde {x}}}$ given ${\displaystyle \mathbf {X} }$ is calculated by marginalizing the distribution of ${\displaystyle {\tilde {x}}}$ given ${\displaystyle \theta }$ over the posterior distribution of ${\displaystyle \theta }$ given ${\displaystyle \mathbf {X} }$:

Because it accounts for uncertainty about ${\displaystyle \theta }$, the posterior predictive distribution will in general be wider than a predictive distribution which plugs in a single best estimate for ${\displaystyle \theta }$.

The prior predictive distribution, in a Bayesian context, is the distribution of a data point marginalized over its prior distribution ${\displaystyle G}$. That is, if ${\displaystyle {\tilde {x}}\sim F({\tilde {x}}|\theta )}$ and ${\displaystyle \theta \sim G(\theta |\alpha )}$, then the prior predictive distribution is the corresponding distribution ${\displaystyle H({\tilde {x}}|\alpha )}$, where

This is similar to the posterior predictive distribution except that the marginalization (or equivalently, expectation) is taken with respect to the prior distribution instead of the posterior distribution.