Posterior predictive distribution

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 ${\displaystyle \theta }$:

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 predicted distribution will be too narrow. Extreme values of ${\displaystyle {\tilde {x}}}$ will occur more often than the posterior distribution suggests.

...
Wikipedia