Posterior predictive distribution

In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values. 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 X = { x 1 , … , x N } {displaystyle mathbf {X} ={x_{1},dots ,x_{N}}} , a new value x ~ {displaystyle { ilde {x}}} will be drawn from a distribution that depends on a parameter θ ∈ Θ {displaystyle heta in Theta } : It may seem tempting to plug in a single best estimate θ ^ {displaystyle {hat { heta }}} for θ {displaystyle heta } , but this ignores uncertainty about θ {displaystyle heta } , and because a source of uncertainty is ignored, the predicted distribution will be too narrow. Extreme values of x ~ {displaystyle { ilde {x}}} will occur more often than the posterior distribution suggests. A posterior predictive distribution accounts for uncertainty about θ {displaystyle heta } . The posterior distribution of possible θ {displaystyle heta } values depends on X {displaystyle mathbf {X} } : And the posterior predictive distribution of x ~ {displaystyle { ilde {x}}} given X {displaystyle mathbf {X} } is calculated by marginalizing the distribution of x ~ {displaystyle { ilde {x}}} given θ {displaystyle heta } over the posterior distribution of θ {displaystyle heta } given X {displaystyle mathbf {X} } : Because it accounts for uncertainty about θ {displaystyle heta } , the posterior predictive distribution will in general be wider than a predictive distribution which plugs in a single best estimate for θ {displaystyle heta } . The prior predictive distribution, in a Bayesian context, is the distribution of a data point marginalized over its prior distribution. That is, if x ~ ∼ F ( x ~ | θ ) {displaystyle { ilde {x}}sim F({ ilde {x}}| heta )} and θ ∼ G ( θ | α ) {displaystyle heta sim G( heta |alpha )} , then the prior predictive distribution is the corresponding distribution H ( x ~ | α ) {displaystyle H({ ilde {x}}|alpha )} , where