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Part of a series on |
Bayesian statistics |
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Posterior = Likelihood × Prior ÷ Evidence |
Background |
Model building |
Posterior approximation |
Estimators |
Evidence approximation |
Model evaluation |
In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values.[1][2]
Given a set of N i.i.d. observations , a new value will be drawn from a distribution that depends on a parameter , where is the parameter space.
It may seem tempting to plug in a single best estimate for , but this ignores uncertainty about , and because a source of uncertainty is ignored, the predictive distribution will be too narrow. Put another way, predictions of extreme values of 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 . The posterior distribution of possible values depends on :
And the posterior predictive distribution of given is calculated by marginalizing the distribution of given over the posterior distribution of given :
Because it accounts for uncertainty about , the posterior predictive distribution will in general be wider than a predictive distribution which plugs in a single best estimate for .