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 \mathbf{X} = {x_1, \dots, x_N}, a new value \tilde{x} will be drawn from a distribution that depends on a parameter \theta \in \Theta, where \Theta is the parameter space. :p(\tilde{x}|\theta) It may seem tempting to plug in a single best estimate \hat{\theta} for \theta, but this ignores uncertainty about \theta, and because a source of uncertainty is ignored, the predictive distribution will be too narrow. Put another way, predictions of extreme values of \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 \theta

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