Posterior predictive distribution ( $p (y^{*} ∣ y)$ )

The distribution of a new or missing quantity integrates its model over the Bayesian posterior:

p (y^{*} ∣ y) = \int p (y^{*} ∣ ϕ, y) p (ϕ ∣ y) d ϕ

. Both parameter and residual uncertainty should enter predictive draws used for multiple imputation.

Posterior predictive repeated-success identity ( $P (Y_{1} = \dots = Y_{k} = 1 ∣ D) = E [π (Θ)^{k} ∣ D]$ )

If future Bernoulli random variables are conditionally independent given a shared parameter

Θ

, with success probability

π (Θ)

, their all-success conditional probability is

π (Θ)^{k}

. Taking a posterior expectation proves the identity. For

k \geq 1

0 \leq π \leq 1

give

(E [π ∣ D])^{k} \leq E [π^{k} ∣ D] \leq E [π ∣ D]

. For two distinct future trials the posterior covariance is

E [π^{2} ∣ D] - (E [π ∣ D])^{2} = Var (π ∣ D)

. Thus shared parameter uncertainty can induce positive dependence even when trials are conditionally independent.

A posterior predictive check compares a summary of the observed data with the same summary in replicated data drawn from a model's posterior predictive distribution. The replication must match the target: replicating an existing group's observations conditional on its effect differs from predicting a new group with a newly drawn effect. Such checks expose mismatches in dispersion, tails or other features; using the observations to fit and check the model means they are not automatically calibrated frequentist tests.

In a hierarchical Bayesian model, replication conditional on fitted observation-level random effects checks the sampling layer. Replication with fresh random effects drawn from their population law also checks the population mean and variation. To assess a dose-response curve, sample the global parameters from their posterior distribution, draw fresh effects for new plates at each existing dose, and compare replicated means, spreads and trend discrepancies with the observed values. Reusing the fitted plate effects can conceal a misspecified mean curve by reproducing deviations those effects already absorbed.

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The posterior predictive distribution is a concept in Bayesian statistics used to make predictions about future observations based on a model that has been updated with observed data. It combines information about the uncertainty of the model parameters (as described by the posterior distribution) with the likelihood of new data given those parameters. Here’s a breakdown of the concept: 1. **Posterior Distribution**: After observing data, we update our beliefs about the model parameters using Bayes' theorem.