C5. Posterior prediction

Topic: Bayesian foundations · more questions on this topic
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Why average over the posterior when predicting? What uncertainty does a plug-in prediction omit?

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Definitions/notation. Let \tilde Y be a future response and let \theta contain all unknown model parameters; with covariates suppressed and future and observed data conditionally independent given \theta, p(\tilde y\mid x)=\int p(\tilde y\mid\theta)p(\theta\mid x)\,d\theta. This averages over posterior parameter uncertainty; a plug-in distribution p(\tilde y\mid\hat\theta) retains response randomness but treats the fitted parameters as known.