Averaging Out Your Doubts

In full Bayesian model averaging for a classifier, you start with a prior distribution $p(w)$ over network weights, observe training data $D$ , and want a predictive distribution for a new input $x$ . Which expression correctly describes the Bayesian model averaging prediction, and what does the posterior $p(w|D)$ represent relative to the prior?

$p(y|x,D) = \int p(y|x,w) p(w) , dw$ , where the posterior is discarded once the prior is chosen

$p(y|x,D) = p(w|D)$ , directly using the posterior as the prediction without a likelihood term

$p(y|x,D) = \int p(y|x,w) p(w|D) , dw$ , where the posterior is the prior updated by the likelihood of the observed data via Bayes' rule

$p(y|x,D) = p(y|x,w_{MAP})$ , where the posterior is just the single weight setting with highest prior density

$p(y|x,D) = \frac{1}{K} \sum_k p(y|x,w_k)$ for $w_k$ sampled uniformly at random, independent of $D$