Empirical Bayes

target and input on test case

precision of output noise

training data

precision of prior

• The equation above is the right predictive distribution (assuming we do not have hyperpriors for alpha and beta). • The equation below is a more tractable approximation that works well if the posterior distributions for alpha and beta are highly peaked (so the distributions are well approximated by their most likely values)

point estimates of alpha and beta that maximize the evidence