Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 34 / 4 / d

Model $m$ gives the sequential forecast $f_{mt}(y_t)=p_m(y_t\mid y_1,\dots ,y_{t-1})$. The probability chain rule yieldsApplication of Bayes theorem makes the factors successive posterior predictive distributions, and their product is the Bayesian model evidence. Under conditionally independent sampling it isTaking logarithms gives the prequential log score identityTherefore the Bayes factor isThis solves the second half of part (d), which is missing from the TeX. Proper priors and finite positive evidences are needed; unrelated improper-prior normalizing constants do not cancel.