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

Let $M\in \{1,2\}$ be one fixed model indicator with equal prior probabilities, and put $w_{m,t}=\mathbb{P}(M=m\mid y_1,\dots ,y_{t-1})$. Within each model the parameters are already updated using the same past data. The Bayesian model averaging forecast is $f_t(y)={\sum }_m{w}_{m,t}{f}_{mt}(y)$. Upon observing $y_t$, Bayes theorem givesTaking the ratio cancels the denominator and gives the prescribed update because $f_{mt}(y_t)=e^{L_{mt}}$. Iteration yieldsThe weights are posterior model probabilities and their mixture is the full Bayesian predictive density. The indicator is fixed across days, rather than choosing a fresh model independently each morning.