= Solution
Let $M\in\{1,2\}$ be one fixed model indicator with equal prior probabilities, and put $w_{m,t}=\mathbb P(M=m\mid y_1,\ldots,y_{t-1})$. Within each model the parameters are already updated using the same past data. The <Bayesian model averaging> forecast is $f_t(y)=\sum_mw_{m,t}f_{mt}(y)$. Upon observing $y_t$, <Bayes theorem> gives
$$
\boxed{w_{m,t+1}
=\frac{w_{m,t}f_{mt}(y_t)}
{\sum_jw_{j,t}f_{jt}(y_t)}.}
$$
Taking the ratio cancels the denominator and gives the prescribed update because $f_{mt}(y_t)=e^{L_{mt}}$. Iteration yields
$$
\boxed{\frac{w_{1,T+1}}{w_{2,T+1}}=e^{T_1-T_2}=B_{12}.}
$$
The 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.
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