= Solution
Model $m$ gives the sequential forecast $f_{mt}(y_t)=p_m(y_t\mid y_1,\ldots,y_{t-1})$. The probability chain rule yields
$$
p_m(y_1,\ldots,y_T)=\prod_{t=1}^Tf_{mt}(y_t).
$$
Application of <Bayes theorem> makes the factors successive <posterior predictive distributions>, and their product is the <Bayesian model evidence>. Under conditionally independent sampling it is
$$
\int\prod_{t=1}^Tp_m(y_t\mid\theta_m)\,
p_m(\theta_m)\,d\theta_m.
$$
Taking logarithms gives the <prequential log score identity>
$$
T_m=\sum_t\log f_{mt}(y_t)=\log p_m(y_1,\ldots,y_T).
$$
Therefore the <Bayes factor> is
$$
\boxed{B_{12}
=\frac{p_1(y_1,\ldots,y_T)}{p_2(y_1,\ldots,y_T)}
=e^{T_1-T_2}.}
$$
This 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.
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