= Solution
Integrate each supplied predictive density to get its <cumulative distribution function> $F_t$, then form the <sequential probability integral transform>
$$
\boxed{U_t=F_t(y_t).}
$$
For a correct continuous one-step conditional predictive model, $\mathbb P(U_t\le u\mid y_1,\ldots,y_{t-1})=u$. Iterating this identity shows that the $U_t$ are independent uniform variables. A histogram or quantile plot checks uniformity; serial plots and autocorrelations check for temporal structure left unexplained by the forecasts.
Also inspect empirical coverage of central <prediction intervals>, tail exceedances and interval widths, assessing calibration together with sharpness. These checks need only the supplied forecasts and observations. Compare chosen <predictive discrepancy statistics> with simulated uniform reference sequences. A total log score alone is a relative reward and does not provide a universal absolute goodness-of-fit threshold.
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