Suppose that I have a predictive posterior, which is an attempt to predict some one-step ahead forecasted value $\hat{y}_{T+1}$.

How do I assess if my posterior has done a good job or not? If we had two distributions to compare, then we could use some distance measure (e.g, KL) but we only have a point and a density?

I have heard about the "log predictive likelihood", but I do not agree that this is the best out there, as deciding on a bin size is nonsensical.



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