In the previous article in this sequence, I conducted a thought experiment in which simple probability was not sufficient to choose how to act. Rationality required reasoning about meta-probabilities, the probabilities of probabilities.

Relatedly, lukeprog has a brief post that explains how this matters; a long article by HoldenKarnofsky makes meta-probability  central to utilitarian estimates of the effectiveness of charitable giving; and Jonathan_Lee, in a reply to that, has used the same framework I presented.

In my previous article, I ran thought experiments that presented you with various colored boxes you could put coins in, gambling with uncertain odds.

The last box I showed you was blue. I explained that it had a fixed but unknown probability of a twofold payout, uniformly distributed between 0 and 0.9. The overall probability of a payout was 0.45, so the expectation value for gambling was 0.9—a bad bet. Yet your optimal strategy was to gamble a bit to figure out whether the odds were good or bad.

Let’s continue the experiment. I hand you a black box, shaped rather differently from the others. Its sealed faceplate is carved with runic inscriptions and eldritch figures. “I find this one particularly interesting,” I say.

This article is the first in a sequence that will consider situations where probability estimates are not, by themselves, adequate to make rational decisions. This one introduces a "meta-probability" approach, borrowed from E. T. Jaynes, and uses it to analyze a gambling problem. This situation is one in which reasonably straightforward decision-theoretic methods suffice. Later articles introduce increasingly problematic cases.