Rule of Succession - Interpretation

Interpretation

Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had observed one success and one failure for sure before we even started the experiments. In a sense we made n + 2 observations (known as pseudocounts) with s+1 successes. Beware: although this may seem the simplest and most reasonable assumption, which also happens to be true, so is a useful mnemonic, it still requires a proof! Indeed, assuming a pseudocount of one per possibility is one way to generalise the binary result, but has unexpected consequences — see Generalization to any number of possibilities, below.

Nevertheless, if we had not known from the start that both success and failure are possible, then we would have had to assign

But see Mathematical details, below, for an analysis of its validity. In particular it is not valid when, or .

If the number of observations increases, and get more and more similar, which is intuitively clear: the more data we have, the less importance should be assigned to our prior information.