The sunrise problem, Laplace's rule of succession

Question

A newly built trading signal has fired 7 times so far and been right all 7 times. A colleague wants to assign a 100% probability that the next call is correct.

Using a uniform prior, compute the probability the next call is right, and explain why it is not 1.

Accepted Answer

The naive estimate is overconfident. With 7 successes in 7 tries, the raw frequency is 7/7 = 1. But claiming certainty from 7 observations ignores that you have barely tested the signal, a run of 7 is entirely possible even for a merely-good (not perfect) signal. Laplace's rule of succession. Put a uniform prior, Beta(1,1), on the unknown success probability p. After s successes in n trials the posterior is Beta(1 + s,\ 1 + n - s), whose mean, the predictive probability that the next trial succeeds, is P(	ext{next is a success}) = \frac{s + 1}{n + 2}. For s = n = 7: \frac{7 + 1}{7 + 2} = \fr…

The sunrise problem, Laplace's rule of succession

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