Bayes, Laplace and the Sun

William M Briggs mentions Laplace's Rule of Succession in a recent blog post. Briggs' is a blog about statistics and related matters that I highly recommend. Laplace used the rule, which relies on Bayes' Theorem, to calculate the probability that the sun will rise tomorrow. It is an elegant and fascinating bit of analysis. According to Wikipedia, Laplace's method give odds 1826250:1 in favour of the sun rising tomorrow.

But I beg to differ! Using Bayesian analysis, I calculate that the probability that the sun will rise tomorrow is 1/2!

Here is my argument. Let N+1 be the number of days from the start of history through tomorrow. Suppose at first that we know nothing about the sun, neither the related physics nor the past history of its rising. This was Laplace's assumption as well. Suppose we only know that the sun rises on some subset of the N+1 days in question. With only this knowledge, we assume a uniform prior probability distribution on this subset. Thus we assume that all 2^(N+1) possible subsets of the N+1 days are equally likely to be the sun-rising subset, each having probability 2^(-N-1). Now suppose we are given additional knowledge, specifically that the sun rose on the first N days. There are then only two possibilities for the sun-rising subset: the set of all N+1 days, and the set that contains only the first N days. By our prior assumption, and a trivial application of Bayes Rule, we see that each of these possibilities now has the posterior probability of 1/2. Thus the probability of the sun rising tomorrow is 1/2.

Put that in your pipe and smoke it! Related post here.