Laplace's Law of Succession

Laplace Law (or Rule) of Succession is an early example of an application of Bayesian thinking (a clear explanation here). Laplace (naively) calculated an estimate for the probability that the sun will rise tomorrow. He took a uniform prior on the Binomial probability to represent indifference or ignorance. The calculated has been widely criticised.

Reading about this made me think about the ad-hoc way in which the expert belief is include in the IMPACT Clinical model. There are 2 dodgy elements to this:

1. The data we have about the transition probabilities in the graph are changed directly by the expert, mixing-up the data and belief.
2. Having seen the outputs of the model using the modified probabilities the data is then changed again by the expert in light of this.

I always felt uncertain about the1) because I was taught that the data and prior belief should be kept separate. This is also mixed in 2) since the expert has seen the data via the output too.

I have tried to address 2) separately and will revisit this in a later post.

When I started to think about 1) in terms of Laplace's Law of Succession, the similarity became apparent. If we think about Laplace's expected probability of the sun rising tomorrow as the updated transition probabilities then this could be used as a more formal justification of the IMPACT calibration process.

Since Laplace assumes that he is ignorant of the chances of the sun rising tomorrow he in effect adds in 2 new data points to the sample, one of a success and one of a failure. Since the sun has risen for all the previously N observed days, the posterior expected probability is given by N+1/(N+2).

For the IMPACT model we need to generalise this to n<N previous successes, a mulitnomial distribution and some thing other than a uniform prior. We can set a Beta prior or, for the multinomial cae the Dirichlet or conversely we can work backwards from the posterior expected probability to identify the required prior.

I'll writeup a more detailed, formal explanation in the near future...