Estimate of the mean number of events per period l
We discuss two approaches: Bayesian and classical statistics
Assuming an uninformed prior p(l) = 1/ l and the Poisson likelihood function for observing a events in period t:
The proportional statement is acceptable because we can ignore terms that don't involve l, and we then get the posterior distribution:
which by comparison with a Gamma density function is a Gamma(0,1/t,a) distribution. The Gamma distribution can also be used to describe our uncertainty about l if we start off with an informed opinion and then observe a events in time t. If we can reasonably describe our prior belief with a Gamma(0,b,a) distribution, the posterior is given by a Gamma(0, b/ (1 + b t),a + a) distribution.
Various classic statistics approaches to estimating l are discussed here.
3. Comparison of classical and Bayesian methods
A comparison of the different approaches to estimating l are discussed here.