...
Assuming an uninformed prior p(l) = 1/ l and the Poisson likelihood function for observing a events in period t:
...
| LaTeX Math Block | |
|---|---|
|
...
| |
l(\alpha \bigg|\lambda,t)=\frac{e^{-\lambda t}\big(\lambda t\big)^\alpha}{\alpha !}\propto e^{-\lambda t}(\lambda )^\alpha |
The proportional statement is acceptable because we can ignore terms that don't involve l, and we then get the posterior distribution:
| LaTeX Math Block |
|---|
...
|
...
| |
p( \lambda\bigg|\alpha) \propto e^{-\lambda t}(\lambda )^{\alpha-1} |
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.
...