The poisson model measures the count of something where the number of opportunities or trials is not known. That is, , and

where is the mean number of events. The likelihood is

and for multiple counts ,

The 🥂 Conjugate prior distribution is

which is

The posterior is then

which is another Gamma distribution,

Interpretation#

The posterior mean is

We can view and as pseudo-counts, where is the number of prior observations and is the sum of those observation values.

Special Priors#

A non-informative prior has . The improper non-informative prior has . 💁‍♂️ Jeffrey's Prior is .