Accident rate over several months, Gamma-Poisson

Question

You model accidents at an intersection as a Poisson process with unknown rate $\lambda$ per month. Your prior is $\text{Gamma}(\text{shape } 3, \text{ rate } 2)$ , where the rate parameter counts months already observed. You then monitor the intersection for 5 months and record 8 accidents.

Using Bayesian updating, what is the posterior mean estimate of the accident rate $\lambda$ (per month)?

Accepted Answer

Intuition first. This is a Gamma-Poisson update where the exposure is a clean number of months. Add total events to the shape and total months of exposure to the rate parameter, which you can read as "months already watched." The rule. With prior 	ext{Gamma}(\alpha, \beta), total events \sum xi, and total exposure t months: 	ext{posterior} = 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + t\Big). Plug in. Here \alpha = 3, \beta = 2, events \sum xi = 8, exposure t = 5 months: 	ext{posterior} = 	ext{Gamma}(3 + 8,\ 2 + 5) = 	ext{Gamma}(11,\ 7). Posterior mean = shape over rate: \mathbb{E}[\la…

Accident rate over several months, Gamma-Poisson

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