Sequential Gamma-Poisson updating with exposure

Question

You model events as a Poisson process with unknown rate $\lambda$ per hour. You start from a $\text{Gamma}(\text{shape } 1, \text{ rate } 1)$ prior. Data arrive in three sessions of differing length:

Session 1: 5 events over 2 hours
Session 2: 4 events over 3 hours
Session 3: 3 events over 1 hour

You update after each session, feeding one session's posterior in as the next session's prior.

After all three sessions, what is the posterior mean estimate of the rate $\lambda$ (per hour)?

Accepted Answer

Intuition first. As always with Bayesian updating, processing the sessions one at a time gives the same belief as pooling them, because each posterior becomes the next prior. For Gamma-Poisson with exposure, every update adds that session's events to the shape and its hours to the rate, so the sequence just accumulates total events and total hours. Update step by step. Gamma prior 	ext{Gamma}(\alpha, \beta) plus a session of s events over h hours gives 	ext{Gamma}(\alpha+s, \beta+h). - Start: 	ext{Gamma}(1, 1). - After Session 1 (5 over 2): 	ext{Gamma}(1+5,\ 1+2) = 	ext{Gamma}(6, 3). - A…

Sequential Gamma-Poisson updating with exposure

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