Sequential Gamma-Poisson updating over a week

Question

You track support tickets per day, modeled as a Poisson process with unknown rate $\lambda$ . You start from a $\text{Gamma}(\text{shape } 2, \text{ rate } 1)$ prior. The counts arrive in three batches:

Mon to Wed: 12 tickets over 3 days
Thu to Fri: 10 tickets over 2 days
Weekend: 4 tickets over 2 days

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

After all three batches, what is the posterior mean estimate of the ticket rate $\lambda$ ?

Accepted Answer

Intuition first. Bayesian updating gives the same answer whether you process the data in batches or all at once, because each posterior becomes the next prior. For a Gamma with Poisson data, every update adds that batch's tickets to the shape and its days to the rate, so the whole sequence collapses to adding up all tickets and all days. Update step by step. Gamma prior 	ext{Gamma}(\alpha, \beta) plus a batch of s tickets over d days gives 	ext{Gamma}(\alpha+s, \beta+d). - Start: 	ext{Gamma}(2, 1). - After Mon to Wed (12 over 3): 	ext{Gamma}(2+12,\ 1+3) = 	ext{Gamma}(14, 4). - After Thu …

Sequential Gamma-Poisson updating over a week

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