Fabric defects per square meter, Gamma-Poisson

Question

You model fabric defects as a Poisson process with unknown rate $\lambda$ per square meter. Your prior is $\text{Gamma}(\text{shape } 4, \text{ rate } 4)$ , where the rate parameter counts square meters already inspected. You then inspect 6.5 square meters and find 10 defects.

Using Bayesian updating, what is the posterior mean estimate of the defect rate $\lambda$ (per square meter)?

Accepted Answer

Intuition first. Here the exposure is area, 6.5 square meters, rather than time, but the Gamma-Poisson rule is identical: add total defects to the shape and total area to the rate parameter, reading the rate parameter as "square meters already inspected." The rule. With prior 	ext{Gamma}(\alpha, \beta), total events \sum xi, and total exposure t (square meters): 	ext{posterior} = 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + t\Big). Plug in. Here \alpha = 4, \beta = 4, events \sum xi = 10, exposure t = 6.5 square meters: 	ext{posterior} = 	ext{Gamma}(4 + 10,\ 4 + 6.5) = 	ext{Gamma}(14,\ …

Fabric defects per square meter, Gamma-Poisson

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