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There is one additional kind of inference that we should discuss, however. For this, let us return to Eq. (3.60) and recall its origins: we assumed a Poisson process, conditioned on the value of the rate parameter and assumed that the rate parameter had a gamma distribution. So, we can ask the question: given that we have observed k events, what does this tell us about the rate parameter? Formally, we want to find Pr{1 < A < 1 + d1|kevents in 0 to t}; we call this the posterior distribution of the rate parameter, given the data, and denote it by the symbol fp(1 |(k, t))d1. We apply the definitions of conditional probability and Bayes's theorem:

Poisson

Poisson

Figure 3.8. The cumulative distribution function for the Poisson (mean = 10) and negative binomial (mean = 10, two different values of the overdispersion parameter) plotted, for convenience, as continuous functions.

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