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Assuming an uninformed prior p(l) = 1/ l  and the Poisson likelihood function for observing a events in period t:

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l(\alpha \bigg|\lambda,t)=\frac{e^{-\lambda t}\big(\lambda t\big)^\alpha}{\alpha !}\propto e^{-\lambda t}(\lambda )^\alpha

The proportional statement is acceptable because we can ignore terms that don't involve l, and we then get the posterior distribution:

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p( \lambda\bigg|\alpha) \propto e^{-\lambda t}(\lambda )^{\alpha-1}

which by comparison with a Gamma density function is a Gamma(0,1/t,a) distribution. The Gamma distribution can also be used to describe our uncertainty about l if we start off with an informed opinion and then observe a events in time t. If we can reasonably describe our prior belief with a Gamma(0,b,a) distribution, the posterior is given by a Gamma(0, b/ (1 + b t),a + a) distribution.

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