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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 )^{\alpha1} 
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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