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I know that the Gamma distribution is the conjugate prior of the Poisson distribution, such that given $\alpha$ and $\beta$ that describe the prior distribution, the posterior distribution is $Gamma(\alpha + n\bar{y}, \beta + n)$, where $n$ is the total number of observations and $\bar{y}$ is the mean of the observations.

Of course, the Poisson distribution is discrete; the Gamma distribution, however, is continuous. My very basic question is, is it appropriate to discretize the Gamma posterior distribution by passing in simply discrete values? E.g., does Gamma(2) yield the probability of 2 counts, or is the probability of two counts given instead by something like Gamma(1.5 - 2.5)?

For context, I'm working on a Python package for Bayesian A/B testing and am hoping to discretize my Poisson plots.

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When $X\sim\mathcal P(\lambda)$, a conjugate family of priors on $\lambda\in\mathbb R^*_+$ is the Gamma family, $\lambda\sim\mathcal G(\alpha,\beta)$ since$$\lambda|x\sim\mathcal G(\alpha+x,\beta+1)$$

The fact that the Poisson $P(\lambda)$ distribution has a countable support, $\mathbb N$, does not restrict the range of the possible values of $\lambda$, $\mathbb R^*_+$. Therefore, there is no argument for discretising the support of the prior/posterior distribution.

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