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I have discrete data that comes from the distribution of a discrete random variable Y. The data appears to follow the normal distribution (ie if I make a kde plot it looks like a normal distribution). I've conducted Bayesian inference and assumed that the mean mu follows a normal prior. Thus, the posterior of mu also follows a normal distribution.

Now I'm interested in conducting predictions for future samples from the pdf of Y given the samples I've already seen, but I'm unsure how to proceed. It doesn't make sense to use a normal distribution for prediction because all samples can only take non-negative Interger values. To me it doesn't seem to make sense to use a Poisson because the variance doesn't equal the mean nor does it make sense to use a Binomial because the value that the sample can take is theoretically unbounded (but in practice it is often bounded by ~25).

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This sounds to me like an over-dispersion problem.
1. For the Poisson, you can "prove" than the distribution approximates the Normal given the right values of $\lambda$ and $n$. 2. It's quite common (I think McCullough and Nelder said safer to assume as default) that count data IRL have variance bigger than mean, and there is a whole world of over-dispersion models to use. You can use a form of the Negative Binomial (for example) to model your data which respects the discrete non-negative property of your data, but doesn't restrict you to having variance equal to the mean. One way of thinking about the Negative Binomial is a Poisson with a the location parameter $\lambda$ assumed to follow a Gamma distribution. I think (more by accident than design) I've tended to assume a log-Normal distribution for $\lambda$ whenever I've done this kind of modelling, but the nice thing about the Poisson-Gamma in a Bayesian framework is you get a nice simple (well, analytically tractable) posterior distribution. That sounds perfect for your problem.

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If they are integer-valued samples, then the probability of getting each sample value $V$ would logically be the total density of the Gaussian prior you've fitted over $(V-0.5; V+0.5)$.

Anything greater - you're falling into the next bucket, less - previous one, as you draw the samples.

Since this is Gauss, you can just calculate a difference between two cdf at the two endpoints of the interval even in, like, Excel or whatever.

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