I have some data that I believe comes from the binomial distribution. I also have some old data from a past-experiment that I would like to base my prior beliefs on. The old data observations are: $$6, 5, 4, 2, 2, 1, 5, 4, 5, 1 $$ I want to base the mean and variance of a beta prior on these observations. Now, I could use the MLE for this data as the mean, but what could I use as the variance? If I were to take the variance of this data and set that to be my prior variance, this wouldn't work as we would have a negative value of $\alpha$ or $\beta$. So just based on this prior data, what is the correct way to decide the variance of my beta prior?
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$\begingroup$ If those are samples from a binomial distribution with parameters $n\in \mathbb N$ and $p\in [0,1]$, you might use a beta prior for $p$ if you know the value of $n$. Do you? $\endgroup$– HenryCommented Dec 20, 2023 at 11:47
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Bayesian analysis is iterative --- to get a "prior" for a new analysis based on older data you use the "posterior" from that older data based on an even more remote "prior-prior". So, if you want to form a prior based on past data, you can do this by first formulating an even more remote prior (belief prior to seeing this past data) and then use the resulting posterior as the prior for your new analysis. For example, you might start with a uniform prior-prior and then update this to obtain the posterior from the data you have listed (you haven't told us the sample sizes so I can't do this part for you) and then you will use that posterior as the prior for your new analysis formed with that older data.
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$\begingroup$ Interesting, I hadn't thought of it like that. So I am free to use my 'old' posterior as a 'new prior' under the Bayesian framework? $\endgroup$ Commented Dec 20, 2023 at 10:07
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7$\begingroup$ Yes, that is correct. Indeed, that is exactly how Bayesian analysis works. $\endgroup$– BenCommented Dec 20, 2023 at 10:08