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Not sure how to determine hyper-prior for prior distributions, specifically using historical data. First what I am doing: I want to estimate parameters for a normal likelihood function using Bayesian approach so that I can use historical data as my priors. (which is exactly the advantage of Bayesian approach)

I have a normal likelihood function, determined by the nature of my research questions $P(data|\mu,\sigma)\sim Norm(\mu,\sigma)$ and want to estimate the distributions of $\mu$ and $\sigma$. Besides my own data, I also have a historical database from which I want to estimate the prior distribution. I know with a normal likelihood function, If I choose also a normal prior over $\mu\sim Norm(\mu|\mu0 ,\sigma0)$ I will get a close-formed solution (Murphy K. 2007). Then the question becomes: how do I estimate the hyper-priors ($\mu0$ and $\sigma0$) for the prior of $\mu$?

Secondly, I am not quite sure if my model is univariate or bivariate normal. By looking at my data and the historical data they do have different variance. For the bivariate case, I need to consider a normal-gamma prior to get a conjuguated distribution. Then the same problem applies here, how do I estimate hyper priors for the gamma distribution($\alpha , \beta)$? I have no clue where to start from so If you know of any good reference I am much appreciated.

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  • $\begingroup$ I think I have found solution to my first question regarding known variance. $\endgroup$ – Zheng Zhou Jun 13 at 0:15
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I think I have found solution to my first question regarding known variance. In Alber and Lee (2014) they stated the "standard model by Gelman" is $\mu \sim N(\mu0, \sigma /n_{\mu})$, where $\mu0$ is mean of prior which is the empirical mean of the historical data in my case, $\sigma$ is the known variance, $n_{\mu}$ is the hypothetical size of prior. Since I have real historical data, $n_{\mu}$ can just be the sample size of the historical dataset.

In the same paper the authors also examined the case of unknown variance $\sigma$. However, I don't quite understand their statement about how to estimate the parameters (a and b) for the inverse gamma prior on $\sigma$.

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