First I want to state that I got unexpected feedback from a reviewer in regard to my question and I am simply interested in others' views in this regard (I have already sent in my rebuttal).
Suppose that we have $N$ observations of the random variable $Y$ and we know that they are iid. In addition, we believe them to be normally distributed. As such we have the following model:
$$Y \sim N(\mu,\sigma)$$
Now, let's assume that $\sigma$ is known and for simplicity equal to 1.
We then have a sample of $Y$, $y$.
Now, if we assume the following model:
$$y \sim N(\mu,1)$$
$$\mu \sim N(\mu_o+\eta\sigma_\mu,\sigma_\mu)$$
$$\eta\sim N(0,1)$$
and suppose that $\sigma_\mu$ is known.
My question is: would it be circular to assume that $\mu_0=\bar{y}$?
That is, we assume that mean of our prior on the mean is the observed sample mean.
To me, it would seem like the sufficient statistic of $\mu$ would maximize the likelihood, but I am not sure. What are others' opinions?
Edit: also, in my actual model, the prior on the variance inhibits a closed form solution. But I tried to simplify things.
Best Answer
Yes, setting $\mu_0=\bar y$ absolutely is circular. The Bayesian model cannot be justified unless you genuinely do have information about $\mu$ that is separate to the data. Based on what you have said here, the reviewer is correct to say that your approach is not statistically valid. You will be overstating the amount of data information that you have because you entering the same data twice, once as the prior mean and once as the observed mean.
It is possible in some contexts to estimate prior distributions from the data using empirical Bayes approaches, but empirical Bayes cannot be done with one sample in isolation as you are doing here.