Bayesian framework - Prior and Likelihood independence

Question

I've a question on how, if ever needed, one can deal with the possible overlapping dependence of prior information and current data.

Basically, I was given the output parameters of a regression that was completed. While I have the context of the model (i.e., experimental design, research question, input, and output variables, etc.) I was not given anything but qualitative data on the samples (e.g., demographics and such).

My end goal is to use these parameters as priors for an updated analysis of a similar nature I'm conducting. The problem I'm having, however, is that due to my sample space, there's going to be some overlap between the samples collected in the previous experiment and the samples collected now. I can assume, with great confidence, that other than possible measurement errors, the information garnered from these overlapped samples will be essentially the same.

Since there is going to be some overlap in the information from the prior, and from the likelihood I will eventually form, is there anything particular I need to do?

My thoughts are that the prior merely provides information on the possible uncertainty regarding the information I'm collecting, and therefore, as long as I'm not replicating the experiment with all of the exact samples, then my analysis should be fine.

Thoughts?

Answer 1

If the prior is built as the posterior over a sample $(x_1,x_0)$, namely $$\pi_a(\theta)\propto\pi_0(\theta)f_1(x_1|\theta)f_0(x_0|\theta)$$ and if one considers the (final) posterior over a sample $(x_2,x_0)$ that mistakenly contains the same $x_0$, $$\pi_b(\theta)\propto \pi_a(\theta)f_2(x_2|\theta)f_0(x_0|\theta)$$ then $$\pi_b(\theta)\propto \pi_0(\theta)f_1(x_1|\theta)f_2(x_2|\theta)f_0(x_0|\theta)^2$$ incorrectly uses the subsample $x_0$ twice.