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I have data from two sources; the two data sets were obtained from different geographical locations and at different historical times. I do not want to pool them into one data set sample and lose information about differing values of parameters that generated those collections, but I also cannot use Bayesian hierarchical model, since there would be just two unobservables from which I could not learn anything about hyperparameters. Is there any possible models that could be used in such a problems?

I know that the question is a bit vague, but I am open to various possibilities.
EDIT
The data are counts, distributed as Poisson with intensities $\lambda _{1}$ and $\lambda _{2}$. Suppose I have two data samples that comes from similar objects and represents number of failures over some period of time. I want to estimate both rates. I don't want to pool them because, for example, the operating conditions (environmental conditions) were different for those two data sources so that intensities were not identical. But I want to be able to borrow some information from each data sample like in hierarchical Bayesian model. But, as I already said, for hierarchical model there is just not enough data sources.

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  • $\begingroup$ Well, you should at least tell what type of data you have and what to want to do with it... $\endgroup$
    – nico
    Commented Sep 18, 2012 at 17:52
  • $\begingroup$ Updated the question. Hope, it will be more informative. $\endgroup$
    – Tomas
    Commented Sep 18, 2012 at 18:03
  • $\begingroup$ I did a bunch of editing; I hope it is now what you wanted to ask $\endgroup$
    – Peter Flom
    Commented Sep 18, 2012 at 23:34
  • $\begingroup$ Where's the hierarchy? Why Bayesian? If you know that a variable is distributed as Poisson, it is simple to estimate the mean - just find the mean. $\endgroup$
    – Peter Flom
    Commented Sep 18, 2012 at 23:36
  • $\begingroup$ Thanks for those few edits. I think it was clear that I do not want to estimate the mean in the simplest way, i.e. by pooling data into one sample. If I would have more data sources, say 10 or 20, then I would just use Bayesian hierarchical model (why Bayesian? Because for me it is the most natural way to do statistical inference. Why hierarchical? Because it is very nice way to account for cour-to-source variability). But hierarchical for two data sources would be of no use. That is way I am looking for another posible way to analyze these two data sets without complete pooling. $\endgroup$
    – Tomas
    Commented Sep 19, 2012 at 7:06

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Channeling Andrew Gelman:

I think you do want to use a hierarchical model. There are two hyperparameters so there's some non-identifiability, but that's the reality, and just means you'll have to make stronger assumptions about the hyper parameters to get something reasonable. You can learn something about the hyper parameters, just not a lot. The data will rule out some combinations of hyperparameters.

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