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Context: I am working on a calibration problem involving a 1D function of parameter $\theta$ for which I derived a Jeffreys prior (in fact a 2D but I have an informative prior for one of the parameters).

Observation: Using this prior gives me in practice very bad inference results. Paralelly when using a default choice of the form : $\theta \sim \mathcal{N}(o,1000^2)$ where $o$ is a typical value I get pretty good results.

Question:

What to do in such a situation (expect checking formulas and code) ? what does it means about my model/data ?

I guess this question is quite abstract but any suggestion or remark would help me.

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    $\begingroup$ How do you derive the Jeffreys prior on the other parameter once you have given a prior to the other parameter ? $\endgroup$ – Stéphane Laurent Mar 30 '14 at 11:59
  • $\begingroup$ @StéphaneLaurent, the Jeffreys prior is computed independently from other parameters. $\endgroup$ – peuhp Mar 31 '14 at 12:33
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    $\begingroup$ I don't understand exactly what you are doing. A valuable way consists in integrating the likelihood with respect to the first prior and then to take the Jeffreys prior for the marginal (integrated) likelihood. I could explain this in an answer if you precise what you are doing in your question (the model and your method). $\endgroup$ – Stéphane Laurent Mar 31 '14 at 12:40
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There's no reason Jeffreys's prior is "supposed" to give you "good" results. The chief motivation behind it is that posterior inferences are invariant to re-parameterization of the model. Is that a major concern for your problem? I can't say. But if it is not, then an alternative choice could improve posterior inferences.

In your case, what these results means is that if you need a more precise posterior that you will need to build a model that accounts for more relevant information about the data generation process, or that includes more information about plausible parameter values.

Alternatively, you could take the opposite view and conclude that the nonspecific posterior is the most coherent representation of the data, given the set of assumptions you are willing to make, the likelihood, and the data. Then the nonspecific posterior is the just what happens when you have limited information. It's nearly a tautology that having less information implies that you can be less confident that some values are more plausible than others.

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    $\begingroup$ For a (regular) one-parameter model, there's a reason for Jeffreys prior to provide "good" results, because it achieves an optimal frequentist-matching property. $\endgroup$ – Stéphane Laurent Mar 30 '14 at 12:01

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