I'm trying to get intuition into the relationship between low identifiability and prior weight in Bayesian model. Is it true to say that in lowly identifiable model + data the prior will have a higher weight?
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Yes, in the limit of no identification, the posterior is the same as the prior, see https://stats.stackexchange.com/questions/322990/when-a-prior-distribution-would-not-be-overwhelmed-by-data-regardless-of-the-sa/323317#323317. Under "low" identifiability, the likelihood is, at least in relevant regions, "flat" in $\theta$, so that $p(\theta|y)\approx p(\theta)$, as the likelihood is "almost" constant in $\theta$. – Christoph Hanck Sep 02 '20 at 06:42
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I have answered a similar question here:
What's the problem with model identifiability?
When your model+data are unidentifiable, that means that you do not get any information about the model from this specific data, therefore, all information comes into the posterior from the prior.
On the contrary, for more and more informative data, less and less weight is given to the prior, and the posterior is dominated by the likelihood instead.
Costas Argyris
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This is probably better suited for a comment than a full-fledged answer. – Sycorax Jan 15 '16 at 19:00