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What are the differences in the inferences using non-informative (specifically Jeffrys) priors as compared to the conjugate prior?

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The difference is as between any other two priors: Bayes theorem is

$$ p(\theta|X) \propto p(X|\theta)\; p(\theta) $$

so whatever you multiply the likelihood with, it will have some impact on the posterior. The idea of "uniformative" priors is that they have as little impact on the posterior as possible, though it is not true that they convey "no" information. Conjugate priors are just priors that can be used with some distributions, so that posterior has a closed-form solution.

Moreover, you can have conjugate priors that are "uninformative", for example for binomial likelihood, beta distribution with parameters $\alpha=\beta=1$ (uniform), or $\alpha=\beta=1/2$ (Jeffreys prior).

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