Ahh, I see. Apologies for the lazy reply. Well, with these issues I usually first consider loss due to floating point representation. In this case, is the Bayesian estimate of the odds ratio computed at each iteration? If so, then the procedure does not simply exponentiate the estimate at the end of the sampling, but rather uses a running average. As the average is computed, loss due to floating point representation may cause the observed difference. For the frequentist approach, the odds ratio is exactly exp(estimate). Just a guess though.
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