I performed a logistic regression using Stata's
bayes: wrapper and obtain the following histogram from 10,000 posterior distribution samples of the log(odds) of my exposure of interest. I also obtained the standard, frequentist distribution of the log(odds) ratio (solid red line).
As a frequentist, I could report that the p-value for the estimate as >0.05 and the 95% confidence intervals include an estimate of 0 (OR 1.0) or no effect. This is essentially a non-conclusion. I am tempted to use the Bayes estimates to make further statements about the association between exposure and outcome, but it's unclear to me what I would say that is fundamentally different. An odds ratio of 0.95 or less is an interesting effect size and the probability of this is 0.538. This seems like an effect worth knowing more about when stated in the Bayes framework.
My question: With the Bayes distribution in hand, what can I say about the effect size I couldn't say with the standard frequentist approach? What can I conclude from Bayes that is distinct from standard logistic regression?