# How to choose estimates after Bayesian regression?

In a Bayesian logistic regression with two predictor variables $$x_{1}$$ and $$x_{2}$$, I did MCMC (2000 samples) to estimate posterior distribution. Now it's done, how can I choose the final estimates for $$\beta_{1}$$ and $$\beta_{2}$$? Do I choose the $$\beta_{1}$$ which corresponds to the mode of the 2000 samples of $$\beta_{1}$$ (could be the 47th sample for example) and do the same for $$\beta_{2}$$ (could be the 1628th)?

If yes, I don't understand as I thought the estimates found on sample $$i$$ are not independent of each other, aren't they? I mean, $$\beta_{1}^{i}$$ (the $$\beta_{1}$$ estimate on sample $$i$$) depends on $$\beta_{2}^{i}$$, no?

It seems to me it would be more correct to choose the couple $$(\beta_{1}^{i},\beta_{2}^{i})$$ that maximizes some quantity, am I wrong?

• A Bayes estimate is only defined with respect to a loss function. Once one has selected a loss function, the Bayes estimate is automatically produced. – Xi'an May 10 at 18:57
• Not sure I understand @Xi'an. In a one parameter model, I've read that one can choose the mode or the mean as point estimate. Where is the loss function here? – Patrick May 10 at 19:05
• @Xi'an, do you mean I should measure the utility on each sample and choose the one that maximizes that utility (or minimizes the loss), without regard to the probability of the posterior? – Patrick May 22 at 13:43