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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?

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  • $\begingroup$ 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. $\endgroup$ – Xi'an May 10 at 18:57
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    $\begingroup$ 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? $\endgroup$ – Patrick May 10 at 19:05
  • $\begingroup$ @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? $\endgroup$ – Patrick May 22 at 13:43

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