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I read some Bayesian references recently and came up with some basic or naive questions about Bayesian inference. My research question is to test the association between predictors and outcome, and it is a multiple testing problem.

I know some people use Bayes factor or posterior probability of association. John Kruschke shows "Bayes factor can accept null with poor precision" in his book Doing Bayesian Data Analysis and some people are against using it. It seems very pessimistic to do association testing or power analysis. So my questions are:

  1. The credible interval does not have the frequentist property. What conclusion can we make for a coefficient $\beta$ in a regression model (with the credible interval)? Or how can we judge if there is association between a predictor and the outcome?
  2. Do Bayesian people do simulations in their paper? What metrics do they show to compare two models? For a frequentist paper, we normally focus on confidence interval coverage rate and power.
  3. It says that "when the prior distributions are non-informative, the Bayesian estimates are equivalent to maximum likelihood estimates." I know some of you may disagree with this, but if that is the case, what does it mean by "equivalent"? Does it mean we can do frequentist inference using the Bayesian estimates?

Any thoughts on any point would be greatly appreciated.

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