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In frequentist statistics, the validity of the inference depends on the assumption that the model is correctly specified as well as pre-specified. Violations of these assumptions (i.e. we only specify a linear relationship but the true relationship is non-linear or we perform model selection procedures) undermine the validity of inference. Are there similar assumptions in Bayesian statistics or how are they solved in Bayesian statistics? I assume the prior is our belief about the correct model and therefore there is no misspecification?

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    $\begingroup$ I just found this comment from Andrew Gelman, which seems to discuss my problem but I still would appreciate some more formal explanations: andrewgelman.com/2013/03/14/… $\endgroup$ – Stats_Monkey Nov 9 '17 at 17:14
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Model specification in the Bayesian paradigm is just as important and potentially just has harmful as in frequentist statistics. The main difference with Bayes is that you can incorporate uncertainties instead of picking a simple model and hoping you're right. Examples:

  1. Instead of assuming normality, include a non-normality parameter in the model (indicating skewness, excess kurtosis, etc.)
  2. Instead of choosing whether an interaction term is in or out of the model, put it "half in" by having a prior for the interaction that favors "no interaction" but allows larger samples to override this.
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  • $\begingroup$ Thank you for your helpful answer. I was asking this question because I was reading Richard Berk's comment about statistical inference with misspecified models. He explains that a parameter should be seen as the average parameter over realizations of the dataset and not as a fixed one as in frequentist statistics assumed. I tried to get my head around if this interpretation would be the same in a Bayesian approach but could not find anything about model misspecification. Reference: Berk et al 2017 Working with Misspecified Regression Models. DOI 10.1007/s10940-017-9348-7 $\endgroup$ – Stats_Monkey Nov 10 '17 at 15:01

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