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Background:

I'm very new to Bayesian. I have an effect size similar to R SQUARE ($R^2$) in the regression context. That is, this effect size goes from 0 to 1 (i.e., 0 is taken as the null effect, and 1 as the perfect upper limit). Let's assume that, a $Beta$ prior e.g., $Beta(x, ~\alpha = 2, ~\beta = 5)$ would make a reasonable choice on this effect size. To review, we have a $Beta$ Prior, and an F distribution as the likelihood for this effect size.

Question:

How does Savage-Dickey density ratio to obtain a Bayes Factor become possible in this setting? That is, the density value at the null value (i.e., here 0) taken from the $Beta$ prior divided over the density value at the null value (i.e., here 0) taken from the Posterior distribution? Do I need to pick a different prior that allows some weight on the null value?

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    $\begingroup$ A general comment: Many modern Bayesians don't consider Bayes factors that test against point-null hypothesis to be useful. They suffer from a number of problems in practice (e.g. Lindley's paradox). I would recommend considering a different null hypothesis, such as an effect size less than $\epsilon$, for some $\epsilon$ say $0.1$. Even better would be considering an effect size less than 0 as one hypothesis, and greater than 0 as the other. This is obviously not possible for the [0,1] statistic you mention, but might be possible generally $\endgroup$ – AaronDefazio Apr 4 '17 at 3:38
  • $\begingroup$ @AaronDefazio, thanks, you're right. So, the thing about Lindley's paradox is directly related to picking unrealistically vague priors. Here we are dealing with an effect size which in my research area is highly used and has some intrinsic scale to it. So, I just want to put some weight on zero, and basically nothing on 1. So seems like I need a customized prior? $\endgroup$ – rnorouzian Apr 4 '17 at 4:10
  • $\begingroup$ I would not say that Bayes factors "suffer a number of problems", e.g. Lindley's paradox is not a problem but rather it points out that Bayes factors and pvalues are measuring different things. Just like with pvalues, you need to know what you are calculating to understand how to interpret the results. Thus, you should feel free to construct a Bayes factor with point-null hypotheses, but you should be aware of how your choices affect the results and interpret those results appropriately. $\endgroup$ – jaradniemi Apr 4 '17 at 14:23
  • $\begingroup$ The issue is way more complex than can be discussed in the comments here. I actually wrote a whole blog post on the subtleties awhile back. I would still strongly caution against using point nulls. They rarely give sensible results. $\endgroup$ – AaronDefazio Apr 5 '17 at 3:07
  • $\begingroup$ @AaronDefazio, I respectfully agree with AaronDefazio. This issue is fairly complex. By the way, nice blog. $\endgroup$ – rnorouzian Apr 5 '17 at 3:12

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