Is it sensible to use Bayes Factors to examine probability that each coefficient in a mixed effects model is > 0? I am fitting the following model using the "brms" package in R:
prior<-c(set_prior("cauchy(0, 2.5)",class = "b", coef = ""),
set_prior("cauchy(0, 10)", class = "Intercept", coef = ""))


m1<-brm(DV~IV1*IV2*IV3+(1+IV1*IV2*IV3|Subject)+(1+IV3|Nonword),data=data, 
family="bernoulli", prior = prior, warmup = 1000, iter = 2000, chains = 20)

Instead of looking at the confidence intervals and making an all or nothing decision, I would like to get a sense of the extent to which the data support each of the estimated parameters having an effect.
I will use the hypothesis() function in "brms" which, as described in the manual, does the following: 
Among others, hypothesis computes an evidence ratio (Evid.Ratio) for each hypothesis. For a directed hypothesis, this is just the posterior probability under the hypothesis against its alternative. That is, when the hypothesis if of the form a > b, the evidence ratio is the ratio of the posterior probability of a > b and the posterior probability of a < b. In this example, values greater than one indicate that the evidence in favour of a > b is larger than evidence in favour of a < b. For an undirected (point) hypothesis, the evidence ratio is a Bayes factor between the hypothesis and its alternative computed via the Savage-Dickey density ratio method. That is the posterior density at the point of interest divided by the prior density at that point.
So, for each of the parameters I had planned to examine the amount of evidence that they are greater than zero.
My questions are:


*

*Are these acceptable priors for this type of analysis?

*Instead of examining whether they are all greater than zero, is there a better comparison that I should be making? My question is essentially: do these predictors have any effect on the outcome. With that being said, perhaps I should compare them to a ROPE around 0?

 A: 1) Any prior that models your belief before seeing the data is an acceptable prior.  If it is truly a Bernoulli likelihood, then a Cauchy prior is a very unusual choice.  By the nature of your parameter, you are granting the greatest weight on 0 and the smallest weight on 1.  Did you intend that?
2) You should be using combinatoric hypotheses.  So, for example, if your largest model is $y=\beta_1x_1+\beta_2x_2+\beta_3x_3+\alpha$, then  you should be testing each of the following hypotheses (unless scientific logic excludes some):


*

*$y=\beta_1x_1+\beta_2x_2+\beta_3x_3+\alpha$

*$y=\alpha$

*$y=\beta_1x_1+\alpha$

*$y=\beta_2x_2+\alpha$

*$y=\beta_3x_3+\alpha$

*$y=\beta_1x_1+\beta_2x_2+\alpha$

*$y=\beta_2x_2+\beta_3x_3+\alpha$

*$y=\beta_1x_1+\beta_3x_3+\alpha$


You also need to assign a prior weight to each of these hypotheses.
You should not use the default burn-in or iterations.  I just did a regression that needed an over two million cycle burn-in.  You need to look at the behavior of the model over the parameter space is.
The question of "is $\beta_x=0$?" is a frequentist way of viewing the world.  Remember that the null is that it is equal to zero.  This does not matter, what if it is very close to zero, but not zero.  You should be doing Bayesian model selection instead.  If a variable does not matter, it will fall out of the selection process.  What if $\beta_2\ne{0}$, but including $x_2$ results in a bad model?  You can be statistically significant, but irrelevant. 
