Naive Savage Dickey ratio on correlations with LKJ prior on correlation matrix suitable?

General
1

I am trying to find correlations between multiple variables. To do so, I model them jointly from a multivariate normal (based on this). I placed LKJ(1) on the correlation matrix (to be unbiased)

vars <- colnames(dfcorradj) 
form <- as.formula(
  paste0("mvbind(", paste(vars, collapse = ","), ") ~ 1")
)

fit_corr <- brm(
  data = dfcorradj,
  formula = bf(form) + set_rescor(TRUE),
  family = gaussian(),
  sample_prior = "yes"
)

My question is, can I calculate the naive savage dickey ratio to test whether a correlation is zero? I wonder because if I set a correlation to zero, the derived marginals of the other correlations will change. But Savage–Dickey requires that the prior under (H_0) matches the conditional prior induced under (H_1) for the nuisance parameters, and I am having trouble understanding what the corresponding (H_0) prior would actually look like here.
Does the dependence between the correlations make naive savage dickey unsuitable, and if so, is there an alternate way?

2

I believe that this paper addresses the issue: https://doi.org/10.1111/bmsp.12150

3

Sorry I didn’t get to this sooner, but may I ask why you’re trying to test if a correlation is zero? It’s not something we usually do in Bayesian statistics, because we’ll never get an exact zero estimate in a Bayesian posterior in a natural example.

4

Thank you for the response :)
My understanding was that with the Bayes factor, I could compare evidence for two models, one including the parameter and the other excluding the parameter and assuming it to be zero, and Savage Dickey is a shortcut to do that under certain conditions, and I wondered if I could do the same for correlationsand get evidence for a model with the correlation being estimated and the other excluding that particular correlation. I intended to be able to quantify evidence for one over the other. I am not sure if I understood your question correctly. I agree that the posterior would not have all the mass at zero. Could you explain again?

5

Thank you for sharing :)

6

I just wrote a response to another post suggesting the OP read Gelman et al.'s chapter on model comparison in BDA3. The link is to the book’s home page with a free pdf version. We prefer to use posterior predictive checks (like chi-square goodness of fit tests in regressions, which is one place the frequentists use the model of interest as the null) and cross-validation (which is the posterior predictive analogue of the prior predictive that Bayes factors measure).