# Bimodal posterior of ATE predicted via Bayesias Additive Regression Trees

I am using BART to estimate the ATE on a large (10k obs x 224 p) dataset with a binomial outcome.

In short, I first model the risk of the outcome $$Y$$ given the $$(Z,X)$$ covariates, then, for a chosen predictor Z, I estimate the outcome risks setting Z to a value of interest, e.g. $$\hat{Y}_i(0) = f(Z = 0, X_i)$$ and $$\hat{Y}_i(1) = f(Z = 1, X_i)$$ for each $$i$$ observation, with $$f()$$ being the BART model.
To compute the ATE I then take the averages of the indivudual predicted risk for each value of Z and subtract them: $$\frac{1}{n}\sum_{i=1}^n\hat{Y}_i(1) - \frac{1}{n}\sum_{i=1}^n\hat{Y}_i(0)$$. This is repeated for each predicted posterior value of $$\hat{Y}$$.

What I noticed is that the posterior of average predicted risks (e.g. $$\frac{1}{n}\sum_{i=1}^n\hat{Y}_i(z)$$) sometimes has more than one mode and consequently so has the final ATE.

Here is the posterior density of the average predicted risk for some Z:

If I take the density of the individual predictions for each single posterior sample I didn't see any multimodal pattern (only 4 posteriors had 2 mode out of 5000), so it doesn't to be related to a mixure of risks in the sample.

Also, the sequence of posterior produced by BART is interesting, since it seems highly autocorrelated to me (I have not idea how BartMachine, the R package I use, sorts the final samples):

The question is, how should I interpret multimodal patterns in predictions made via BART? Does it imply non-identifiability, or it has some epistemological implication in the phenomenon I'm studying?

• Is the posterior of the treatment effect bimodal? – Noah Oct 6 '20 at 5:38
• yes, with more than two modes for some variables. – Bakaburg Oct 7 '20 at 14:02