This question might be silly, but I thought I should ask anyway. Is it possible to implement BART with stan? If it is possible, has someone done it or plan to do it? If it is not possible, could someone give me a high-level explanation of why?

If your python-fu is sharp and youâ€™re really willing to use HMC-type algorithms to fit your models you may try Discontinuous HMC, an implementation of which lives here. Of course, @betanalpha may have his own geometric qualms with that, but it seems promising to me.

Like many other non-parametric models, nothing can do full Bayesian inference for BARTâ€”itâ€™s combinatorially intractable. The best thatâ€™s likely to happen is that youâ€™ll find a local mode or even a few thousand and noodel around there.

Like neural networks (this decadeâ€™s replacement for additive regression trees) or latent Dirichlet allocation, which also canâ€™t be fit with full Bayes, you might still find BART useful in applications. Just make sure to call it â€śapproximate Bayesâ€ť (MCMC isnâ€™t approximate in the same senseâ€”with enough computation, you can almost surely get any precision you want out of MCMC).

Did you have something concrete in mind? The priors are over things like discrete tree structures and are too intractable to marginalize. The whole point of these tree things is to divide the space up into ever finer discrete subregions then use branching to decide which one to use for a given tree, then sum a bunch together.