trees. In essence, a binary regression tree in BART may be viewed as a penalized form of an Analysis of Variance (ANOVA) model. When the binary regression tree only splits on one variable for the whole tree, a main effects model is obtained. When the regression tree involve splits on many different variables, a multiple-way interaction model is obtained. BART combines all $m$ regression trees together in an additive manner to obtain non-linear estimates of the main and interaction effects. This additive procedure is done by first ‘breaking’ $Y$ into $m$ equal ‘pieces’ and fitting a regression tree to each piece. Subsequently, the regression tree in each $m$ piece is then estimated by looking at the residual produced by the other $m-1$ most updated regression trees. MCMC procedures are then used to obtain the posterior distribution of $f(X)$. When the default priors of BART suggested by [@chipman_bart] are assumed, the MCMC ensures that the eventual distribution of the the sum of regression trees is concentrated around the true distribution of the model [@rockova].

For binary outcomes, BART uses a probit link where $$\label{chap3:bin_bart_eq}
    P(Y=1|X)=\Phi(\sum_{j=1}^mg[X,T_j,M_j])$$ where $\Phi(.)$ is the cdf of a standard normal distribution. Estimation of the posterior distribution is 