BART Procedure

As described by Chipman, George, and McCulloch (2010), a probit BART model can be used to analyze binary response data.

The probit BART model has the following components:

The BART probit model assumes that the probability of an event is given by

The regularization prior for the probit model inherits many properties of the prior for the Gaussian model. For more information about the regularization prior for Gaussian BART models, see the section BART Model for Normal Data.

Like the Gaussian BART model, the probit BART model assumes independent tree structures and conditionally independent terminal node parameters, given the tree structure. The prior probability of a node being split is also given by the same function of the node’s depth. For more information about the tree prior, see the section Tree Prior. The differences between the Gaussian and probit BART models are related to the variance parameter , the prior that is used for the leaf parameters , the presence of an offset variable c, and the algorithm that is used to generate posterior samples of the sum-of-trees ensemble. The remainder of this section elaborates on these differences.

As described by Chipman, George, and McCulloch (2010), samples of the sum-of-trees ensemble for probit models are produced by a Gibbs sampling algorithm that is similar to the algorithm that is used for Gaussian models but involves the addition of latent variables , . The latent variables are assumed to be independent and identically distributed (iid) normal random variables with mean and variance 1. Moreover, the latent variables are assumed to be related to the observed response by

Conditional on the observed response values, the latent variables therefore follow the truncated normal distributions

With these assumptions, the latent variables are used to sample updates to the tree structures and leaf parameters. The algorithm for sampling these updates follows the same process that is used for Gaussian models, but the observed response values are replaced by the sampled values of the latent variables. For more information about the sampling algorithm that is used for Gaussian models, see the section Sampling the Posterior.

For the latent variables, the probit BART model assumes a prior of the form

Like the Gaussian model, the probit model uses a normal prior for the leaf parameters that enforces the weak learner property. But in contrast to the Gaussian model, the prior mean and variance do not depend on the range of the observed data. For a value of , the prior mean and variance for the probit model leaf parameters are

This choice for the prior mean and variance places a greater prior probability on values of within the range and applies regularization to , shrinking its values toward 0. You can specify the value of k that determines the prior variance of the leaf parameter by using the LEAFSIGMAK= option in the PROC BART statement.

Each iteration of the Gibbs sampling algorithm for probit models begins by sampling values of for each observation . Let denote the vector of sampled values, and let denote the tree structures for the trees in the ensemble, excluding the jth tree. Similarly, let denote the set of all leaf parameters, excluding the jth tree parameters. Then, for each tree and set of leaf parameters , , samples are taken from

As with the algorithm that is used for Gaussian models, the updates for the tree structure and can be made in two steps. First, an update to the tree structure is sampled, and then an update to the leaf parameters, given the updated tree structure, is sampled. The proposal density for modifying the tree structure and the probability of accepting a proposed modification to the tree structure are determined by using the same expressions that are described in the section Sampling the Posterior.

By default, for probit models, the BART procedure determines the value of the offset variable c on the basis of the proportion of events that occur in the training data. Alternatively, you can use the OFFSET= option in the MODEL statement to specify an offset variable, in which case the offset values can be different for each observation.