Probit flexBART for binary outcomes

Fit a BART model of a binary responses using the the Albert and Chib (1993) data augmentation for probit models.

Usage

probit_flexBART(formula, train_data, test_data = NULL,...)

Arguments

formula: an object of class formula (or one that can be coerced to the class): a symbolic description of the model to be fitted. The details of model specification are given under 'Details'.
train_data: an object of class data.frame containing data used to train the model. As usual, rows (resp. columns) correspond to observations (resp. variables)
test_data: an optional object of class data.frame containing test-set (i.e., out-of-sample) data. Default is NULL.
...: Additional arguments for setting prior hyperparameters (e.g., number of trees, \(\mu_{0}\), \(\tau\), etc.) and MCMC control parameters (e.g., number of chains, iterations, etc.) See flexBART for details about additional arguments.

Details

For a binary response \(Y\), \(p\) predictors \(X_{1}, \ldots, X_{p}\), probit_flexBART models \(P(Y=1 \vert X = x) = \Phi(f(x)\), where \(\Phi\) is the standard normal cumulative distribution function. probit_flexBART combines the Albert & Chib (1993) data augmentation strategy for probit regression with the usual Bayesian backfitting used to fit (VC)BART models.

The formula argument

Currently, probit_flexBART only supports fitting single ensemble probit BART models. So, the only valid formula will look something like Y~bart(.) or Y ~ bart(x1+x2). As with flexBART, you must include the string “bart” on the right-hand side of the formula object.

Prior specification

probit_flexBART approximates the function \(f(x)\) with an ensemble of binary regression trees. It also specifies independent priors on the trees in the ensemble that are essentially identical to those deployed by flexBART. That is, the tree structure is generated using a branching process in which the probability that a node at depth \(d\) is non-terminal is \(\alpha \times (1 + d)^{-\beta}\). Then, decision rules are drawn sequentially from the root down to each leaf. Finally, independent \(N(\mu_0, \tau^2)\) priors are specified for the outputs in each leaf.

With this specification, the marginal prior of any evaluation of the regression function \(f(x)\) is \(N(M \times \mu_{0}, \tau^{2} \times M)\), where \(M\) is the number of trees in the ensemble. Thus, for each \(x\), the induced prior for \(P(Y = 1 \vert X = x)\) places 95% probability on the interval \([\Phi(M \times \mu_{0} - 2 \times \tau \times \sqrt{M}), \Phi(M \times \mu_{0} + 2 \times \tau \times \sqrt{M})]\). By default, probit_flexBART sets \(\tau = 1/\sqrt{M}\) and \(\Phi^{-1}(\overline{y})/M\) (i.e., qnorm(mean(Y))/M). Use the mu0_vec and tau_vec arguments to set other hyperparameter values.

Value

An object of class “flexBART” (essentially a list) containing

dinfo: Essentially a list containing information about the input and output variables. Used by predict.flexBART.
trees: A list (or length nd) of character vectors (of lenght M) containing textual representations of the regression trees. These strings are parsed by predict.flexBART to reconstruct the C++ representations of the sampled trees.
scaling_info: Essentially a list containing information for re-scaling raw MCMC output to the original outcome scale. Used by predict.flexBART.
M: A copy of the argument M_vec. Used by predict.flexBART.
cov_ensm: An \(p \times R\) binary matrix encoding whose (j,r)-element is 1 if trees in the ensemble for \(\beta_{r}(X)\) can split on \(X_{j}\).
is.probit: Logical with value TRUE. Used by predict.flexBART.
prob.train.mean: Vector containing posterior mean of \(P(Y=1 \vert X = x)\) for the training data.
prob.train: Matrix with nd rows and length(Y_train) columns containing posterior samples of \(P(Y=1 \vert X = x)\) for the training data. Each row corresponds to a posterior sample of the regression functionand each column corresponds to a training observation. Only returned if save_samples = TRUE.
prob.test.mean: Vector containing posterior mean of \(P(Y=1 \vert X = x)\) on testing data, if testing data is provided.
prob.test: If testing data was supplied, matrix containing posterior samples of the regression function evaluated on the testing data. Structure is similar to that of prob.train. Only returned if testing data is passed and save_samples = TRUE.
varcounts: Matrix that counts the number of times a variable was used in a decision rule in each MCMC iteration. Structure is similar to that of prob.train, with rows corresponding to MCMC iteration and columns corresponding to predictors
timing: Vector of runtimes for each chain

References

Albert, J.H. and Chib, S. (1993) Bayesian analysis of binary and polychotomous data. Journal of the American Statistical Association. 88(422):669–679. doi:10.2307/2290350 .