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Density Regression with Bayesian Additive Regression Trees (DR-BART)

This package implements the DR-BART model for density regression as described in Density Regression with Bayesian Additive Regression Trees. Code was written by Jared Murray and Vittorio Orlandi and relies on code originally accompanying Pratola et. al (2013).

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Installation

You can install drbart from GitHub via:

# install.packages("devtools")
devtools::install_github('vittorioorlandi/drbart', ref = 'main')

Density Regression

Most regression methods focus on modeling the conditional mean of a joint distribution E(y ∣ x). While this may be an appropriate summary of the joint in some cases, there are many instances where other aspects of the density – like the variance or the skew – change as a function of covariates and can reveal interesting features of the data. Density regression is a method for modeling the entire conditional density p(y ∣ x) and DR-BART is a Bayesian method for doing so via Bayesian Additive Regression Trees. The method is highly flexible, capable of estimating arbitrary functionals of the conditional densities that may be of interest, and also yields proper uncertainty quantification about those estimates.

An Example

We’ll illustrate the package functionality by mimicking one of the simulations in the paper. We’ll simulate data according to the below:

# Simulate data from a covariate dependent mixture of a normal and a log gamma
gamma_shape <- function(x) 0.5 + x ^ 2
m <- function(x) 1 + 2 * (x - 0.8)
p <- function(x) exp(-10 * (x - 0.8) ^ 2)
mu0 <- function(x) {
  return(5 * exp(15 * (x - 0.5)) / (1 + exp(15 * (x - 0.5))) - 4 * x)
}

r <- function(x) {
  n <- length(x)
  z <- rbinom(n, 1, p(x))
  return(z * rnorm(n, m(x), 0.3) + 
           (1 - z) * log(rgamma(n, gamma_shape(x), 1)) + mu0(x))
}

n <- 1000
x <- runif(n)
y <- r(x)

We can fit a model to estimate p(y ∣ x) by calling the drbart() function. Below, we call the function with default parameters, though sampling details, prior parameters, and other aspects of the model can also be user-specified.

library(drbart)
fit <- drbart(y, x)

Given the fitted model, we can estimate the conditional densities we’re interested in via the predict.drbart() method:

xpred <- matrix(c(0.1, 0.5, 0.8), ncol = 1)
ygrid <- seq(-12, 5, length.out = 500)
preds <- predict(fit, xpred, ygrid)

In this case, we’re estimating p(y ∣ x = 0.1), p(y ∣ x = 0.5), and p(y ∣ x = 0.8) and the argument ygrid specifies the values at which the conditional densities are to be evaluated. The predict() method returns an object of class predict.drbart, which is a list that includes information about the conditional densities in its $preds slot. In this case, the predictions take the form of an array with dimensions 3 (conditional densities) x 500 (evaluation points) x 5000 (posterior samples). Via the type argument, predict.drbart() can also be used to return conditional distribution functions, quantiles, and means.

Lastly, the plot.predict.drbart() method plots the estimated conditional densities:

plot(preds)

<img src="man/figures/README-plot-1.png" width="100%" /> We see that though the mean E(y ∣ x) does change as a function of x, it alone is an insufficient descriptor of the impact of x on the response y. DR-BART additionally reveals that with increasing x, the conditional density becomes more sharply peaked and less skewed.