Calibrate the scale parameter on an inverse gamma prior for the global error variance as in Chipman et al (2022)

Chipman, H., George, E., Hahn, R., McCulloch, R., Pratola, M. and Sparapani, R. (2022). Bayesian Additive Regression Trees, Computational Approaches. In Wiley StatsRef: Statistics Reference Online (eds N. Balakrishnan, T. Colton, B. Everitt, W. Piegorsch, F. Ruggeri and J.L. Teugels). https://doi.org/10.1002/9781118445112.stat08288

Usage

calibrateInverseGammaErrorVariance(
  y,
  X,
  W = NULL,
  nu = 3,
  quant = 0.9,
  standardize = TRUE
)

Arguments

y

Outcome to be modeled using BART, BCF or another nonparametric ensemble method.

X

Covariates to be used to partition trees in an ensemble or series of ensemble.

W

(Optional) Basis used to define a "leaf regression" model for each decision tree. The "classic" BART model assumes a constant leaf parameter, which is equivalent to a "leaf regression" on a basis of all ones, though it is not necessary to pass a vector of ones, here or to the BART function. Default: NULL.

nu

The shape parameter for the global error variance's IG prior. The scale parameter in the Sparapani et al (2021) parameterization is defined as nu*lambda where lambda is the output of this function. Default: 3.

quant

(Optional) Quantile of the inverse gamma prior distribution represented by a linear-regression-based overestimate of sigma^2. Default: 0.9.

standardize

(Optional) Whether or not outcome should be standardized ((y-mean(y))/sd(y)) before calibration of lambda. Default: TRUE.

Value

Value of lambda which determines the scale parameter of the global error variance prior (sigma^2 ~ IG(nu,nu*lambda))

Examples

n <- 100
p <- 5
X <- matrix(runif(n*p), ncol = p)
y <- 10*X[,1] - 20*X[,2] + rnorm(n)
nu <- 3
lambda <- calibrateInverseGammaErrorVariance(y, X, nu = nu)
sigma2hat <- mean(resid(lm(y~X))^2)
mean(var(y)/rgamma(100000, nu, rate = nu*lambda) < sigma2hat)
#> [1] 0.90026