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SeBR: Semiparametric Bayesian Regression

Overview. Data transformations are a useful companion for parametric regression models. A well-chosen or learned transformation can greatly enhance the applicability of a given model, especially for data with irregular marginal features (e.g., multimodality, skewness) or various data domains (e.g., real-valued, positive, or compactly-supported data).

Given paired data (xi,yi) for i = 1, …, n, SeBR implements efficient and fully Bayesian inference for semiparametric regression models that incorporate (1) an unknown data transformation

g(yi) = zi

and (2) a useful parametric regression model

$$ z_i \stackrel{indep}{\sim} P_{Z \mid \theta, X = x_i} $$

with unknown parameters θ.

Examples. We focus on the following important special cases of PZ ∣ θ, X:

  1. The linear model is a natural starting point:

$$ z_i = x_i'\theta + \epsilon_i, \quad \epsilon_i \stackrel{iid}{\sim} N(0, \sigma_\epsilon^2) $$

The transformation g broadens the applicability of this useful class of models, including for positive or compactly-supported data, while PZ ∣ θ, X = x = N(x′θ,σϵ2).

  1. The quantile regression model replaces the Gaussian assumption in the linear model with an asymmetric Laplace distribution (ALD)

$$ z_i = x_i'\theta + \epsilon_i, \quad \epsilon_i \stackrel{iid}{\sim} ALD(\tau) $$

to target the τth quantile of z at x, or equivalently, the g−1(τ)th quantile of y at x. The ALD is quite often a very poor model for real data, especially when τ is near zero or one. The transformation g offers a pathway to significantly improve the model adequacy, while still targeting the desired quantile of the data.

  1. The Gaussian process (GP) model generalizes the linear model to include a nonparametric regression function,

$$ z_i = f_\theta(x_i) + \epsilon_i, \quad \epsilon_i \stackrel{iid}{\sim} N(0, \sigma_\epsilon^2) $$

where fθ is a GP and θ parameterizes the mean and covariance functions. Although GPs offer substantial flexibility for the regression function fθ, this model may be inadequate when y has irregular marginal features or a restricted domain (e.g., positive or compact).

Challenges: The goal is to provide fully Bayesian posterior inference for the unknowns (g,θ) and posterior predictive inference for future/unobserved data ỹ(x). We prefer a model and algorithm that offer both (i) flexible modeling of g and (ii) efficient posterior and predictive computations.

Innovations: Our approach (https://arxiv.org/abs/2306.05498) specifies a nonparametric model for g, yet also provides Monte Carlo (not MCMC) sampling for the posterior and predictive distributions. As a result, we control the approximation accuracy via the number of simulations, but do not require the lengthy runs, burn-in periods, convergence diagnostics, or inefficiency factors that accompany MCMC. The Monte Carlo sampling is typically quite fast.

Using SeBR

The package SeBR is installed and loaded as follows:

# CRAN version:
# install.packages("SeBR")

# Development version: 
# devtools::install_github("drkowal/SeBR")
library(SeBR)

The main functions in SeBR are:

  • sblm(): Monte Carlo sampling for posterior and predictive inference with the semiparametric Bayesian linear model;

  • sbsm(): Monte Carlo sampling for posterior and predictive inference with the semiparametric Bayesian spline model, which replaces the linear model with a spline for nonlinear modeling of x ∈ ℝ;

  • sbqr(): blocked Gibbs sampling for posterior and predictive inference with the semiparametric Bayesian quantile regression; and

  • sbgp(): Monte Carlo sampling for predictive inference with the semiparametric Bayesian Gaussian process model.

Each function returns a point estimate of θ (coefficients), point predictions at some specified testing points (fitted.values), posterior samples of the transformation g (post_g), and posterior predictive samples of ỹ(x) at the testing points (post_ypred), as well as other function-specific quantities (e.g., posterior draws of θ, post_theta). The calls coef() and fitted() extract the point estimates and point predictions, respectively.

Note: The package also includes Box-Cox variants of these functions, i.e., restricting g to the (signed) Box-Cox parametric family g(t;λ) = {sign(t)|t|λ − 1}/λ with known or unknown λ. The parametric transformation is less flexible, especially for irregular marginals or restricted domains, and requires MCMC sampling. These functions (e.g., blm_bc(), etc.) are primarily for benchmarking.

Detailed documentation and examples are available at https://drkowal.github.io/SeBR/.