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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 \((x_i,y_i)\) for \(i=1,\ldots,n\), SeBR implements efficient and fully Bayesian inference for semiparametric regression models that incorporate (1) an unknown data transformation

\[ g(y_i) = z_i \]

and (2) a useful parametric regression model

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

with unknown parameters \(\theta\).

Examples. We focus on the following important special cases of \(P_{Z \mid \theta, X}\):

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 \(P_{Z \mid \theta, X=x} = N(x'\theta, \sigma_\epsilon^2)\).

2. 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 \(\tau\)th quantile of \(z\) at \(x\), or equivalently, the \(g^{-1}(\tau)\)th quantile of \(y\) at \(x\). The ALD is quite often a very poor model for real data, especially when \(\tau\) 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.

3. 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 \(f_\theta\) is a GP and \(\theta\) parameterizes the mean and covariance functions. Although GPs offer substantial flexibility for the regression function \(f_\theta\), 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 \((g, \theta)\) and posterior predictive inference for future/unobserved data \(\tilde y(x)\). 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:

# install.packages("devtools")
# 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 \(x \in \mathbb{R}\);

• 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 \(\theta\) (coefficients), point predictions at some specified testing points (fitted.values), posterior samples of the transformation \(g\) (post_g), and posterior predictive samples of \(\tilde y(x)\) at the testing points (post_ypred), as well as other function-specific quantities (e.g., posterior draws of \(\theta\), 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 \(g(t; \lambda) = \{\mbox{sign}(t) \vert t \vert^\lambda - 1\}/\lambda\) with known or unknown \(\lambda\). 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/.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.