A note on Bayesian R-squared for generalized additive mixed models

TL;DR

This paper introduces a new Bayesian framework for decomposing variance in generalized additive mixed models, providing a rigorous and extended definition of Bayesian R-squared and a method to assess individual model contributions.

Contribution

It proposes a novel Bayesian variance decomposition and extends the Bayesian R-squared concept to a broader class of models, including a partial R-squared for individual terms.

Findings

The new Bayesian R-squared aligns with previous intuitive definitions.

The framework allows for variance decomposition into explained and unexplained parts.

A partial R-squared quantifies the contribution of individual model terms.

Abstract

We present a novel Bayesian framework to decompose the posterior predictive variance in a fitted Generalized Additive Mixed Model (GAMM) into explained and unexplained components. This decomposition enables a rigorous definition of Bayesian $R^{2}$. We show that the new definition aligns with the intuitive Bayesian $R^{2}$ proposed by Gelman, Goodrich, Gabry, and Vehtari (2019) [\emph{The American Statistician}, \textbf{73}(3), 307-309], but extends its applicability to a broader class of models. Furthermore, we introduce a partial Bayesian $R^{2}$ to quantify the contribution of individual model terms to the explained variation in the posterior predictions