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

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}, $\mathbf{\text{73}}$(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