On the Gaussian process limit of Bayesian Additive Regression Trees (2410.20289v1)

Abstract: Bayesian Additive Regression Trees (BART) is a nonparametric Bayesian regression technique of rising fame. It is a sum-of-decision-trees model, and is in some sense the Bayesian version of boosting. In the limit of infinite trees, it becomes equivalent to Gaussian process (GP) regression. This limit is known but has not yet led to any useful analysis or application. For the first time, I derive and compute the exact BART prior covariance function. With it I implement the infinite trees limit of BART as GP regression. Through empirical tests, I show that this limit is worse than standard BART in a fixed configuration, but also that tuning the hyperparameters in the natural GP way yields a competitive method, although a properly tuned BART is still superior. The advantage of using a GP surrogate of BART is the analytical likelihood, which simplifies model building and sidesteps the complex BART MCMC. More generally, this study opens new ways to understand and develop BART and GP regression. The implementation of BART as GP is available in the Python package https://github.com/Gattocrucco/lsqfitgp .

• The paper derives the exact BART prior covariance function under the infinite tree limit, bridging BART with GP regression.
• Empirical tests reveal that unadjusted GP implementations underperform compared to standard BART, but tuned hyperparameters enable competitive performance.
• The analytical likelihood available in the GP framework simplifies model construction by reducing the need for complex MCMC procedures.

Overview of the Gaussian Process Limit of Bayesian Additive Regression Trees

This paper presents a paper on the Gaussian process (GP) limit of Bayesian Additive Regression Trees (BART), a method notable for its nonparametric Bayesian regression capabilities. BART functions by combining decision trees, and in the theoretical limit of an infinite number of trees, it converges to GP regression. This limit, although known, had not previously yielded substantial analytical or practical insights.

The author derives the exact BART prior covariance function and explores its implementation in the GP framework. Despite empirical results showing that this infinite tree limit is inferior to a standard BART with fixed configurations, optimally tuning hyperparameters in the GP setting can yield a competitive methodology. Nevertheless, a well-adjusted standard BART remains superior.

The advantage of transitioning BART into a GP setting is the availability of an analytical likelihood, which simplifies model construction and negates the complexity of BART's MCMC procedures. This work proposes novel directions for enhancing BART and GP regression methods.

Theoretical Development

The paper offers a formal derivation of the BART prior covariance function. Through recursive computation, an expression characterizing the correlation function of BART is provided, addressing the need for efficient and accurate evaluation to be practical in GP regression. This marks a significant advance over existing approximations that oversimplify these calculations.

Empirical Comparisons

Empirical evaluations demonstrate that the BART infinite tree GP implementation, without additional tuning, performs worse than standard BART configurations. However, incorporating hyperparameter tuning—particularly in the GP manner—results in a performance competitive with BART. A surprising insight is that mean and covariance function hyperparameters critically influence the balance between model flexibility and fit.

Practical Implications

The GP characterization of BART offers practical utility by potentially alleviating the computational demands typically associated with MCMC methods in hierarchical models. The transition to GP allows for a more straightforward model expansion and integration with other statistical methods, offering a practical alternative to standard BART in situations demanding explicit likelihood formulations.

Future Directions

Investigating kernel development further, particularly examining if BART's complex structure can be approached or exceeded through new, sophisticated GP kernels, holds promise. There are also implications for integrating broader GP methodologies into areas traditionally dominated by BART, offering a more nuanced assessment of model structures like the impact of tree depth on predictive performance.

In conclusion, while the GP surrogate does not outperform the finely tuned standard BART, this research represents a step towards a more comprehensive understanding of BART's theoretical underpinnings and further extends its applicability in regression tasks. The potential for a unified approach to tuning hyperparameters in both BART and GP frameworks could offer new insights into the adaptability and optimization of nonparametric Bayesian models.