arXiv subjects

Matthew T. Pratola

Publications and source records attributed to Matthew T. Pratola.

11 recordsLinked to original sources

arXiv2026

Combining Climate Models using Bayesian Regression Trees and Random Paths

General circulation models (GCMs) are essential tools for climate studies. Such climate models may have varying accuracy across the input domain, but no model is uniformly best. One can improve climate model prediction performance by integrating multiple models using input-dependent weights. Weight functions modeled using Bayesian Additive Regression Trees (BART) were recently shown to be useful in nuclear physics applications. However, a restriction of that approach was the piecewise constant weight functions. To smoothly integrate multiple climate models, we propose a new tree-based model, Random Path BART (RPBART), that incorporates random path assignments in BART to produce smooth weight functions and smooth predictions, all in a matrix-free formulation. RPBART requires a more complex prior specification, for which we introduce a semivariogram to guide hyperparameter selection. This approach is easy to interpret, computationally cheap, and avoids expensive cross-validation. Finally, we propose a posterior projection technique to enable detailed analysis of the fitted weight functions. This allows us to identify a sparse set of climate models that recovers the underlying system within a given spatial region as well as quantifying model discrepancy given the available model set. Our method is demonstrated on an ensemble of 8 GCMs modeling average monthly surface temperature.

stat.ME↗

arXiv2025

Bayesian model-data comparison incorporating theoretical uncertainties

Accurate comparisons between theoretical models and experimental data are critical for scientific progress. However, inferred physical model parameters can vary significantly with the chosen physics model, highlighting the importance of properly accounting for theoretical uncertainties. In this Letter, we present a Bayesian framework that explicitly quantifies these uncertainties by statistically modeling theory errors, guided by qualitative knowledge of a theory's varying reliability across the input domain. We demonstrate the effectiveness of this approach using two systems: a simple ball drop experiment and multi-stage heavy-ion simulations. In both cases incorporating model discrepancy leads to improved parameter estimates, with systematic improvements observed as additional experimental observables are integrated.

hep-ph↗

arXiv2025

Estimating Shapley Effects in Big-Data Emulation and Regression Settings using Bayesian Additive Regression Trees

Shapley effects are a particularly interpretable approach to assessing how a function depends on its various inputs. The existing literature contains various estimators for this class of sensitivity indices in the context of nonparametric regression where the function is observed with noise, but there does not seem to be an estimator that is computationally tractable for input dimensions in the hundreds scale. This article provides such an estimator that is computationally tractable on this scale. The estimator uses a metamodel-based approach by first fitting a Bayesian Additive Regression Trees model which is then used to compute Shapley-effect estimates. This article also establishes a theoretical guarantee of posterior consistency on a large function class for this Shapley-effect estimator. Finally, this paper explores the performance of these Shapley-effect estimators on four different test functions for various input dimensions, including $p=500$.

stat.ME↗

arXiv2023

Sharded Bayesian Additive Regression Trees

In this paper we develop the randomized Sharded Bayesian Additive Regression Trees (SBT) model. We introduce a randomization auxiliary variable and a sharding tree to decide partitioning of data, and fit each partition component to a sub-model using Bayesian Additive Regression Tree (BART). By observing that the optimal design of a sharding tree can determine optimal sharding for sub-models on a product space, we introduce an intersection tree structure to completely specify both the sharding and modeling using only tree structures. In addition to experiments, we also derive the theoretical optimal weights for minimizing posterior contractions and prove the worst-case complexity of SBT.

stat.ML↗

arXiv2023

Influential Observations in Bayesian Regression Tree Models

BCART (Bayesian Classification and Regression Trees) and BART (Bayesian Additive Regression Trees) are popular Bayesian regression models widely applicable in modern regression problems. Their popularity is intimately tied to the ability to flexibly model complex responses depending on high-dimensional inputs while simultaneously being able to quantify uncertainties. This ability to quantify uncertainties is key, as it allows researchers to perform appropriate inferential analyses in settings that have generally been too difficult to handle using the Bayesian approach. However, surprisingly little work has been done to evaluate the sensitivity of these modern regression models to violations of modeling assumptions. In particular, we will consider influential observations, which one reasonably would imagine to be common -- or at least a concern -- in the big-data setting. In this paper, we consider both the problem of detecting influential observations and adjusting predictions to not be unduly affected by such potentially problematic data. We consider three detection diagnostics for Bayesian tree models, one an analogue of Cook's distance and the others taking the form of a divergence measure and a conditional predictive density metric, and then propose an importance sampling algorithm to re-weight previously sampled posterior draws so as to remove the effects of influential data in a computationally efficient manner. Finally, our methods are demonstrated on real-world data where blind application of the models can lead to poor predictions and inference.

stat.ME↗

arXiv2023

Model Mixing Using Bayesian Additive Regression Trees

In modern computer experiment applications, one often encounters the situation where various models of a physical system are considered, each implemented as a simulator on a computer. An important question in such a setting is determining the best simulator, or the best combination of simulators, to use for prediction and inference. Bayesian model averaging (BMA) and stacking are two statistical approaches used to account for model uncertainty by aggregating a set of predictions through a simple linear combination or weighted average. Bayesian model mixing (BMM) extends these ideas to capture the localized behavior of each simulator by defining input-dependent weights. One possibility is to define the relationship between inputs and the weight functions using a flexible non-parametric model that learns the local strengths and weaknesses of each simulator. This paper proposes a BMM model based on Bayesian Additive Regression Trees (BART). The proposed methodology is applied to combine predictions from Effective Field Theories (EFTs) associated with a motivating nuclear physics application.

stat.ME↗

arXiv2022

Optimal Design Emulators: A Point Process Approach

Design of experiments is a fundamental topic in applied statistics with a long history. Yet its application is often limited by the complexity and costliness of constructing experimental designs, which involve searching a high-dimensional input space and evaluating computationally expensive criterion functions. In this work, we introduce a novel approach to the challenging design problem. We will take a probabilistic view of the problem by representing the optimal design as being one element (or a subset of elements) of a probability space. Given a suitable distribution on this space, a generative point process can be specified from which stochastic design realizations can be drawn. In particular, we describe a scenario where the classical entropy-optimal design for Gaussian Process regression coincides with the mode of a particular point process. We conclude with outlining an algorithm for drawing such design realizations, its extension to sequential designs, and applying the techniques developed to constructing designs for Stochastic Gradient Descent and Gaussian process regression.

stat.ME↗

arXiv2021

Using BART to Perform Pareto Optimization and Quantify its Uncertainties

Techniques to reduce the energy burden of an industrial ecosystem often require solving a multiobjective optimization problem. However, collecting experimental data can often be either expensive or time-consuming. In such cases, statistical methods can be helpful. This article proposes Pareto Front (PF) and Pareto Set (PS) estimation methods using Bayesian Additive Regression Trees (BART), which is a non-parametric model whose assumptions are typically less restrictive than popular alternatives, such as Gaussian Processes (GPs). These less restrictive assumptions allow BART to handle scenarios (e.g. high-dimensional input spaces, nonsmooth responses, large datasets) that GPs find difficult. The performance of our BART-based method is compared to a GP-based method using analytic test functions, demonstrating convincing advantages. Finally, our BART-based methodology is applied to a motivating engineering problem. Supplementary materials, which include a theorem proof, algorithms, and R code, for this article are available online.

cs.LG↗

arXiv2020

Assessing variable activity for Bayesian regression trees

Bayesian Additive Regression Trees (BART) are non-parametric models that can capture complex exogenous variable effects. In any regression problem, it is often of interest to learn which variables are most active. Variable activity in BART is usually measured by counting the number of times a tree splits for each variable. Such one-way counts have the advantage of fast computations. Despite their convenience, one-way counts have several issues. They are statistically unjustified, cannot distinguish between main effects and interaction effects, and become inflated when measuring interaction effects. An alternative method well-established in the literature is Sobol' indices, a variance-based global sensitivity analysis technique. However, these indices often require Monte Carlo integration, which can be computationally expensive. This paper provides analytic expressions for Sobol' indices for BART posterior samples. These expressions are easy to interpret and are computationally feasible. Furthermore, we will show a fascinating connection between first-order (main-effects) Sobol' indices and one-way counts. We also introduce a novel ranking method, and use this to demonstrate that the proposed indices preserve the Sobol'-based rank order of variable importance. Finally, we compare these methods using analytic test functions and the En-ROADS climate impacts simulator.

stat.ME↗

arXiv2020

Sparse Additive Gaussian Process Regression

In this paper we introduce a novel model for Gaussian process (GP) regression in the fully Bayesian setting. Motivated by the ideas of sparsification, localization and Bayesian additive modeling, our model is built around a recursive partitioning (RP) scheme. Within each RP partition, a sparse GP (SGP) regression model is fitted. A Bayesian additive framework then combines multiple layers of partitioned SGPs, capturing both global trends and local refinements with efficient computations. The model addresses both the problem of efficiency in fitting a full Gaussian process regression model and the problem of prediction performance associated with a single SGP. Our approach mitigates the issue of pseudo-input selection and avoids the need for complex inter-block correlations in existing methods. The crucial trade-off becomes choosing between many simpler local model components or fewer complex global model components, which the practitioner can sensibly tune. Implementation is via a Metropolis-Hasting Markov chain Monte-Carlo algorithm with Bayesian back-fitting. We compare our model against popular alternatives on simulated and real datasets, and find the performance is competitive, while the fully Bayesian procedure enables the quantification of model uncertainties.

math.ST↗

arXiv2013

Parallel Bayesian Additive Regression Trees

Bayesian Additive Regression Trees (BART) is a Bayesian approach to flexible non-linear regression which has been shown to be competitive with the best modern predictive methods such as those based on bagging and boosting. BART offers some advantages. For example, the stochastic search Markov Chain Monte Carlo (MCMC) algorithm can provide a more complete search of the model space and variation across MCMC draws can capture the level of uncertainty in the usual Bayesian way. The BART prior is robust in that reasonable results are typically obtained with a default prior specification. However, the publicly available implementation of the BART algorithm in the R package BayesTree is not fast enough to be considered interactive with over a thousand observations, and is unlikely to even run with 50,000 to 100,000 observations. In this paper we show how the BART algorithm may be modified and then computed using single program, multiple data (SPMD) parallel computation implemented using the Message Passing Interface (MPI) library. The approach scales nearly linearly in the number of processor cores, enabling the practitioner to perform statistical inference on massive datasets. Our approach can also handle datasets too massive to fit on any single data repository.

stat.CO↗