Assessing variable activity for Bayesian regression trees
Akira Horiguchi, Matthew T. Pratola, Thomas J. Santner
备注：
46 pages, 8 figures, submitted to the special issue "Recent Advances in Sensitivity Analysis of Model Outputs" in the Reliability Engineering and Safety System journal
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.
Selection bias is a serious potential problem for inference about relationships of scientific interest based on samples without well-defined probability sampling mechanisms. Motivated by the potential for selection bias in (a) estimated relationships of polygenic scores (PGSs) with phenotypes in genetic studies of volunteers, and (b) estimated differences in subgroup means in surveys of smartphone users, we derive novel measures of selection bias for estimates of the coefficients in linear and probit regression models fitted to non-probability samples, when aggregate-level auxiliary data are a…
Large scale study of primary school student performance relative to their LMS activity and socioeconomic demographics using a Bayesian Additive Regression Trees containing random effects
We propose two bounded comparison metrics that may be implemented to arbitrary dimensions in regression tasks. One quantifies the structure of uncertainty and the other quantifies the distribution of uncertainty. The structure metric assesses the similarity in shape and location of uncertainty with the true error, while the distribution metric quantifies the supported magnitudes between the two. We apply these metrics to Gaussian Processes (GPs), Ensemble Deep Neural Nets (DNNs), and Ensemble Deep Neural Operators (DNOs) on high-dimensional and nonlinear test cases. We find that comparing a mo…
Stellar photospheric activity is known to limit the detection and characterisation of extra-solar planets. In particular, the study of Earth-like planets around Sun-like stars requires data analysis methods that can accurately model the stellar activity phenomena affecting radial velocity (RV) measurements. Gaussian Process Regression Networks (GPRNs) offer a principled approach to the analysis of simultaneous time-series, combining the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian Processes. Using HARPS-N solar spectroscopic observations enc…
This paper evaluates the dynamic response of economic activity to shocks in uncertainty as percieved by agents.The study focuses on the comparison between the perception of economic uncertainty by manufacturers and consumers.Since uncertainty is not directly observable, we approximate it using the geometric discrepancy indicator of Claveria et al.(2019).This approach allows us quantifying the proportion of disagreement in business and consumer expectations of eleven European countries and the Euro Area.First, we compute three independent indices of discrepancy corresponding to three dimensions…
We consider Bayesian inference in inverse regression problems where the objective is to infer about unobserved covariates from observed responses and covariates. We establish posterior consistency of such unobserved covariates in Bayesian inverse regression problemsunder appropriate priors in a leave-one-out cross-validation setup. We relate this to posterior consistency of inverse reference distributions (Bhattacharya (2013)) for assessing model adequacy. We illustrate our theory and methods with various examples of Bayesian inverse regression, along with adequate simulation experiments.
The R package BNSP provides a unified framework for semiparametric location-scale regression and stochastic search variable selection. The statistical methodology that the package is built upon utilizes basis function expansions to represent semiparametric covariate effects in the mean and variance functions, and spike-slab priors to perform selection and regularization of the estimated effects. In addition to the main function that performs posterior sampling, the package includes functions for assessing convergence of the sampler, summarizing model fits, visualizing covariate effects and obt…
University of Edinburgh（爱丁堡大学）；California Institute of Technology（加州理工学院）
Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires O(n^2) space and O(n^3) time for a dataset of n examples. Several approximation methods have been proposed, but there is a lack of understanding of the relative merits of the different approximations, and in what situations they are most useful. We recommend assessing the quality of the predictions obtained as a function of the compute time taken, and comparing to standard baselines (e.g., Subse…
Additive models are flexible regression tools that handle linear as well as nonlinear terms. The latter are typically modelled via smoothing splines. Additive mixed models extend additive models to include random terms when the data are sampled according to cluster designs (e.g., longitudinal). These models find applications in the study of phenomena like growth, certain disease mechanisms and energy consumption in humans, when repeated measurements are available. In this paper, we propose a novel additive mixed model for quantile regression. Our methods are motivated by an application to phys…
We consider the counting rate estimation of an unknown radioactive source, which emits photons at times modeled by an homogeneous Poisson process. A spectrometer converts the energy of incoming photons into electrical pulses, whose number provides a rough estimate of the intensity of the Poisson process. When the activity of the source is high, a physical phenomenon known as pileup effect distorts direct measurements, resulting in a significant bias to the standard estimators of the source activities used so far in the field. We show in this paper that the problem of counting rate estimation c…
Variable Selection in Functional Linear Quantile Regression for Identifying Associations between Daily Patterns of Physical Activity and Cognitive Function
Wearable devices such as the ActiGraph are now commonly used in health studies to monitor or track physical activity. This trend aligns well with the growing need to accurately assess the effects of physical activity on health outcomes such as obesity. When accessing the association between these device-based physical activity measures with health outcomes such as body mass index, the device-based data is considered functions, while the outcome is a scalar-valued. The regression model applied in these settings is the scalar-on-function regression (SoFR). Most estimation approaches in SoFR assu…
This paper presents a novel nonlinear regression model for estimating heterogeneous treatment effects from observational data, geared specifically towards situations with small effect sizes, heterogeneous effects, and strong confounding. Standard nonlinear regression models, which may work quite well for prediction, have two notable weaknesses when used to estimate heterogeneous treatment effects. First, they can yield badly biased estimates of treatment effects when fit to data with strong confounding. The Bayesian causal forest model presented in this paper avoids this problem by directly in…
Accelerometers enable an objective measurement of physical activity levels among groups of individuals in free-living environments, providing high-resolution detail about physical activity changes at different time scales. Current approaches used in the literature for analyzing such data typically employ summary measures such as total inactivity time or compositional metrics. However, at the conceptual level, these methods have the potential disadvantage of discarding important information from recorded data when calculating these summaries and metrics since these typically depend on cut-offs …
The causal effect of an intervention (treatment/exposure) on an outcome can be estimated by: i) specifying knowledge about the data-generating process; ii) assessing under what assumptions a target quantity, such as for example a causal odds ratio, can be identified given the specified knowledge (and given the measured data); and then, iii) using appropriate statistical estimation techniques to estimate the desired parameter of interest. As regression is the cornerstone of statistical analysis, it seems obvious to ask: is it appropriate to use estimated regression parameters for causal effect …
Sibling fixed effects (FE) models are useful for estimating causal treatment effects while offsetting unobserved sibling-invariant confounding. However, treatment estimates are biased if an individual's outcome affects their sibling's outcome. We propose a robustness test for assessing the presence of outcome-to-outcome interference in linear two-sibling FE models. We regress a gain-score--the difference between siblings' continuous outcomes--on both siblings' treatments and on a pre-treatment observed FE. Under certain restrictions, the observed FE's partial regression coefficient signals the…
Spatial Capture-Recapture With Penalized Regression Splines to Flexibly Model Wildlife Density and Distribution
Utrecht University（乌得勒支大学）；University of Glasgow（格拉斯哥大学）；University of St Andrews（圣安德鲁斯大学）；World Wide Fund for Nature（世界自然基金会）；University of Auckland（奥克兰大学）
