arXiv subjects

Veronika Rockova

Publications and source records attributed to Veronika Rockova.

27 records · Page 2Linked to original sources

arXiv2019

Posterior Concentration for Bayesian Regression Trees and Forests

Since their inception in the 1980's, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction of the regression surface, where the bins correspond to terminal nodes of recursive partitioning. Trees are powerful, yet susceptible to over-fitting. Strategies against overfitting have traditionally relied on pruning greedily grown trees. The Bayesian framework offers an alternative remedy against overfitting through priors. Roughly speaking, a good prior charges smaller trees where overfitting does not occur. While the consistency of random histograms, trees and their ensembles has been studied quite extensively, the theoretical understanding of the Bayesian counterparts has been missing. In this paper, we take a step towards understanding why/when do Bayesian trees and their ensembles not overfit. To address this question, we study the speed at which the posterior concentrates around the true smooth regression function. We propose a spike-and-tree variant of the popular Bayesian CART prior and establish new theoretical results showing that regression trees (and their ensembles) (a) are capable of recovering smooth regression surfaces, achieving optimal rates up to a log factor, (b) can adapt to the unknown level of smoothness and (c) can perform effective dimension reduction when p>n. These results provide a piece of missing theoretical evidence explaining why Bayesian trees (and additive variants thereof) have worked so well in practice.

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arXiv2019

On Semi-parametric Bernstein-von Mises Theorems for BART

Few methods in Bayesian non-parametric statistics/ machine learning have received as much attention as Bayesian Additive Regression Trees (BART). While BART is now routinely performed for prediction tasks, its theoretical properties began to be understood only very recently. In this work, we continue the theoretical investigation of BART initiated by Rockova and van der Pas (2017). In particular, we study the Bernstein-von Mises (BvM) phenomenon (i.e. asymptotic normality) for smooth linear functionals of the regression surface within the framework of non-parametric regression with fixed covariates. As with other adaptive priors, the BvM phenomenon may fail when the regularities of the functional and the truth are not compatible. To overcome the curse of adaptivity under hierarchical priors, we induce a self-similarity assumption to ensure convergence towards a single Gaussian distribution as opposed to a Gaussian mixture. Similar qualitative restrictions on the functional parameter are known to be necessary for adaptive inference. Many machine learning methods lack coherent probabilistic mechanisms for gauging uncertainty. BART readily provides such quantification via posterior credible sets. The BvM theorem implies that the credible sets are also confidence regions with the same asymptotic coverage. This paper presents the first asymptotic normality result for BART priors, providing another piece of evidence that BART is a valid tool from a frequentist point of view.

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arXiv2018

Dynamic Sparse Factor Analysis

Its conceptual appeal and effectiveness has made latent factor modeling an indispensable tool for multivariate analysis. Despite its popularity across many fields, there are outstanding methodological challenges that have hampered practical deployments. One major challenge is the selection of the number of factors, which is exacerbated for dynamic factor models, where factors can disappear, emerge, and/or reoccur over time. Existing tools that assume a fixed number of factors may provide a misguided representation of the data mechanism, especially when the number of factors is crudely misspecified. Another challenge is the interpretability of the factor structure, which is often regarded as an unattainable objective due to the lack of identifiability. Motivated by a topical macroeconomic application, we develop a flexible Bayesian method for dynamic factor analysis (DFA) that can simultaneously accommodate a time-varying number of factors and enhance interpretability without strict identifiability constraints. To this end, we turn to dynamic sparsity by employing Dynamic Spike-and-Slab (DSS) priors within DFA. Scalable Bayesian EM estimation is proposed for fast posterior mode identification via rotations to sparsity, enabling Bayesian data analysis at scales that would have been previously time-consuming. We study a large-scale balanced panel of macroeconomic variables covering multiple facets of the US economy, with a focus on the Great Recession, to highlight the efficacy and usefulness of our proposed method.

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arXiv2018

Variance prior forms for high-dimensional Bayesian variable selection

Consider the problem of high dimensional variable selection for the Gaussian linear model when the unknown error variance is also of interest. In this paper, we show that the use of conjugate shrinkage priors for Bayesian variable selection can have detrimental consequences for such variance estimation. Such priors are often motivated by the invariance argument of Jeffreys (1961). Revisiting this work, however, we highlight a caveat that Jeffreys himself noticed; namely that biased estimators can result from inducing dependence between parameters a priori. In a similar way, we show that conjugate priors for linear regression, which induce prior dependence, can lead to such underestimation in the Bayesian high-dimensional regression setting. Following Jeffreys, we recommend as a remedy to treat regression coefficients and the error variance as independent a priori. Using such an independence prior framework, we extend the Spike-and-Slab Lasso of Rockova and George (2018) to the unknown variance case. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on simulated data. On the protein activity dataset of Clyde and Parmigiani (1998), the Spike-and-Slab Lasso with unknown variance achieves lower cross-validation error than alternative penalized likelihood methods, demonstrating the gains in predictive accuracy afforded by simultaneous error variance estimation.

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arXiv2018

On Theory for BART

Ensemble learning is a statistical paradigm built on the premise that many weak learners can perform exceptionally well when deployed collectively. The BART method of Chipman et al. (2010) is a prominent example of Bayesian ensemble learning, where each learner is a tree. Due to its impressive performance, BART has received a lot of attention from practitioners. Despite its wide popularity, however, theoretical studies of BART have begun emerging only very recently. Laying the foundations for the theoretical analysis of Bayesian forests, Rockova and van der Pas (2017) showed optimal posterior concentration under conditionally uniform tree priors. These priors deviate from the actual priors implemented in BART. Here, we study the exact BART prior and propose a simple modification so that it also enjoys optimality properties. To this end, we dive into branching process theory. We obtain tail bounds for the distribution of total progeny under heterogeneous Galton-Watson (GW) processes exploiting their connection to random walks. We conclude with a result stating the optimal rate of posterior convergence for BART.

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arXiv2018

The Median Probability Model and Correlated Variables

The median probability model (MPM) Barbieri and Berger (2004) is defined as the model consisting of those variables whose marginal posterior probability of inclusion is at least 0.5. The MPM rule yields the best single model for prediction in orthogonal and nested correlated designs. This result was originally conceived under a specific class of priors, such as the point mass mixtures of non-informative and g-type priors. The MPM rule, however, has become so very popular that it is now being deployed for a wider variety of priors and under correlated designs, where the properties of MPM are not yet completely understood. The main thrust of this work is to shed light on properties of MPM in these contexts by (a) characterizing situations when MPM is still safe under correlated designs, (b) providing significant generalizations of MPM to a broader class of priors (such as continuous spike-and-slab priors). We also provide new supporting evidence for the suitability of g-priors, as opposed to independent product priors, using new predictive matching arguments. Furthermore, we emphasize the importance of prior model probabilities and highlight the merits of non-uniform prior probability assignments using the notion of model aggregates.

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arXiv2018

Simultaneous Variable and Covariance Selection with the Multivariate Spike-and-Slab Lasso

We propose a Bayesian procedure for simultaneous variable and covariance selection using continuous spike-and-slab priors in multivariate linear regression models where q possibly correlated responses are regressed onto p predictors. Rather than relying on a stochastic search through the high-dimensional model space, we develop an ECM algorithm similar to the EMVS procedure of Rockova & George (2014) targeting modal estimates of the matrix of regression coefficients and residual precision matrix. Varying the scale of the continuous spike densities facilitates dynamic posterior exploration and allows us to filter out negligible regression coefficients and partial covariances gradually. Our method is seen to substantially outperform regularization competitors on simulated data. We demonstrate our method with a re-examination of data from a recent observational study of the effect of playing high school football on several later-life cognition, psychological, and socio-economic outcomes.

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arXiv2018

Posterior Concentration for Sparse Deep Learning

Spike-and-Slab Deep Learning (SS-DL) is a fully Bayesian alternative to Dropout for improving generalizability of deep ReLU networks. This new type of regularization enables provable recovery of smooth input-output maps with unknown levels of smoothness. Indeed, we show that the posterior distribution concentrates at the near minimax rate for $α$-Hölder smooth maps, performing as well as if we knew the smoothness level $α$ ahead of time. Our result sheds light on architecture design for deep neural networks, namely the choice of depth, width and sparsity level. These network attributes typically depend on unknown smoothness in order to be optimal. We obviate this constraint with the fully Bayes construction. As an aside, we show that SS-DL does not overfit in the sense that the posterior concentrates on smaller networks with fewer (up to the optimal number of) nodes and links. Our results provide new theoretical justifications for deep ReLU networks from a Bayesian point of view.

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arXiv2017

Bayesian Dyadic Trees and Histograms for Regression

Many machine learning tools for regression are based on recursive partitioning of the covariate space into smaller regions, where the regression function can be estimated locally. Among these, regression trees and their ensembles have demonstrated impressive empirical performance. In this work, we shed light on the machinery behind Bayesian variants of these methods. In particular, we study Bayesian regression histograms, such as Bayesian dyadic trees, in the simple regression case with just one predictor. We focus on the reconstruction of regression surfaces that are piecewise constant, where the number of jumps is unknown. We show that with suitably designed priors, posterior distributions concentrate around the true step regression function at a near-minimax rate. These results do not require the knowledge of the true number of steps, nor the width of the true partitioning cells. Thus, Bayesian dyadic regression trees are fully adaptive and can recover the true piecewise regression function nearly as well as if we knew the exact number and location of jumps. Our results constitute the first step towards understanding why Bayesian trees and their ensembles have worked so well in practice. As an aside, we discuss prior distributions on balanced interval partitions and how they relate to an old problem in geometric probability. Namely, we relate the probability of covering the circumference of a circle with random arcs whose endpoints are confined to a grid, a new variant of the original problem.

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