P. Richard Hahn

Nikolay Krantsevich, Jingyu He, P. Richard Hahn

Determining subgroups that respond especially well (or poorly) to specific interventions (medical or policy) requires new supervised learning methods tailored specifically for causal inference. Bayesian Causal Forest (BCF) is a recent method that has been documented to perform well on data generating processes with strong confounding of the sort that is plausible in many applications. This paper develops a novel algorithm for fitting the BCF model, which is more efficient than the previously available Gibbs sampler. The new algorithm can be used to initialize independent chains of the existing Gibbs sampler leading to better posterior exploration and coverage of the associated interval estimates in simulation studies. The new algorithm is compared to related approaches via simulation studies as well as an empirical analysis.

Demetrios Papakostas, Andrew Herren, P. Richard Hahn et al.

Causal inference has gained much popularity in recent years, with interests ranging from academic, to industrial, to educational, and all in between. Concurrently, the study and usage of neural networks has also grown profoundly (albeit at a far faster rate). What we aim to do in this blog write-up is demonstrate a Neural Network causal inference architecture. We develop a fully connected neural network implementation of the popular Bayesian Causal Forest algorithm, a state of the art tree based method for estimating heterogeneous treatment effects. We compare our implementation to existing neural network causal inference methodologies, showing improvements in performance in simulation settings. We apply our method to a dataset examining the effect of stress on sleep.

Andrew Herren, P. Richard Hahn

A straightforward application of semi-supervised machine learning to the problem of treatment effect estimation would be to consider data as "unlabeled" if treatment assignment and covariates are observed but outcomes are unobserved. According to this formulation, large unlabeled data sets could be used to estimate a high dimensional propensity function and causal inference using a much smaller labeled data set could proceed via weighted estimators using the learned propensity scores. In the limiting case of infinite unlabeled data, one may estimate the high dimensional propensity function exactly. However, longstanding advice in the causal inference community suggests that estimated propensity scores (from labeled data alone) are actually preferable to true propensity scores, implying that the unlabeled data is actually useless in this context. In this paper we examine this paradox and propose a simple procedure that reconciles the strong intuition that a known propensity functions should be useful for estimating treatment effects with the previous literature suggesting otherwise. Further, simulation studies suggest that direct regression may be preferable to inverse-propensity weight estimators in many circumstances.

Jingyu He, P. Richard Hahn

This paper develops a novel stochastic tree ensemble method for nonlinear regression, which we refer to as XBART, short for Accelerated Bayesian Additive Regression Trees. By combining regularization and stochastic search strategies from Bayesian modeling with computationally efficient techniques from recursive partitioning approaches, the new method attains state-of-the-art performance: in many settings it is both faster and more accurate than the widely-used XGBoost algorithm. Via careful simulation studies, we demonstrate that our new approach provides accurate point-wise estimates of the mean function and does so faster than popular alternatives, such as BART, XGBoost and neural networks (using Keras). We also prove a number of basic theoretical results about the new algorithm, including consistency of the single tree version of the model and stationarity of the Markov chain produced by the ensemble version. Furthermore, we demonstrate that initializing standard Bayesian additive regression trees Markov chain Monte Carlo (MCMC) at XBART-fitted trees considerably improves credible interval coverage and reduces total run-time.

Jingyu He, Saar Yalov, P. Richard Hahn

Bayesian additive regression trees (BART) (Chipman et. al., 2010) is a powerful predictive model that often outperforms alternative models at out-of-sample prediction. BART is especially well-suited to settings with unstructured predictor variables and substantial sources of unmeasured variation as is typical in the social, behavioral and health sciences. This paper develops a modified version of BART that is amenable to fast posterior estimation. We present a stochastic hill climbing algorithm that matches the remarkable predictive accuracy of previous BART implementations, but is many times faster and less memory intensive. Simulation studies show that the new method is comparable in computation time and more accurate at function estimation than both random forests and gradient boosting.

Ruocheng Guo, Lu Cheng, Jundong Li et al.

This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.

P. Richard Hahn, Jingyu He, Hedibert Lopes

This paper develops a slice sampler for Bayesian linear regression models with arbitrary priors. The new sampler has two advantages over current approaches. One, it is faster than many custom implementations that rely on auxiliary latent variables, if the number of regressors is large. Two, it can be used with any prior with a density function that can be evaluated up to a normalizing constant, making it ideal for investigating the properties of new shrinkage priors without having to develop custom sampling algorithms. The new sampler takes advantage of the special structure of the linear regression likelihood, allowing it to produce better effective sample size per second than common alternative approaches.

Tom LaGatta, P. Richard Hahn

We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, we construct a version of the Ordinary Least Squares estimator in absolute generality. Our Gauss-Markov theorem demonstrates that OLS is a "best linear unbiased estimator", extending the classical result. We construct a stochastic version of the OLS estimator, which is a continuous disintegration exactly for the class of "uncorrelated implies independent" (UII) measures. As a consequence, Gaussian measures always exhibit continuous disintegrations through continuous linear maps, extending a theorem of the first author. Applying this framework to some problems in machine learning, we prove a useful representation theorem for covariance tensors, and show that OLS defines a good kriging predictor for vector-valued arrays on general index spaces. We also construct a support-vector machine classifier in this setting. We hope that our article shines light on some deeper connections between probability theory, statistics and machine learning, and may serve as a point of intersection for these three communities.