A global priority for the behavioural sciences is to develop cost-effective, scalable interventions that could improve the academic outcomes of adolescents at a population level, but no such interventions have so far been evaluated in a population-generalizable sample. Here we show that a short (less than one hour), online growth mindset intervention—which teaches that intellectual abilities can be developed—improved grades among lower-achieving students and increased overall enrolment to advanced mathematics courses in a nationally representative sample of students in secondary education in the United States. Notably, the study identified school contexts that sustained the effects of the growth mindset intervention: the intervention changed grades when peer norms aligned with the messages of the intervention. Confidence in the conclusions of this study comes from independent data collection and processing, pre-registration of analyses, and corroboration of results by a blinded Bayesian analysis.

This paper develops a semi-parametric Bayesian regression model for estimating heterogeneous treatment effects from observational data. Standard nonlinear regression models, which may work quite well for prediction, can yield badly biased estimates of treatment effects when fit to data with strong confounding. Our Bayesian causal forest model avoids this problem by directly incorporating an estimate of the propensity function in the specification of the response model, implicitly inducing a covariate-dependent prior on the regression function. This new parametrization also allows treatment heterogeneity to be regularized separately from the prognostic effect of control variables, making it possible to informatively "shrink to homogeneity", in contrast to existing Bayesian non-and semi-parametric approaches. We illustrate the benefits of this approach via the reanalysis of an observational study assessing the causal effects of smoking on medical expenditures as well as extensive simulation studies.

This paper presents a novel nonlinear regression model for estimating heterogeneous treatment effects, geared specifically towards situations with small effect sizes, heterogeneous effects, and strong confounding by observables. 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 incorporating an estimate of the propensity function in the specification of the response model, implicitly inducing a covariatedependent prior on the regression function. Second, standard approaches to response surface modeling do not provide adequate control over the strength of regularization over effect heterogeneity. The Bayesian causal forest model permits treatment effect heterogeneity to be regularized separately from the prognostic effect of control variables, making it possible to informatively "shrink to homogeneity". While we focus on observational data, our methods are equally useful for inferring heterogeneous treatment effects from randomized controlled experiments where careful regularization is somewhat less complicated but no less important. We illustrate these benefits via the reanalysis of an observational study assessing the causal effects of smoking on medical expenditures as well as extensive simulation studies.

Selecting a subset of variables for linear models remains an active area of research. This paper reviews many of the recent contributions to the Bayesian model selection and shrinkage prior literature. A posterior variable selection summary is proposed, which distills a full posterior distribution over regression coefficients into a sequence of sparse linear predictors.1. Introduction. This paper revisits the venerable problem of variable selection in linear models. The vantage point throughout is Bayesian: a normal likelihood is assumed and inferences are based on the posterior distribution, which is arrived at by conditioning on observed data.In applied regression analysis, a "high-dimensional" linear model can be one which involves tens or hundreds of variables, especially when seeking to compute a full Bayesian posterior distribution.Our review will be from the perspective of a data analyst facing a problem in this "moderate" regime. Likewise, we focus on the situation where the number of predictor variables, p, is fixed.In contrast to other recent papers surveying the large body of literature on Bayesian variable selection [Liang et al., 2008, Bayarri et al., 2012 and shrinkage priors [O'Hara and Sillanpää, 2009, Polson andScott, 2012], our review focuses specifically on the relationship between variable selection priors and shrinkage priors. Selection priors and shrinkage priors are related both by the statistical ends they attempt to serve (e.g., strong regularization and efficient estimation) and also in the technical means they use to achieve these goals (hierarchical priors with local scale parameters).We also compare these approaches on computational considerations.Finally, we turn to variable selection as a problem of posterior summarization. We argue that if variable selection is desired primarily for parsimonious communication of linear trends in the data, that this can be accomplished as a post-inference operation irrespective of the choice of prior distribution. To this end, we introduce a posterior variable selection summary, which distills a full

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.

IN LINEAR REGRESSION FOR TREATMENT EFFECT ESTIMATIONThis paper investigates the use of regularization priors in the context of treatment effect estimation using observational data where the number of control variables is large relative to the number of observations. First, the phenomenon of "regularization-induced confounding" is introduced, which refers to the tendency of regularization priors to adversely bias treatment effect estimates by over-shrinking control variable regression coefficients. Then, a simultaneous regression model is presented which permits regularization priors to be specified in a way that avoids this unintentional "re-confounding".The new model is illustrated on synthetic and empirical data.1. Introduction. This paper considers the use of Bayesian regularized linear regression models for the purpose of estimating a treatment effect from observational data. Treatment effects -the amount some response variable would change if the value of the treatment variable were changed by a given amount -can only be properly estimated from observational data by taking into account all of the various explanatory factors that may otherwise account for the observed correlation between the treatment and response variables. In the case of a linear regression model (assuming it to be correct) this "adjustment for confounding" means that the model includes a sufficient set of control variables as regressors in addition to the treatment

Digital health data are multimodal and high-dimensional. A patient’s health state can be characterized by a multitude of signals including medical imaging, clinical variables, genome sequencing, conversations between clinicians and patients, and continuous signals from wearables, among others. This high volume, personalized data stream aggregated over patients’ lives has spurred interest in developing new artificial intelligence (AI) models for higher-precision diagnosis, prognosis, and tracking. While the promise of these algorithms is undeniable, their dissemination and adoption have been slow, owing partially to unpredictable AI model performance once deployed in the real world. We posit that one of the rate-limiting factors in developing algorithms that generalize to real-world scenarios is the very attribute that makes the data exciting—their high-dimensional nature. This paper considers how the large number of features in vast digital health data can challenge the development of robust AI models—a phenomenon known as “the curse of dimensionality” in statistical learning theory. We provide an overview of the curse of dimensionality in the context of digital health, demonstrate how it can negatively impact out-of-sample performance, and highlight important considerations for researchers and algorithm designers.

A Bayesian framework is attractive in the context of prediction, but a fast recursive update of the predictive distribution has apparently been out of reach, in part because Monte Carlo methods are generally used to compute the predictive. This paper shows that online Bayesian prediction is possible by characterizing the Bayesian predictive update in terms of a bivariate copula, making it unnecessary to pass through the posterior to update the predictive. In standard models, the Bayesian predictive update corresponds to familiar choices of copula but, in nonparametric problems, the appropriate copula may not have a closed-form expression. In such cases, our new perspective suggests a fast recursive approximation to the predictive density, in the spirit of Newton's predictive recursion algorithm, but without requiring evaluation of normalizing constants. Consistency of the new algorithm is shown, and numerical examples demonstrate its quality performance in finite-samples compared to fully Bayesian and kernel methods.