Contributed Discussion on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Briol, François‐Xavier; Teymur, Onur; Yoo, William Weimin; Cockayne, Jon; Schober, Michael; Hennig, Philipp

doi:10.1214/16-ba1017a

Cited by 3 publications

(5 citation statements)

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“…Different prior choices allow us to encode properties of the integrand, such as smoothness or periodicity, in a straightforward manner, leading to algorithms which respect these properties. The most common model is a Gaussian process (GP); although more recent work also considers alternatives such as Bayesian additive regression trees (BART) (Zhu et al, 2020) or multi-output Gaussian processes (Xi et al, 2018;Gessner et al, 2019).…”

Section: Supplementary Materialsmentioning

confidence: 99%

Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects (with Discussion)

2020

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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.

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Section: Supplementary Materialsmentioning

confidence: 99%

Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects (with Discussion)

2020

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“…This restricts possible trajectories to piecewise linear functions with derivative discontinuities over the sampling grid. This special case also illustrates the answer for a question of Briol et al (2016) about how to best exploit sparsity of the covariance structures. In this example, the updating problem is reduced to a linear extrapolation at each solver step with a random component with predictive variance.…”

Section: Relationship To Numerical Solversmentioning

confidence: 87%

“…When Gaussian processes are defined directly, as in UQDE, Briol et al (2016) point out that differentiation is often simpler than integration as a technique of defining covariances over the state, u, and any derivatives. However, in this case, it is not straightforward to choose a covariance corresponding to an anisotropic prior that enforces u(0) = u 0 with probability one.…”

Section: Prior Specificationmentioning

confidence: 99%

“…ODE boundary value problems, this is not strictly required as boundary conditions can be enforced by conditioning on the known boundary values. However, as pointed out by Briol et al (2016), this technique breaks down for PDE problems with functional boundaries.…”

Section: Prior Specificationmentioning

confidence: 99%

“…Next, we discuss the points made by Briol et al (2016) and Schober and Hennig (2016) and give an example of how a special case of UQDE can emulate a numerical method. We agree that analogies between numerical and probabilistic methods can provide insights into both techniques.…”

Section: Relationship To Numerical Solversmentioning

confidence: 99%

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Chkrebtii¹,

Campbell²,

Calderhead³

2016

Bayesian Anal.

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Probabilistic Integration: A Role in Statistical Computation?

Briol¹,

Oates²,

Girolami³

et al. 2019

Statist. Sci.

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153

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A research frontier has emerged in scientific computation, wherein discretisation error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly deterministic) computational work-flow, in order to assess the impact of discretisation error on the computer output. This paper examines the case for probabilistic numerical methods in routine statistical computation. Our focus is on numerical integration, where a probabilistic integrator is equipped with a full distribution over its output that reflects the fact that the integrand has been discretised. Our main technical contribution is to establish, for the first time, rates of posterior contraction for one such method. Several substantial applications are provided for illustration and critical evaluation, including examples from statistical modelling, computer graphics and a computer model for an oil reservoir. arXiv:1512.00933v6 [stat.ML] 18 Oct 2017 2 computer code to reproduce experiments reported in this paper can be downloaded from http://www. warwick.ac.uk/fxbriol/probabilistic_integration.

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Contributed Discussion on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Abstract: We commend the authors for an exciting paper which provides a strong contribution to the emerging field of probabilistic numerics (PN). Below, we discuss aspects of prior modelling which need to be considered thoroughly in future wor

Cited by 3 publications

References 5 publications

Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects (with Discussion)

Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects (with Discussion)

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Probabilistic Integration: A Role in Statistical Computation?

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