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Interventional Processes for Causal Uncertainty Quantification

Hugh Dance, Peter Orbanz, Arthur Gretton

TL;DR

The paper tackles principled uncertainty quantification for causal effects under continuous treatments in nonparametric settings by introducing IMPspec, a Gaussian process framework that places priors on RKHS-represented causal functions through a spectral RKHS expansion. This yields tractable training, closed-form posterior moments, and calibrated credible intervals, while avoiding underfitting and variance collapse seen in prior GP-on-RKHS methods. Its spectral-prior construction reduces the infinite-dimensional problem to scalar GP coordinates, preserving consistency with state-of-the-art kernel-based estimators and enabling robust calibration and posterior-based causal Bayesian optimization. Empirically, IMPspec delivers state-of-the-art uncertainty quantification and improves optimization of interventions in synthetic benchmarks and healthcare scenarios. The work advances practical causal inference by providing reliable uncertainty quantification and decision-making tools for continuous treatments in high-stakes domains.

Abstract

Reliable uncertainty quantification for causal effects is crucial in various applications, but remains difficult in nonparametric models, particularly for continuous treatments. We introduce IMPspec, a Gaussian process (GP) framework for modeling uncertainty over interventional causal functions under continuous treatments, which can be represented using reproducing Kernel Hilbert Spaces (RKHSs). By using principled function class expansions and a spectral representation of RKHS features, IMPspec yields tractable training and inference, a spectral algorithm to calibrate posterior credible intervals, and avoids the underfitting and variance collapse pathologies of earlier GP-on-RKHS methods. Across synthetic benchmarks and an application in healthcare, IMPspec delivers state-of-the-art performance in causal uncertainty quantification and downstream causal Bayesian optimization tasks.

Interventional Processes for Causal Uncertainty Quantification

TL;DR

The paper tackles principled uncertainty quantification for causal effects under continuous treatments in nonparametric settings by introducing IMPspec, a Gaussian process framework that places priors on RKHS-represented causal functions through a spectral RKHS expansion. This yields tractable training, closed-form posterior moments, and calibrated credible intervals, while avoiding underfitting and variance collapse seen in prior GP-on-RKHS methods. Its spectral-prior construction reduces the infinite-dimensional problem to scalar GP coordinates, preserving consistency with state-of-the-art kernel-based estimators and enabling robust calibration and posterior-based causal Bayesian optimization. Empirically, IMPspec delivers state-of-the-art uncertainty quantification and improves optimization of interventions in synthetic benchmarks and healthcare scenarios. The work advances practical causal inference by providing reliable uncertainty quantification and decision-making tools for continuous treatments in high-stakes domains.

Abstract

Reliable uncertainty quantification for causal effects is crucial in various applications, but remains difficult in nonparametric models, particularly for continuous treatments. We introduce IMPspec, a Gaussian process (GP) framework for modeling uncertainty over interventional causal functions under continuous treatments, which can be represented using reproducing Kernel Hilbert Spaces (RKHSs). By using principled function class expansions and a spectral representation of RKHS features, IMPspec yields tractable training and inference, a spectral algorithm to calibrate posterior credible intervals, and avoids the underfitting and variance collapse pathologies of earlier GP-on-RKHS methods. Across synthetic benchmarks and an application in healthcare, IMPspec delivers state-of-the-art performance in causal uncertainty quantification and downstream causal Bayesian optimization tasks.

Paper Structure

This paper contains 70 sections, 12 theorems, 177 equations, 7 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Background
  4. Causal Inference and Do-Calculus
  5. Causal DAGs
  6. Identification of Causal Effects
  7. Kernel Methods for Causal Functions
  8. GPs for Uncertainty Quantification
  9. Causal Uncertainty with IMPspec
  10. From Bounded to Unbounded Functions
  11. Spectral Priors on Features
  12. IMPspec Model
  13. Posterior Inference for IMPspec
  14. Posterior Moments for Causal Effects
  15. Credible Interval Construction
  16. ...and 55 more sections

Key Result

Theorem 1

Under eq:isogp–eq:Pmodel2, the posterior mean and variance of $\gamma(w,z)\mid \mathcal{X}^n$ are given by where and, for $x \in \{w,v,z\}$ we use the definitions

Figures (7)

  • Figure 1: causal graph where $(V,Z)$ satisfies the back-door criterion w.r.t. $(A,Y)$ (left), and where $M$ satisfies the front-door criterion w.r.t. $(A,Y)$ (right). Dashed edges = effect of unobserved confounders $U$.
  • Figure 2: Samples from a standard GP prior (left) and IMPspec prior on $\gamma$ (right) for fixed $W=w$.
  • Figure 3: (Toy Example) Left and Middle Left: Estimated $\mathbb E[Y|\mathrm{do}(A)]$ and RMSE, using BayesIMP chau2021bayesimp (red) and our method IMPspec (green). True $\mathbb E[Y|\mathrm{do}(A)]$ = black line, posterior mean = colored lines, light shading = 90 % credible intervals, dark shading = interquartile range. All quantities are averaged over 50 trials. BayesIMP underfits the causal function due to the non-stationarity of the kernel used (see \ref{['sec:background:related_work']} and \ref{['appendix:details']}). Right and Middle Right: Calibration plots of different methods over 50 trials, with 95% confidence intervals (shaded regions) estimated using 100 bootstrap replications. IMPspec is best calibrated and is significantly improved by optimizing the spectral representation.
  • Figure 4: (Synthetic Benchmark) Left + Middle-Left = posterior mean and credible intervals (50% and 95%) for $\mathbb E[Y|\mathrm{do}(D), b=0]$ (black) using BayesIMP (green) and our IMPspec (red), along with total RMSE. Grey region = support of $\mathbb P_D$. Right + Middle-Right = calibration error of different methods in- and out-of-distribution. BayesIMP's posterior variance collapses out-of-distribution. IMPspec is best calibrated overall.
  • Figure 5: Basis functions of the Gaussian kernel $k(x,{\,\vcenter{\hbox{\tiny$\bullet$}}\,}) = \exp(-\lVert x-{\,\vcenter{\hbox{\tiny$\bullet$}}\,}\rVert)$ (left) and nuclear dominant kernel $r(x,{\,\vcenter{\hbox{\tiny$\bullet$}}\,}) = \int k(x,t)k({\,\vcenter{\hbox{\tiny$\bullet$}}\,},t)\mathcal{N}(dt|0,1)$ (right) for $x$ grid-spaced on $\{-5...,5\}$. $r(x,{\,\vcenter{\hbox{\tiny$\bullet$}}\,})$ collapses exponentially fast as $|x| \to \infty$.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: Back-door criterion pearl1995causal
  • Definition 2: Front-door criterion pearl1995causal
  • Remark 4: Case $W = \emptyset$
  • Proposition 1: Uniform tail limit for the nuclear-dominant kernel basis function
  • Proposition 2: Decay rate for log-convex, integrable $h$ and density of $\nu$
  • Remark 5
  • ...and 25 more