# Semi-Exact Control Functionals From Sard's Method

arXiv: https://arxiv.org/abs/2002.00033 · HTML: https://mlsanity.com/paper/2002.00033/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2020.

## Built on by (most cited first, 9 of 9)

- [Positively Weighted Kernel Quadrature via Subsampling](https://mlsanity.com/paper/2107.09597/) (2021)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [Meta-learning Control Variates: Variance Reduction with Limited Data](https://mlsanity.com/paper/2303.04756/) (2023)
- [The Polynomial Stein Discrepancy for Assessing Moment Convergence](https://mlsanity.com/paper/2412.05135/) (2024)
- [Conditional neural control variates for variance reduction in Bayesian inverse problems](https://mlsanity.com/paper/2602.21357/) (2026)
- [Bayesian Quadrature](https://mlsanity.com/paper/2602.16218/) (2026)
- [Feynman-Kac Operator Expectation Estimator](https://mlsanity.com/paper/2407.02010/) (2024)
- [Convex-Geometric Error Bounds for Positive-Weight Kernel Quadrature](https://mlsanity.com/paper/2605.05705/) (2026)
- [λ-PSD: Scalable Approximate SNR-Optimised Polynomial Stein Discrepancies](https://mlsanity.com/paper/2606.26621/) (2026)

## Abstract (arXiv)

The numerical approximation of posterior expected quantities of interest is considered. A novel control variate technique is proposed for post-processing of Markov chain Monte Carlo output, based both on Stein's method and an approach to numerical integration due to Sard. The resulting estimators are proven to be polynomially exact in the Gaussian context, while empirical results suggest the estimators approximate a Gaussian cubature method near the Bernstein-von-Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results are presented across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV.