mlsanityData to 14 Oct 2026

On Nesting Monte Carlo Estimators

arXiv 1709.06181 ↗First posted 18 Sep 2017stat.CO stat.ME stat.MLcited by 83

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Abstract (arXiv)

Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistical implications of nesting MC estimators, including cases of multiple levels of nesting, and establish the conditions under which they converge. We derive corresponding rates of convergence and provide empirical evidence that these rates are observed in practice. We further establish a number of pitfalls that can arise from naive nesting of MC estimators, provide guidelines about how these can be avoided, and lay out novel methods for reformulating certain classes of nested expectation problems into single expectations, leading to improved convergence rates. We demonstrate the applicability of our work by using our results to develop a new estimator for discrete Bayesian experimental design problems and derive error bounds for a class of variational objectives.

Key facts. Cited by 83 papers in the map (data through 14 Oct 2026). First posted on arXiv 18 Sep 2017.

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Citing papers per year: 2021: 10; 2022: 14; 2023: 18; 2024: 12; 2025: 13; 2026: 16.

83 papers.

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  1. 2023Modern Bayesian Experimental Designcited by 125
  2. 2021Deep Adaptive Design: Amortizing Sequential Bayesian Experimental Designcited by 106
  3. 2021Decomposing 3D Scenes into Objects via Unsupervised Volume Segmentationcited by 88
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  6. 2021Implicit Deep Adaptive Design: Policy-Based Experimental Design without Likelihoodscited by 61
  7. 2022Interventions, Where and How? Experimental Design for Causal Models at Scalecited by 49
  8. 2022Optimizing Sequential Experimental Design with Deep Reinforcement Learningcited by 46
  9. 2023Contrastive Difference Predictive Codingcited by 36
  10. 2022Active Bayesian Causal Inferencecited by 35
  11. 2023MINDE: Mutual Information Neural Diffusion Estimationcited by 31
  12. 2025BED-LLM: Intelligent Information Gathering with LLMs and Bayesian Experimental Designcited by 24
38 more
  1. 2021Test Distribution-Aware Active Learning: A Principled Approach Against Distribution Shift and Outlierscited by 22
  2. 2024Amortized Bayesian Experimental Design for Decision-Makingcited by 21
  3. 2021Tight Mutual Information Estimation With Contrastive Fenchel-Legendre Optimizationcited by 18
  4. 2025BoxingGym: Benchmarking Progress in Automated Experimental Design and Model Discoverycited by 18
  5. 2021Learning non-Gaussian graphical models via Hessian scores and triangular transportcited by 16
  6. 2022Policy-Based Bayesian Experimental Design for Non-Differentiable Implicit Modelscited by 16
  7. 2023Differentiable Multi-Target Causal Bayesian Experimental Designcited by 15
  8. 2023On the Expected Size of Conformal Prediction Setscited by 13
  9. 2024Variational Bayesian Optimal Experimental Design with Normalizing Flowscited by 13
  10. 2021Targeted Active Learning for Bayesian Decision-Makingcited by 12
  11. 2022Illusory Attacks: Information-Theoretic Detectability Matters in Adversarial Attackscited by 12
  12. 2023DiscoBAX: Discovery of Optimal Intervention Sets in Genomic Experiment Designcited by 12
  13. 2024Nesting Particle Filters for Experimental Design in Dynamical Systemscited by 12
  14. 2025Step-DAD: Semi-Amortized Policy-Based Bayesian Experimental Designcited by 12
  15. 2022Bayesian model calibration for block copolymer self-assembly: Likelihood-free inference and expected information gain computation via measure transportcited by 11
  16. 2024Bayesian Experimental Design via Contrastive Diffusionscited by 11
  17. 2021Efficient Semi-Implicit Variational Inferencecited by 9
  18. 2022Approximate Bayesian Computation with Domain Expert in the Loopcited by 9
  19. 2024Conditional Bayesian Quadraturecited by 9
  20. 2021A Scalable Gradient-Free Method for Bayesian Experimental Design with Implicit Modelscited by 7
  21. 2022Design Amortization for Bayesian Optimal Experimental Designcited by 7
  22. 2023On the Properties and Estimation of Pointwise Mutual Information Profilescited by 7
  23. 2023Human-in-the-Loop Visual Re-ID for Population Size Estimationcited by 7
  24. 2022Constructing unbiased gradient estimators with finite variance for conditional stochastic optimizationcited by 6
  25. 2023Statistically Efficient Bayesian Sequential Experiment Design via Reinforcement Learning with Cross-Entropy Estimatorscited by 6
  26. 2023Bayesian Active Learning in the Presence of Nuisance Parameterscited by 6
  27. 2023Zero Grads: Learning Local Surrogate Losses for Non-Differentiable Graphicscited by 5
  28. 2023On Estimating the Gradient of the Expected Information Gain in Bayesian Experimental Designcited by 5
  29. 2025Nested Expectations with Kernel Quadraturecited by 5
  30. 2023Optimal randomized multilevel Monte Carlo for repeatedly nested expectationscited by 4
  31. 2023CO-BED: Information-Theoretic Contextual Optimization via Bayesian Experimental Designcited by 4
  32. 2024Automated Efficient Estimation using Monte Carlo Efficient Influence Functionscited by 4
  33. 2024Expected information gain estimation via density approximations: Sample allocation and dimension reductioncited by 4
  34. 2022Uphill Roads to Variational Tightness: Monotonicity and Monte Carlo Objectivescited by 3
  35. 2022Estimators of Entropy and Information via Inference in Probabilistic Modelscited by 3
  36. 2022Efficient Real-world Testing of Causal Decision Making via Bayesian Experimental Design for Contextual Optimisationcited by 3
  37. 2025Performance Comparisons of Reinforcement Learning Algorithms for Sequential Experimental Designcited by 3
  38. 2025A Geometric Approach to Optimal Experimental Designcited by 3

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