On Nesting Monte Carlo Estimators
By Tom Rainforth, R. Cornish, Hongseok Yang, Andrew Warrington. 2017
167 citations recorded by the source corpus.
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.
Subfields: Artificial Intelligence, Numerical Analysis, Statistics and Probability, Probabilistic & Bayesian Methods
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
- Sergey Levine
- Yoshua Bengio
- Frank D. Wood
- Ruslan Salakhutdinov
- Quoc V. Le
- Pieter Abbeel
- Yee Whye Teh
- Gilles Louppe
In Numerical Analysis
In Statistics and Probability
- Tuan Anh Le
- Frank D. Wood
- Kyle Cranmer
- Gilles Louppe
- Atilim Gunes Baydin
- Hedvig Kjellström
- Cheng Zhang
- Tom Rainforth
In Probabilistic & Bayesian Methods
- Gilles Louppe
- Frank D. Wood
- Kyle Cranmer
- Tuan Anh Le
- Yee Whye Teh
- Atilim Gunes Baydin
- Tom Rainforth
- George Tucker
Citations in the corpus
- On Variational Bounds of Mutual Information (2019, background): “Recent work has studied the convergence and asymptotic consistency of such nested Monte-Carlo estimators, but does not address the problem of building bounds that hold with finite samples (Rainforth et al., 2018; Mathieu et al., 2018).”
- Ultra-large library docking for discovering new chemotypes (2019, unknown): “To select the most informative compounds to test, we evaluated the Shannon information gain of six candidate designs, defined as the expected difference in posterior minus prior entropy over the prior-predictive distribution, by nested Monte Carlo 72,73 .”
- Tighter Variational Bounds are Not Necessarily Better (2018, background): “ o(1) (Z+ o(1)) Z^ 1;K Z r ;˚ Z^ 1;K Z Z^ 1;K Z 2 (r ;˚Z+ o(1)) 1This approach follows similar lines to the derivation of nested Monte Carlo convergence bounds in (Rainforth, 2017; Rainforth et al., 2018; Fort et al., 2017) and the derivation of the mean squared error for self-normalized imp”
- Efficient Probabilistic Inference in the Quest for Physics Beyond the Standard Model (2018, background): “[77] T.”
- Advances in Variational Inference (2017, background): “As IWAEs introduce a biased estimator, additional steps to obtain potentially better variance-bias trade-offs can be taken, such as in [139], [152], [153] .”
- Nesting Probabilistic Programs (2018, methodology): “Though informally the high-level result follows directly from (Rainforth et al., 2018, Theorem 3), there are three subtleties that require further attention.”