Technology Category
Reasoning under Uncertainty: Probabilistic ProgrammingSearch and Optimization: Mixed Discrete/Continuous SearchMachine Learning: Probabilistic Circuits and Graphical Models
Efficient Importance Sampling for Wrong Exit Probabilities over Combinatorially Many Rare Regions
🤖 AI Summary
This work addresses the problem of estimating the exit probability—i.e., the probability that a high-dimensional light-tailed random walk reaches a target before first hitting any of several mutually exclusive rare sets—a challenge arising from combinatorial explosion in sequential multiple hypothesis testing as dimension increases. Conventional importance sampling suffers from exponential growth in required mixture components and uncontrolled variance in high dimensions. To overcome this, we propose a novel hybrid importance sampling method that combines a small number of optimal exponential tilts with an adaptive auxiliary proposal distribution. Theoretically, the estimator is asymptotically efficient; practically, it drastically reduces the number of mixture components needed. Both theoretical analysis and numerical experiments—including on multidimensional extensions of the Siegmund exit problem—demonstrate that the method achieves high efficiency, scalability, and low variance. It thus establishes a computationally tractable and robust paradigm for estimating high-dimensional rare-event probabilities.
Technology Category
Application Category
📝 Abstract
We consider importance sampling for estimating the probability that a light-tailed $d$-dimensional random walk exits through one of many disjoint rare-event regions before reaching an anticipated target. This problem arises in sequential multiple hypothesis testing, where the number of such regions may grow combinatorially and in some cases exponentially with the dimension. While mixtures over all associated exponential tilts are asymptotically efficient, they become computationally infeasible even for moderate values of $d$. We develop a method for constructing asymptotically efficient mixtures with substantially fewer components by combining optimal tilts for a small number of regions with additional proposals that control variance across a large collection of regions. The approach is applied to the estimation of three probabilities that arise in sequential multiple testing, including a multidimensional extension of Siegmund's classical exit problem, and is supported by both theoretical analysis and numerical experiments.
Problem
Research questions and friction points this paper is trying to address.
Estimating wrong exit probabilities in high-dimensional random walks
Addressing combinatorial growth of rare-event regions in hypothesis testing
Developing efficient importance sampling with reduced computational complexity
Innovation
Methods, ideas, or system contributions that make the work stand out.
Efficient mixtures with fewer components
Combining optimal tilts with additional proposals
Controlling variance across many rare regions
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