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Approximate Inference Suffices for Statistical Distance Estimation

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.6Download Report PDF
statistical distanceprobabilistic inferencereductions
Abstract:

Statistical distance (also known as total variation distance) and probabilistic inference are fundamental notions, widely used in machine learning, information theory, and high-dimensional statistics. While there are efficient algorithms that can estimate statistical distance or probabilistic inference in some specific settings, it has remained an open problem to see whether these two notions can be approximately reduced to each other. In this work, we take the first step in addressing this problem, and show that estimating statistical distance can be reduced to estimating probabilistic inference, via an efficient structure preserving randomized reduction. This allows us to use approximate inference algorithms to multiplicatively estimate statistical distance in directed graphical models.

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Overview

Overall Novelty Assessment

The paper establishes a reduction from statistical distance estimation to approximate probabilistic inference, specifically targeting total variation distance in directed graphical models. It resides in the 'Total Variation Distance Reduction' leaf, which contains only two papers including this work. This sparse population suggests the reduction framework itself is relatively unexplored territory within the broader taxonomy of fifty papers. The contribution sits at the core of the 'Core Reduction Theory and Algorithms' branch, which focuses on theoretical reductions with algorithmic guarantees rather than application-driven distance-based inference methods.

The taxonomy reveals substantial activity in neighboring areas. The sibling leaf 'Statistical Distance Computation' develops algorithms for computing distances without reduction frameworks, while the broader 'Distance-Based Inference Methods' branch encompasses variational inference with optimal transport, approximate Bayesian computation, and robust inference methods. The paper's reduction approach diverges from these directions by treating inference as a computational primitive rather than designing specialized distance estimation algorithms. Nearby work in 'Inference Evaluation and Diagnostics' measures inference accuracy, providing complementary perspectives on the quality-efficiency tradeoffs the reduction exploits.

Among thirty candidates examined, the contribution-level analysis reveals mixed novelty signals. The core reduction from distance estimation to inference and the FPRAS construction each identified one refutable candidate among ten examined, suggesting some overlap with prior theoretical work in this limited search scope. The normalization constant estimation and sampling procedures found no refutable candidates among ten examined, indicating these technical components may be more novel or distinct from the surveyed literature. The statistics reflect a focused semantic search rather than exhaustive coverage, leaving open whether broader literature contains additional overlapping results.

Based on the top-thirty semantic matches and citation expansion, the work appears to occupy a sparsely populated research direction with some prior theoretical foundations. The reduction framework itself has minimal direct precedent in the examined taxonomy, though related distance computation and inference evaluation methods exist in neighboring branches. The analysis does not cover the full landscape of computational complexity theory or graphical model inference, where additional relevant work may reside outside the distance-centric framing of this search.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
30
Contribution Candidate Papers Compared
2
Refutable Paper

Research Landscape Overview

Core task: reducing statistical distance estimation to approximate probabilistic inference. The field encompasses a diverse set of methodological branches that connect distance metrics, inference algorithms, and statistical testing. At the highest level, Core Reduction Theory and Algorithms develops foundational techniques for converting distance estimation problems into inference tasks, often focusing on total variation or Wasserstein-based reductions. Distance-Based Inference Methods explores how various metrics—ranging from sliced Wasserstein variants (Max Sliced Wasserstein[10], Sliced Wasserstein Variational[6]) to robust ABC distances (Robust ABC Distances[8], Adaptive ABC Distances[3])—can guide approximate Bayesian computation and related inference schemes. Inference Evaluation and Diagnostics provides tools for assessing the quality of posterior approximations (Posterior Approximation Bounds[32], Evaluating Bayesian Deep[28]), while Statistical Testing and Hypothesis Testing leverages distance-based frameworks for hypothesis evaluation (Distance ANOVA Approximate[21], Wasserstein Selective Inference[18]). Application-Specific Inference and Specialized Applications branches illustrate how these methods extend to domains such as phylogenetics, model selection, and generative model evaluation, and Methodological Foundations and Theory underpins the entire taxonomy with rigorous statistical and geometric perspectives (Fibrations Statistical Games[5], Metric Statistics[7]). Recent work has intensified around the interplay between computational efficiency and theoretical guarantees in distance-driven inference. A particularly active line investigates how approximate inference suffices for reliable distance estimation, with Approximate Inference Suffices[0] situated within the Total Variation Distance Reduction cluster alongside Total Variation Probabilistic[1] and Approximating Total Variation[13]. These studies contrast with Wasserstein-centric approaches (ABC Wasserstein[4], Constrained Transport Metric[2]) that emphasize geometric structure and robustness but may require more sophisticated optimization. Meanwhile, adaptive and data-driven distance selection (Adaptive Distance Estimation[9], DEER Distance Inference[15]) addresses the challenge of choosing metrics that balance fidelity and tractability. Approximate Inference Suffices[0] contributes to this landscape by demonstrating that even coarse probabilistic inference can yield provably accurate distance estimates, offering a complementary perspective to works like Total Variation Probabilistic[1] that focus on direct variational bounds and to methods such as Adaptive ABC Distances[3] that dynamically tune metric choices during inference.

Claimed Contributions

Efficient reduction from statistical distance estimation to approximate probabilistic inference

The authors establish the first reduction showing that statistical distance estimation between Bayes nets can be efficiently reduced to approximate probabilistic inference queries, rather than requiring exact inference as in prior work. This enables the use of practical approximate inference algorithms for statistical distance problems.

10 retrieved papers
Can Refute
FPRAS for statistical distance estimation using approximate inference

The authors provide a fully polynomial-time randomized approximation scheme that multiplicatively estimates statistical distance between Bayes nets by making calls to approximate probabilistic inference oracles, with running time polynomial in the network size, alphabet size, and accuracy parameters.

10 retrieved papers
Can Refute
Normalization constant estimation and approximate sampling procedures

The authors develop novel algorithmic components including a procedure to estimate normalization constants using approximate inference and a sequential sampling method that generates samples from an approximate target distribution, both handling approximation errors that arise from using only approximate inference oracles.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Efficient reduction from statistical distance estimation to approximate probabilistic inference

The authors establish the first reduction showing that statistical distance estimation between Bayes nets can be efficiently reduced to approximate probabilistic inference queries, rather than requiring exact inference as in prior work. This enables the use of practical approximate inference algorithms for statistical distance problems.

Contribution

FPRAS for statistical distance estimation using approximate inference

The authors provide a fully polynomial-time randomized approximation scheme that multiplicatively estimates statistical distance between Bayes nets by making calls to approximate probabilistic inference oracles, with running time polynomial in the network size, alphabet size, and accuracy parameters.

Contribution

Normalization constant estimation and approximate sampling procedures

The authors develop novel algorithmic components including a procedure to estimate normalization constants using approximate inference and a sequential sampling method that generates samples from an approximate target distribution, both handling approximation errors that arise from using only approximate inference oracles.

Table of Contents

Approximate Inference Suffices for Statistical Distance Estimation | Novelty Validation