Semiparametric Uncertainty Quantification via Isotonized Posterior for Deconvolutions

2026-02-20

AI Summary

This work proposes a nonparametric Bayesian deconvolution method to quantify uncertainty in the signal distribution \(F_0\) for the additive model \(Z = X + Y\) with nonnegative random variables, where the noise distribution is known. The approach places a Dirichlet process prior on the observed data and projects the posterior onto the space of signal cumulative distribution functions (CDFs) via the solution to a Volterra integral equation. To ensure the resulting estimate is a valid CDF, the projection is followed by isotonic regression using the greatest convex minorant, yielding an Isotonic Inverse Posterior. By innovatively combining posterior projection with isotonic adjustment, the method avoids estimating nuisance quantities such as the noise density or its derivatives, yet achieves asymptotically valid frequentist coverage for credible sets. A Bayes–Chernoff calibration is further introduced to enhance finite-sample performance. Simulations demonstrate that the method is computationally efficient and exhibits robust coverage across various noise settings.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We address the problem of uncertainty quantification for the deconvolution model \(Z = X + Y\), where \(X\) and \(Y\) are nonnegative random variables and the goal is to estimate the signal's distribution of \(X \sim F_0\) supported on~\([0,\infty)\), from observations where the noise distribution is known. Existing frequentist methods often produce confidence intervals for $F_0(x)$ that depend on unknown nuisance parameters, such as the density of \(X\) and its derivative, which are difficult to estimate in practice. This paper introduces a novel and computationally efficient nonparametric Bayesian approach, based on projecting the posterior, to overcome this limitation. Our method leverages the solution \(p\) to a specific Volterra integral equation as in \cite{74}, which relates the cumulative distribution function (CDF) of the signal, \(F_0\), to the distribution of the observables. We place a Dirichlet Process prior directly on the distribution of the observed data $Z$, yielding a simple, conjugate posterior. To ensure the resulting estimates for \(F_0\) are valid CDFs, we isotonize posterior draws taking the Greatest Convex Majorant of the primitive of the posterior draws and defining what we term the Isotonic Inverse Posterior. We show that this framework yields posterior credible sets for \(F_0\) that are not only computationally fast to generate but also possess asymptotically correct frequentist coverage after a straightforward recalibration technique for the so-called Bayes Chernoff distribution introduced in \cite{54}. Our approach thus does not require the estimation of nuisance parameters to deliver uncertainty quantification for the parameter of interest $F_0(x)$. The practical effectiveness and robustness of the method are demonstrated through a simulation study with various noise distributions for $Y$.

Problem

• deconvolution
• uncertainty quantification
• nonparametric Bayesian
• nuisance parameters
• CDF estimation

Innovation

• Semiparametric Bayesian
• Deconvolution
• Isotonic Inverse Posterior
• Uncertainty Quantification
• Volterra Integral Equation