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Semiparametric Uncertainty Quantification via Isotonized Posterior for Deconvolutions

Francesco Gili, Geurt Jongbloed

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 $Y$.

Semiparametric Uncertainty Quantification via Isotonized Posterior for Deconvolutions

Abstract

We address the problem of uncertainty quantification for the deconvolution model , where and are nonnegative random variables and the goal is to estimate the signal's distribution of supported on~\([0,\infty)\), from observations where the noise distribution is known. Existing frequentist methods often produce confidence intervals for that depend on unknown nuisance parameters, such as the density of 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 to a specific Volterra integral equation as in \cite{74}, which relates the cumulative distribution function (CDF) of the signal, , to the distribution of the observables. We place a Dirichlet Process prior directly on the distribution of the observed data , yielding a simple, conjugate posterior. To ensure the resulting estimates for 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 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 . The practical effectiveness and robustness of the method are demonstrated through a simulation study with various noise distributions for .
Paper Structure (6 sections, 12 theorems, 129 equations, 4 figures, 1 table)

This paper contains 6 sections, 12 theorems, 129 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Simulation study
  4. Conclusion
  5. Complementary results
  6. Review of the integral equation

Key Result

Theorem 1

Let $p$ satisfy Condition cond:2, $0 < T < \infty$ and let $x \in [0, T)$ be fixed and $F_0$ be such that $F_0$ has a continuous strictly positive derivative $f_0$ in a neighborhood of $x$. Then $n^{1/3}(\hat{F}_n(x) - F_0(x))$ converges in distribution as $n \to \infty$; specifically, $\forall \, z where $\mathbb{W}_1$ is two-sided Brownian motion originating from $0$.

Figures (4)

  • Figure 1: The Bayes--Chernoff distribution $Z_B$ and its inverse function $A^{-1}$, with the identity line for reference (the computations are based on 20000 gathered samples of $Z_B$).
  • Figure 2: Resolvent of proposed kernels.
  • Figure 3: Combined deconvolution plots for $n=200$. In green, we plot draws of $\hat{F}_{{G}}$ based on prior draws from $G \sim \operatorname{DP}(M \alpha)$ with $\alpha \sim \mathrm{Ga}(2,2)$ and prior precision $M =10$; in purple, draws from the isotonized posterior; in red, the average of these draws (approximating the posterior mean); in yellow, the (IIE); and in orange $F_0$.
  • Figure 4: Combined deconvolution plots with calibrated uncertainty quantification for $n=200$. In green, we plot draws of $\hat{F}_{{G}}$ based on prior draws from $G \sim \operatorname{DP}(M \alpha)$ with $\alpha \sim \mathrm{Ga}(2,2)$ and prior precision $M =10$; in pink, the calibrated credible bands from the isotonized posterior; in red, the average of these draws (approximating the posterior mean); in yellow, the (IIE); and in orange $F_0$.

Theorems & Definitions (21)

  • Theorem 1: Theorem 2 in 74
  • Theorem 2
  • proof
  • Lemma 1: Lemma 3.5 in 54
  • Corollary 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 11 more