Quasi-Bayesian sequential deconvolution

TL;DR

This paper introduces a computationally efficient sequential deconvolution method using a quasi-Bayesian approach, suitable for streaming data, with proven asymptotic properties and empirical validation against existing methods.

Contribution

It develops a novel sequential density deconvolution technique based on quasi-Bayesian modeling, with theoretical guarantees and practical advantages for streaming data.

Findings

Sequential estimator is computationally efficient with constant cost.

Asymptotic Gaussian CLTs enable credible intervals and bands.

Method outperforms kernel and Bayesian nonparametric approaches in experiments.

Abstract

Density deconvolution deals with the estimation of the probability density function $f$ of a random signal from $n\geq1$ data observed with independent and known additive random noise. This is a classical problem in statistics, for which frequentist and Bayesian nonparametric approaches are available to estimate $f$ in static or batch domains. In this paper, we consider the problem of density deconvolution in a streaming or online domain, and develop a principled sequential approach to estimate $f$. By relying on a quasi-Bayesian sequential (learning) model for the data, often referred to as Newton's algorithm, we obtain a sequential deconvolution estimate $f_{n}$ of $f$ that is of easy evaluation, computationally efficient, and with constant computational cost as data increase, which is desirable for streaming data. In particular, local and uniform Gaussian central limit theorems for $f_{n}$ are established, leading to asymptotic credible intervals and bands for $f$, respectively. We provide the sequential deconvolution estimate $f_{n}$ with large sample asymptotic guarantees under the quasi-Bayesian sequential model for the data, proving a merging with respect to the direct density estimation problem, and also under a ``true" frequentist model for the data, proving consistency. An empirical validation of our methods is presented on synthetic and real data, also comparing with respect to a kernel approach and a Bayesian nonparametric approach with a Dirichlet process mixture prior.