Quasi-Bayesian sequential deconvolution

Abstract

Density deconvolution is the inverse problem of estimating a probability density from observations contaminated by additive noise. Traditionally studied in static or batch settings, it increasingly arises with streaming data, where existing frequentist and Bayesian procedures face substantial computational bottlenecks. We develop a quasi-Bayesian nonparametric method for sequential density deconvolution based on Newton's recursive algorithm. The resulting estimate is straightforward to evaluate and scalable to massive datasets, as its per-observation computational cost remains constant as new data arrive. The quasi-Bayesian interpretation enables uncertainty quantification: local and uniform central limit theorems yield asymptotic credible intervals and bands, respectively. Under a frequentist data-generating model, we establish $L^1$-consistency for the proposed estimate and show that it asymptotically agrees with the estimate that would be obtained if the uncontaminated variables were directly observed. Further, under additional regularity conditions, we derive an $L^1$-Wasserstein convergence rate and establish merging, at an explicit rate, with the Bayesian nonparametric posterior mean estimate under a Dirichlet process mixture model. Synthetic-data experiments, together with an acquisition-ordered flow-cytometry application, demonstrate accuracy comparable to Bayesian nonparametric and Fourier deconvolution methods, while offering a substantial computational advantage.