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 $Y$.

翻译：本文研究去卷积模型 \(Z = X + Y\) 的不确定性量化问题，其中 \(X\) 和 \(Y\) 为非负随机变量，目标是在噪声分布已知的条件下，从观测数据中估计支撑在 \([0,\infty)\) 上的信号分布 \(X \sim F_0\)。现有的频率学派方法通常产生依赖于未知冗余参数（如 \(X\) 的密度及其导数）的置信区间，这些参数在实际中难以估计。本文提出一种新颖且计算高效的非参数贝叶斯方法，基于后验投影来克服这一局限。我们的方法利用特定 Volterra 积分方程的解 \(p\)（如 \cite{74} 所述），该解将信号的累积分布函数 \(F_0\) 与观测数据的分布联系起来。我们直接在观测数据 $Z$ 的分布上设置 Dirichlet Process 先验，从而得到简单共轭的后验分布。为确保对 \(F_0\) 的估计结果为有效的累积分布函数，我们对后验抽样进行等渗化处理，即取后验抽样原函数的极大凸包络，并定义所谓的“等渗逆后验”。我们证明，该框架生成的 \(F_0\) 后验可信集不仅计算生成速度快，而且通过 \cite{54} 引入的所谓 Bayes Chernoff 分布的简单重校准技术后，具有渐近正确的频率学派覆盖性。因此，我们的方法无需估计冗余参数即可为目标参数 $F_0(x)$ 提供不确定性量化。通过对 $Y$ 采用多种噪声分布的模拟研究，验证了本方法的实际有效性和鲁棒性。