Inverse moments of univariate discrete distributions via the Poisson expansion

Minimax Optimal Rates for Poisson Inverse Problems with Physical Constraints

This paper considers fundamental limits for solving sparse inverse problems in the presence of Poisson noise with physical constraints. Such problems arise in a variety of applications, including photon-limited imaging systems based on compressed sensing. Most prior theoretical results in compressed sensing and related inverse problems apply to idealized settings where the noise is i.i.d., and do not account for signal-dependent noise and physical sensing constraints. Prior results on Poisson compressed sensing with signal-dependent noise and physical constraints provided upper bounds on mean …

Minimax optimal convex methods for Poisson inverse problems under $\ell_q$-ball sparsity

In this paper, we study the minimax rates and provide an implementable convex algorithm for Poisson inverse problems under weak sparsity and physical constraints. In particular we assume the model $y_i \sim \mbox{Poisson}(Ta_i^{\top}f^*)$ for $1 \leq i \leq n$ where $T \in \mathbb{R}_+$ is the intensity, and we impose weak sparsity on $f^* \in \mathbb{R}^p$ by assuming $f^*$ lies in an $\ell_q$-ball when rotated according to an orthonormal basis $D \in \mathbb{R}^{p \times p}$. In addition, since we are modeling real physical systems we also impose positivity and flux-preserving constraints on…

通过泊松点过程的随机泰勒展开

Stochastic Taylor expansion via Poisson point processes

本文基于泊松点过程提出随机泰勒展开，构建非线性回归框架，证明估计量一致几乎必然收敛，并通过模拟和股票数据验证方法。

Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes

We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form $$Y_t = \sum_{k=1}^{N(t)} Z_k,~~~ t \ge 0,$$ where $N(t)$ is a standard Poisson process of intensity $λ$, and $Z_k$ are drawn i.i.d.~from jump measure $μ$. A high-dimensional wavelet series prior for the Lévy measure $ν= λμ$ is devised and the posterior distribution arises from observing discrete samples $Y_Δ, Y_{2Δ}, \dots, Y_{nΔ}$ at fixed observation distance $Δ$, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior…

Bayesian approach for inverse obstacle scattering with Poisson data

We consider an acoustic obstacle reconstruction problem with Poisson data. Due to the stochastic nature of the data, we tackle this problem in the framework of Bayesian inversion. The unknown obstacle is parameterized in its angular form. The prior for the parameterized unknown plays key role in the Bayes reconstruction algorithm. The most popular used prior is the Gaussian. Under the Gaussian prior assumption, we further suppose that the unknown satisfies the total variation prior. With the hybrid prior, the well-posedness of the posterior distribution is discussed. The numerical examples ver…

Coherent prior distributions in univariate finite mixture and Markov-switching models

Finite mixture and Markov-switching models generalize and, therefore, nest specifications featuring only one component. While specifying priors in the two: the general (mixture) model and its special (single-component) case, it may be desirable to ensure that the prior assumptions introduced into both structures are coherent in the sense that the prior distribution in the nested model amounts to the conditional prior in the mixture model under relevant parametric restriction. The study provides the rudiments of setting coherent priors in Bayesian univariate finite mixture and Markov-switching …

Limit Shape of the Generalized Inverse Gaussian-Poisson Distribution

The generalized inverse Gaussian-Poisson (GIGP) distribution proposed by Sichel in the 1970s has proved to be a flexible fitting tool for diverse frequency data, collectively described using the item production model. In this paper, we identify the limit shape (specified as an incomplete gamma function) of the properly scaled diagrammatic representations of random samples from the GIGP distribution (known as Young diagrams). We also show that fluctuations are asymptotically normal and, moreover, the corresponding empirical random process is approximated via a rescaled Brownian motion in invert…

贝叶斯反问题中的似然硬币泊松采样：精确采样与尖锐复杂度

Exact Likelihood-Coin Poisson Sampling for Bayesian Inverse Problems with Sharp Complexity Bounds

本文提出一种基于伯努利事件与泊松点过程细化的贝叶斯反问题精确采样框架，通过似然硬币构造实现后验采样，并给出尖锐的计算复杂度分析，适用于椭圆预解问题。

A data-dependent weighted LASSO under Poisson noise

Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that (a) sidesteps the technical challenges in previous work, (b) admits estimators that can readily be computed using off-the…

Exact confidence intervals and hypothesis tests for parameters of discrete distributions

We study exact confidence intervals and two-sided hypothesis tests for univariate parameters of stochastically increasing discrete distributions, such as the binomial and Poisson distributions. It is shown that several popular methods for constructing short intervals lack strict nestedness, meaning that accepting a lower confidence level not always will lead to a shorter confidence interval. These intervals correspond to a class of tests that are shown to assign differing $p$-values to indistinguishable models. Finally, we show that among strictly nested intervals, fiducial intervals, includin…

Poisson点过程的收敛及最优传输正则化在PET重建变分分析中的应用

Convergence of Poisson point processes and of optimal transport regularization with application in variational analysis of PET reconstruction

本文针对PET等反问题，在信噪比趋于无穷大时，对使用最优传输正则化的动态贝叶斯重建泛函进行变分分析，研究其重建收敛性及与探测器尺寸等参数的相互作用。