Suppose we observe a Poisson process in real time for which the intensity may take on two possible values $λ_0$ and $λ_1$. Suppose further that the priori probability of the true intensity is not given. We solve a minimax version of Bayesian problem of sequential testing of two simple hypotheses to minimize a linear combination of the probability of wrong detection and the expected waiting time in the worst scenario of all possible priori distributions. An equivalent characterization for the least favorable distributions is derived and a sufficient condition for the existence is concluded.
University of Oslo（奥斯陆大学）；University of Neuchatel（纳沙泰尔大学）；Zenseact；Chalmers University of Technology（查尔姆斯理工大学）
While the Bayesian decision-theoretic framework offers an elegant solution to the problem of decision making under uncertainty, one question is how to appropriately select the prior distribution. One idea is to employ a worst-case prior. However, this is not as easy to specify in sequential decision making as in simple statistical estimation problems. This paper studies (sometimes approximate) minimax-Bayes solutions for various reinforcement learning problems to gain insights into the properties of the corresponding priors and policies. We find that while the worst-case prior depends on the s…
Under mild Markov assumptions, sufficient conditions for strict minimax optimality of sequential tests for multiple hypotheses under distributional uncertainty are derived. First, the design of optimal sequential tests for simple hypotheses is revisited and it is shown that the partial derivatives of the corresponding cost function are closely related to the performance metrics of the underlying sequential test. Second, an implicit characterization of the least favorable distributions for a given testing policy is stated. By combining the results on optimal sequential tests and least favorable…
Motivated by applications in cybersecurity and epidemiology, we consider the problem of detecting an abrupt change in the intensity of a Poisson process, characterised by a jump (non transitory change) or a bump (transitory change) from constant. We propose a complete study from the nonasymptotic minimax testing point of view, when the constant baseline intensity is known or unknown. The question of minimax adaptation with respect to each parameter (height, location, length) of the change is tackled, leading to a comprehensive overview of the various minimax separation rate regimes. We exhibit…
We study the problem of estimating the mode and maximum of an unknown regression function in the presence of noise. We adopt the Bayesian approach by using tensor-product B-splines and endowing the coefficients with Gaussian priors. In the usual fixed-in-advanced sampling plan, we establish posterior contraction rates for mode and maximum and show that they coincide with the minimax rates for this problem. To quantify estimation uncertainty, we construct credible sets for these two quantities that have high coverage probabilities with optimal sizes. If one is allowed to collect data sequential…
We consider the identity testing problem - or goodness-of-fit testing problem - in multivariate binomial families, multivariate Poisson families and multinomial distributions. Given a known distribution $p$ and $n$ iid samples drawn from an unknown distribution $q$, we investigate how large $ρ>0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q) \geq ρ$, where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q) = \|p-q\|_t$ for $t \in [1,2]$ where $\|\cdot\|_t$ is the…
This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avo…
This paper considers the problem of joint change detection and identification assuming multiple composite postchange hypotheses. We propose a multihypothesis changepoint detection-identification procedure that controls the probabilities of false alarm and wrong identification. We show that the proposed procedure is asymptotically minimax and pointwise optimal, minimizing moments of the detection delay as probabilities of false alarm and wrong identification approach zero. The asymptotic optimality properties hold for general stochastic models with dependent observations. We illustrate general …
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…
We focus on one-sided, mixture-based stopping rules for the problem of sequential testing a simple null hypothesis against a composite alternative. For the latter, we consider two cases---either a discrete alternative or a continuous alternative that can be embedded into an exponential family. For each case, we find a mixture-based stopping rule that is nearly minimax in the sense of minimizing the maximal Kullback-Leibler information. The proof of this result is based on finding an almost Bayes rule for an appropriate sequential decision problem and on high-order asymptotic approximations for…
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 …
Asymptotic efficiency is proved for the constructed in part 1 procedure, i.e. Pinsker's constant is found in the asymptotic lower bound for the minimax quadratic risk. It is shown that the asymptotic minimax quadratic risk of the constructed procedure coincides with this constant.
We develop a new class of dynamic multivariate Poisson count models that allow for fast online updating and we refer to these models as multivariate Poisson-scaled beta (MPSB). The MPSB model allows for serial dependence in the counts as well as dependence across multiple series with a random common environment. Other notable features include analytic forms for state propagation and predictive likelihood densities. Sequential updating occurs through the updating of the sufficient statistics for static model parameters, leading to a fully adapted particle learning algorithm and a new class of p…
Suppose that local characteristics of several independent compound Poisson and Wiener processes change suddenly and simultaneously at some unobservable disorder time. The problem is to detect the disorder time as quickly as possible after it happens and minimize the rate of false alarms at the same time. These problems arise, for example, from managing product quality in manufacturing systems and preventing the spread of infectious diseases. The promptness and accuracy of detection rules improve greatly if multiple independent information sources are available. Earlier work on sequential chang…
This paper discusses predictive densities under the Kullback--Leibler loss for high-dimensional Poisson sequence models under sparsity constraints. Sparsity in count data implies zero-inflation. We present a class of Bayes predictive densities that attain asymptotic minimaxity in sparse Poisson sequence models. We also show that our class with an estimator of unknown sparsity level plugged-in is adaptive in the asymptotically minimax sense. For application, we extend our results to settings with quasi-sparsity and with missing-completely-at-random observations. The simulation studies as well a…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Microsoft Research New York（微软研究院纽约）
We introduce a new dynamical system for sequentially observed multivariate count data. This model is based on the gamma--Poisson construction---a natural choice for count data---and relies on a novel Bayesian nonparametric prior that ties and shrinks the model parameters, thus avoiding overfitting. We present an efficient MCMC inference algorithm that advances recent work on augmentation schemes for inference in negative binomial models. Finally, we demonstrate the model's inductive bias using a variety of real-world data sets, showing that it exhibits superior predictive performance over othe…
We consider the nonparametric estimation of the intensity function of a Poisson point process in a circular model from indirect observations $N_1,\ldots,N_n$. These observations emerge from hidden point process realizations with the target intensity through contamination with additive error. In case that the error distribution can only be estimated from an additional sample $Y_1,\ldots,Y_m$ we derive minimax rates of convergence with respect to the sample sizes $n$ and $m$ under abstract smoothness conditions and propose an orthonormal series estimator which attains the optimal rate of converg…
We study the nonparametric estimation of the jump density of a compound Poisson process from the discrete observation of one trajectory over $[0,T]$. We consider the microscopic regime when the sampling rate $Δ=Δ_T\rightarrow0$ as $T\rightarrow\infty$. We propose an adaptive wavelet threshold density estimator and study its performance for the $L_p$ loss, $p\geq 1$, over Besov spaces. The main novelty is that we achieve minimax rates of convergence for sampling rates $Δ_T$ that vanish with $T$ at arbitrary polynomial rates. More precicely, our estimator attains minimax rates of convergence pro…
This paper considers the problem of adaptive estimation of a non-homogeneous intensity function from the observation of n independent Poisson processes having a common intensity that is randomly shifted for each observed trajectory. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet b…
The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,γ}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of $\tilde{f}_{n,γ}$ on Besov spaces ${\cal B}^{\ensuremath α}_{p,q}$ are established. Under mild assumptions, we prove that $$…
This work is motivated by the analysis of ecological interaction networks. Poisson stochastic blockmodels are widely used in this field to decipher the structure that underlies a weighted network, while accounting for covariate effects. Efficient algorithms based on variational approximations exist for frequentist inference, but without statistical guaranties as for the resulting estimates. In absence of variational Bayes estimates, we show that a good proxy of the posterior distribution can be straightforwardly derived from the frequentist variational estimation procedure, using a Laplace app…
The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,γ}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a logarithmic term. Oracle inequalities allow to derive the maxiset of $\tilde{f}_{n,γ}$. Then, minimax properties of $\tilde{f}_{n,γ}$ are established. We first prove that the rate…
A compound Poisson process whose jump measure and intensity are unknown is observed at finitely many equispaced times. We construct a purely data-driven estimator of the Lévy density $ν$ through the spectral approach using general Calderon--Zygmund integral operators, which include convolution and projection kernels. Assuming minimal tail assumptions, it is shown to estimate $ν$ at the minimax rate of estimation over Besov balls under the losses $L^p(\mathbb{R})$, $p\in[1,\infty]$, and robustly to the observation regime (high- and low-frequency). To achieve adaptation in a minimax sense, we us…
We consider together the retrospective and the sequential change-point detection in a general class of integer-valued time series. The conditional mean of the process depends on a parameter $θ^*$ which may change over time. We propose procedures which are based on the Poisson quasi-maximum likelihood estimator of the parameter, and where the updated estimator is computed without the historical observations in the sequential framework. For both the retrospective and the sequential detection, the test statistics converge to some distributions obtained from the standard Brownian motion under the …
We study the minimax estimation of $α$-divergences between discrete distributions for integer $α\ge 1$, which include the Kullback--Leibler divergence and the $χ^2$-divergences as special examples. Dropping the usual theoretical tricks to acquire independence, we construct the first minimax rate-optimal estimator which does not require any Poissonization, sample splitting, or explicit construction of approximating polynomials. The estimator uses a hybrid approach which solves a problem-independent linear program based on moment matching in the non-smooth regime, and applies a problem-dependent…
As a general rule of thumb the resolution of a light microscope (i.e. the ability to discern objects) is predominantly described by the full width at half maximum (FWHM) of its point spread function (psf)---the diameter of the blurring density at half of its maximum. Classical wave optics suggests a linear relationship between FWHM and resolution also manifested in the well known Abbe and Rayleigh criteria, dating back to the end of 19th century. However, during the last two decades conventional light microscopy has undergone a shift from microscopic scales to nanoscales. This increase in reso…
We provide an overview of the state-of-the-art in the area of sequential change-point detection assuming discrete time and known pre- and post-change distributions. The overview spans over all major formulations of the underlying optimization problem, namely, Bayesian, generalized Bayesian, and minimax. We pay particular attention to the latest advances in each. Also, we link together the generalized Bayesian problem with multi-cyclic disorder detection in a stationary regime when the change occurs at a distant time horizon. We conclude with two case studies to illustrate the cutting edge of t…
This paper proposes a model of interactions between two point processes, ruled by a reproduction function h, which is considered as the intensity of a Poisson process. In particular, we focus on the context of neurosciences to detect possible interactions in the cerebral activity associated with two neurons. To provide a mathematical answer to this specific problem of neurobiologists, we address so the question of testing the nullity of the intensity h. We construct a multiple testing procedure obtained by the aggregation of single tests based on a wavelet thresholding method. This test has go…
Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $\int Φ(g(x))\,dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Hölder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $Φ$ which are satisfied for $Φ(u)=|u|^p$, $p\ge 1$. As an application, we consider the problem of estimating the $L^p$-norm and derive the minimax separation rates…
We consider a doubly stochastic Poisson process with stochastic intensity $λ_t =n q\left(X_t\right)$ where $X$ is a continuous Itô semimartingale and $n$ is an integer. Both processes are observed continuously over a fixed period $\left[0,T\right]$. An estimation procedure is proposed in a non parametrical setting for the function $q$ on an interval $I$ where $X$ is sufficiently observed using a local polynomial estimator. A method to select the bandwidth in a non asymptotic framework is proposed, leading to an oracle inequality. If $m$ is the degree of the chosen polynomial, the accuracy of o…
Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose single tests whose test statistics are U-statistics based on general kernel functions. The corresponding critical values are constructed from a non-asymptotic wild bootstrap approach, leading to level αtests. Various choices for the kernel functions are possible, including projection, approximation or reproducing kernels. In this last case, we obtain a parametric rate of testing for a weak metric defined in the RKHS associated with the considered reproduc…
Assume that we observe a sample of size n composed of p-dimensional signals, each signal having independent entries drawn from a scaled Poisson distribution with an unknown intensity. We are interested in estimating the sum of the n unknown intensity vectors, under the assumption that most of them coincide with a given 'background' signal. The number s of p-dimensional signals different from the background signal plays the role of sparsity and the goal is to leverage this sparsity assumption in order to improve the quality of estimation as compared to the naive estimator that computes the sum …
