We aim at modelling the appearance of distinct tags in a sequence of labelled objects. Common examples of this type of data include words in a corpus or distinct species in a sample. These sequential discoveries are often summarised via accumulation curves, which count the number of distinct entities observed in an increasingly large set of objects. We propose a novel Bayesian nonparametric method for species sampling modelling by directly specifying the probability of a new discovery, therefore allowing for flexible specifications. The asymptotic behavior and finite sample properties of such an approach are extensively studied. Interestingly, our enlarged class of sequential processes includes highly tractable special cases. We present a subclass of models characterized by appealing theoretical and computational properties. Moreover, due to strong connections with logistic regression models, the latter subclass can naturally account for covariates. We finally test our proposal on both synthetic and real data, with special emphasis on a large fungal biodiversity study in Finland.
We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov-Smirnov statistic based on the difference of the estimated innovation distributions built from the first ns and the last n-ns residuals, respectively. Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change …
The spectral energy distribution (SED) is a relatively easy way for astronomers to distinguish between different astronomical objects such as galaxies, black holes, and stellar objects. By comparing the observations from a source at different frequencies with template models, astronomers are able to infer the type of this observed object. In this paper, we take a Bayesian model averaging perspective to learn astronomical objects, employing a Bayesian nonparametric approach to accommodate the deviation from convex combinations of known log-SEDs. To effectively use telescope time for observation…
This paper describes how to specify probability models for data analysis via a backward induction procedure. The new approach yields coherent, prior-free uncertainty assessment. After presenting some intuition-building examples, the new approach is applied to a kernel density estimator, which leads to a novel method for computing point-wise credible intervals in nonparametric density estimation. The new approach has two additional advantages; 1) the posterior mean density can be accurately approximated without resorting to Monte Carlo simulation and 2) concentration bounds are easily establish…
The problem of maximizing cell type discovery under budget constraints is a fundamental challenge for the collection and analysis of single-cell RNA-sequencing (scRNA-seq) data. In this paper, we introduce a simple, computationally efficient, and scalable Bayesian nonparametric sequential approach to optimize the budget allocation when designing a large scale experiment for the collection of scRNA-seq data for the purpose of, but not limited to, creating cell atlases. Our approach relies on the following tools: i) a hierarchical Pitman-Yor prior that recapitulates biological assumptions regard…
We propose a Bayesian nonparametric utility-based group sequential design for a randomized clinical trial to compare a gel sealant to standard care for resolving air leaks after pulmonary resection. Clinically, resolving air leaks in the days soon after surgery is highly important, since longer resolution time produces undesirable complications that require extended hospitalization. The problem of comparing treatments is complicated by the fact that the resolution time distributions are skewed and multi-modal, so using means is misleading. We address these challenges by assuming Bayesian nonpa…
University of Michigan（密歇根大学）；Duke University（杜克大学）
Analysis of heterogeneous patterns in complex spatio-temporal data finds usage across various domains in applied science and engineering, including training autonomous vehicles to navigate in complex traffic scenarios. Motivated by applications arising in the transportation domain, in this paper we develop a model for learning heterogeneous and dynamic patterns of velocity field data. We draw from basic nonparameric Bayesian modeling elements such as hierarchical Dirichlet process and infinite hidden Markov model, while the smoothness of each homogeneous velocity field element is captured with…
Dynamic treatment regimes in oncology and other disease areas often can be characterized by an alternating sequence of treatments or other actions and transition times between disease states. The sequence of transition states may vary substantially from patient to patient, depending on how the regime plays out, and in practice there often are many possible counterfactual outcome sequences. For evaluating the regimes, the mean final overall time may be expressed as a weighted average of the means of all possible sums of successive transitions times. A common example arises in cancer therapies w…
We derive explicit Bayesian nonparametric analysis for a species sampling model with finitely many types of Gibbs form of type $α= -1$ recently introduced in Gnedin (2009). Our results complement existing analysis under Gibbs priors of type $α\in [0, 1)$ proposed in Lijoi et al. (2008). Calculations rely on a groups sequential construction of Gibbs partitions introduced in Cerquetti (2008).
A nonparametric Bayesian sparse graph linear dynamical system (SGLDS) is proposed to model sequentially observed multivariate data. SGLDS uses the Bernoulli-Poisson link together with a gamma process to generate an infinite dimensional sparse random graph to model state transitions. Depending on the sparsity pattern of the corresponding row and column of the graph affinity matrix, a latent state of SGLDS can be categorized as either a non-dynamic state or a dynamic one. A normal-gamma construction is used to shrink the energy captured by the non-dynamic states, while the dynamic states can be …
Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper we detail two major classes of sequential CRM representations---series representations and superposition representations---within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Si…
We construct a novel class of stochastic blockmodels using Bayesian nonparametric mixtures. These model allows us to jointly estimate the structure of multiple networks and explicitly compare the community structures underlying them, while allowing us to capture realistic properties of the underlying networks. Inference is carried out using MCMC algorithms that incorporates sequentially allocated split-merge steps to improve mixing. The models are illustrated using a simulation study and a variety of real-life examples.
In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or concave. We propose a new Bayesian nonparametric multivariate approach based on characterizing the unknown regression function as the max of a random collection of unknown hyperplanes. This specification induces a prior with large support in a Kullback-Leibler sense on the space …
We propose a sequential nonparametric test for detecting a change in distribution, based on windowed Kolmogorov--Smirnov statistics. The approach is simple, robust, highly computationally efficient, easy to calibrate, and requires no parametric assumptions about the underlying null and alternative distributions. We show that both the false-alarm rate and the power of our procedure are amenable to rigorous analysis, and that the method outperforms existing sequential testing procedures in practice. We then apply the method to the problem of detecting radiological anomalies, using data collected…
A gamma process dynamic Poisson factor analysis model is proposed to factorize a dynamic count matrix, whose columns are sequentially observed count vectors. The model builds a novel Markov chain that sends the latent gamma random variables at time $(t-1)$ as the shape parameters of those at time $t$, which are linked to observed or latent counts under the Poisson likelihood. The significant challenge of inferring the gamma shape parameters is fully addressed, using unique data augmentation and marginalization techniques for the negative binomial distribution. The same nonparametric Bayesian m…
This paper is concerned with the online estimation of a nonlinear dynamic system from a series of noisy measurements. The focus is on cases wherein outliers are present in-between normal noises. We assume that the outliers follow an unknown generating mechanism which deviates from that of normal noises, and then model the outliers using a Bayesian nonparametric model called Dirichlet process mixture (DPM). A sequential particle-based algorithm is derived for posterior inference for the outlier model as well as the state of the system to be estimated. The resulting algorithm is termed DPM based…
There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the ubiquitous Hidden Markov Model for learning from sequential and time-series data. However, in many settings the HDP-HMM's strict Markovian constraints are undesirable, particularly if we wish to learn or encode non-geometric state durations. We can extend the HDP-HMM to capture such structure by drawing upon explicit-duration semi-Markovianity, which has been developed mainly in the parametric frequentist setting, to allow construction of highly interp…
The aim of sequential change-point detection is to issue an alarm when it is thought that certain probabilistic properties of the monitored observations have changed. This work is concerned with nonparametric, closed-end testing procedures based on differences of empirical distribution functions that are designed to be particularly sensitive to changes in the comtemporary distribution of multivariate time series. The proposed detectors are adaptations of statistics used in a posteriori (offline) change-point testing and involve a weighting allowing to give more importance to recent observation…
Many exact Markov chain Monte Carlo algorithms have been developed for posterior inference in Bayesian nonparametric models which involve infinite-dimensional priors. However, these methods are not generic and special methodology must be developed for different classes of prior or different models. Alternatively, the infinite-dimensional prior can be truncated and standard Markov chain Monte Carlo methods used for inference. However, the error in approximating the infinite-dimensional posterior can be hard to control for many models. This paper describes an adaptive truncation method which all…
Time series prediction covers a vast field of every-day statistical applications in medical, environmental and economic domains. In this paper we develop nonparametric prediction strategies based on the combination of a set of 'experts' and show the universal consistency of these strategies under a minimum of conditions. We perform an in-depth analysis of real-world data sets and show that these nonparametric strategies are more flexible, faster and generally outperform ARMA methods in terms of normalized cumulative prediction error.
A Bayesian nonparametric method for unimodal densities on the real line is provided by considering a class of species sampling mixture models containing random densities that are unimodal and not necessarily symmetric. This class of densities generalize the model considered by Brunner (1992, Statist. Probab. Lett.), in which the Dirichlet process is replaced by a more general class of species sampling models. A novel and explicit characterization of the posterior distribution via a finite mixture of two dependent $\mathbf{S}$-paths is derived. This results in a closed-form and tractable Bayes …
Optimization via simulation (OvS) procedures that assume the simulation inputs are generated from the real-world distributions are subject to the risk of selecting a suboptimal solution when the distributions are substituted with input models estimated from finite real-world data -- known as input model risk. Focusing on discrete OvS, this paper proposes a new Bayesian framework for analyzing input model risk of implementing an arbitrary solution, $x$, where uncertainty about the input models is captured by a posterior distribution. We define the $α$-level risk set of solution $x$ as the set o…
Structural regularities in man-made environments reflect in the distribution of their surface normals. Describing these surface normal distributions is important in many computer vision applications, such as scene understanding, plane segmentation, and regularization of 3D reconstructions. Based on the small-variance limit of Bayesian nonparametric von-Mises-Fisher (vMF) mixture distributions, we propose two new flexible and efficient k-means-like clustering algorithms for directional data such as surface normals. The first, DP-vMF-means, is a batch clustering algorithm derived from the Dirich…
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）
We present algorithms for nonparametric regression in settings where the data are obtained sequentially. While traditional estimators select bandwidths that depend upon the sample size, for sequential data the effective sample size is dynamically changing. We propose a linear time algorithm that adjusts the bandwidth for each new data point, and show that the estimator achieves the optimal minimax rate of convergence. We also propose the use of online expert mixing algorithms to adapt to unknown smoothness of the regression function. We provide simulations that confirm the theoretical results,…
Naively trained neural networks tend to experience catastrophic forgetting in sequential task settings, where data from previous tasks are unavailable. A number of methods, using various model expansion strategies, have been proposed recently as possible solutions. However, determining how much to expand the model is left to the practitioner, and often a constant schedule is chosen for simplicity, regardless of how complex the incoming task is. Instead, we propose a principled Bayesian nonparametric approach based on the Indian Buffet Process (IBP) prior, letting the data determine how much to…
We study the problem of estimating the continuous response over time to interventions using observational time series---a retrospective dataset where the policy by which the data are generated is unknown to the learner. We are motivated by applications where response varies by individuals and therefore, estimating responses at the individual-level is valuable for personalizing decision-making. We refer to this as the problem of estimating individualized treatment response (ITR) curves. In statistics, G-computation formula (Robins, 1986) has been commonly used for estimating treatment responses…
State University of New York at Buffalo（纽约州立大学布法罗分校）；University of California, Irvine（加州大学欧文分校）
Current state-of-the-art nonparametric Bayesian text clustering methods model documents through multinomial distribution on bags of words. Although these methods can effectively utilize the word burstiness representation of documents and achieve decent performance, they do not explore the sequential information of text and relationships among synonyms. In this paper, the documents are modeled as the joint of bags of words, sequential features and word embeddings. We proposed Sequential Embedding induced Dirichlet Process Mixture Model (SiDPMM) to effectively exploit this joint document represe…
We develop a nonparametric extension of the sequential generalized likelihood ratio (GLR) test and corresponding time-uniform confidence sequences for the mean of a univariate distribution. By utilizing a geometric interpretation of the GLR statistic, we derive a simple analytic upper bound on the probability that it exceeds any prespecified boundary; these are intractable to approximate via simulations due to infinite horizon of the tests and the composite nonparametric nulls under consideration. Using time-uniform boundary-crossing inequalities, we carry out a unified nonasymptotic analysis …
This paper develops non-parametric rotation invariant CUSUMs suited to the detection of changes in the mean direction as well as changes in the concentration parameter of angular data. The properties of the CUSUMs are illustrated by theoretical calculations, Monte Carlo simulation and application to sequentially observed angular data from health science and astrophysics.
We consider the problem of bandwidth selection by cross-validation from a sequential point of view in a nonparametric regression model. Having in mind that in applications one often aims at estimation, prediction and change detection simultaneously, we investigate that approach for sequential kernel smoothers in order to base these tasks on a single statistic. We provide uniform weak laws of large numbers and weak consistency results for the cross-validated bandwidth. Extensions to weakly dependent error terms are discussed as well. The errors may be α-mixing or L2-near epoch dependent, which …
University of California, San Diego（加州大学圣地亚哥分校）；University of California, Berkeley（加州大学伯克利分校）
We propose a new algorithmic framework for sequential hypothesis testing with i.i.d. data, which includes A/B testing, nonparametric two-sample testing, and independence testing as special cases. It is novel in several ways: (a) it takes linear time and constant space to compute on the fly, (b) it has the same power guarantee as a non-sequential version of the test with the same computational constraints up to a small factor, and (c) it accesses only as many samples as are required - its stopping time adapts to the unknown difficulty of the problem. All our test statistics are constructed to b…
The API economy refers to the widespread integration of API (advanced programming interface) microservices, where software applications can communicate with each other, as a crucial element in business models and functions. The number of possible ways in which such a system could be used is huge. It is thus desirable to monitor the usage patterns and identify when the system is used in a way that was never used before. This provides a warning to the system analysts and they can ensure uninterrupted operation of the system. In this work we analyze both histograms and call graph of API usage to …
The aim of online monitoring is to issue an alarm as soon as there is significant evidence in the collected observations to suggest that the underlying data generating mechanism has changed. This work is concerned with open-end, nonparametric procedures that can be interpreted as statistical tests. The proposed monitoring schemes consist of computing the so-called retrospective CUSUM statistic (or minor variations thereof) after the arrival of each new observation. After proposing suitable threshold functions for the chosen detectors, the asymptotic validity of the procedures is investigated i…
Inference procedures in sequential trial emulation with survival outcomes: comparing confidence intervals based on the sandwich variance estimator, bootstrap and jackknife
The nonparametric test for change-point detection proposed by Gombay and Horváth is revisited and extended in the broader setting of empirical process theory. The resulting testing procedure for potentially multivariate observations is based on a sequential generalization of the functional multiplier central limit theorem and on modifications of Gombay and Horváth's seminal approach that appears to improve the finite-sample behavior of the tests. A large number of candidate test statistics based on processes indexed by lower-left orthants and half-spaces are considered and their performance is…
Pranab Kumar Sen was born on November 7, 1937 in Calcutta, India. His father died when Pranab was 10 years old, so his mother raised the family of seven children. Given his superior performance on an exam, Pranab nearly went into medical school, but did not because he was underage. He received a B.Sc. degree in 1955 and an M.Sc. degree in 1957 in statistics from Calcutta University, topping the class both times. Dr. Sen's dissertation on order statistics and nonparametrics, under the direction of Professor Hari Kinkar Nandi, was completed in 1961. After teaching for three years at Calcutta Uni…
This article introduces a flexible and adaptive nonparametric method for estimating the association between multiple covariates and power spectra of multiple time series. The proposed approach uses a Bayesian sum of trees model to capture complex dependencies and interactions between covariates and the power spectrum, which are often observed in studies of biomedical time series. Local power spectra corresponding to terminal nodes within trees are estimated nonparametrically using Bayesian penalized linear splines. The trees are considered to be random and fit using a Bayesian backfitting Mark…
Flow cytometry is a high-throughput technology used to quantify multiple surface and intracellular markers at the level of a single cell. This enables to identify cell sub-types, and to determine their relative proportions. Improvements of this technology allow to describe millions of individual cells from a blood sample using multiple markers. This results in high-dimensional datasets, whose manual analysis is highly time-consuming and poorly reproducible. While several methods have been developed to perform automatic recognition of cell populations, most of them treat and analyze each sample…
Change point analysis has applications in a wide variety of fields. The general problem concerns the inference of a change in distribution for a set of time-ordered observations. Sequential detection is an online version in which new data is continually arriving and is analyzed adaptively. We are concerned with the related, but distinct, offline version, in which retrospective analysis of an entire sequence is performed. For a set of multivariate observations of arbitrary dimension, we consider nonparametric estimation of both the number of change points and the positions at which they occur. …
Bayesian nonparametric methods are a popular choice for analysing survival data due to their ability to flexibly model the distribution of survival times. These methods typically employ a nonparametric prior on the survival function that is conjugate with respect to right-censored data. Eliciting these priors, particularly in the presence of covariates, can be challenging and inference typically relies on computationally intensive Markov chain Monte Carlo schemes. In this paper, we build on recent work that recasts Bayesian inference as assigning a predictive distribution on the unseen values …
In health-related experiments, treatment effects can be identified using paired data that consist of pre- and post-treatment measurements. In this framework, sequential testing strategies are widely accepted statistical tools in practice. Since performances of parametric sequential testing procedures vitally depend on the validity of the parametric assumptions regarding underlying data distributions, we focus on distribution-free mechanisms for sequentially evaluating treatment effects. In fixed sample size designs, the density-based empirical likelihood (DBEL) methods provide powerful nonpara…
There are many different ways in which change point analysis can be performed, from purely parametric methods to those that are distribution free. The ecp package is designed to perform multiple change point analysis while making as few assumptions as possible. While many other change point methods are applicable only for univariate data, this R package is suitable for both univariate and multivariate observations. Estimation can be based upon either a hierarchical divisive or agglomerative algorithm. Divisive estimation sequentially identifies change points via a bisection algorithm. The aggl…
Classifier predictions often rely on the assumption that new observations come from the same distribution as training data. When the underlying distribution changes, so does the optimal classification rule, and performance may degrade. We consider the problem of detecting such a change in distribution in sequentially-observed, unlabeled classification data. We focus on label shift changes to the distribution, where the class priors shift but the class conditional distributions remain unchanged. We reduce this problem to the problem of detecting a change in the one-dimensional classifier scores…
Nonparametric tests for equality of multivariate distributions are frequently desired in research. It is commonly required that test-procedures based on relatively small samples of vectors accurately control the corresponding Type I Error (TIE) rates. Often, in the multivariate testing, extensions of null-distribution-free univariate methods, e.g., Kolmogorov-Smirnov and Cramer-von Mises type schemes, are not exact, since their null distributions depend on underlying data distributions. The present paper extends the density-based empirical likelihood technique in order to nonparametrically app…
We propose nonparametric open-end sequential testing procedures that can detect all types of changes in the contemporary distribution function of possibly multivariate observations. Their asymptotic properties are theoretically investigated under stationarity and under alternatives to stationarity. Monte Carlo experiments reveal their good finite-sample behavior in the case of continuous univariate, bivariate and trivariate observations. A short data example concludes the work.
