The effectiveness of Bayesian Additive Regression Trees (BART) has been demonstrated in a variety of contexts including non-parametric regression and classification. A BART scheme for estimating the intensity of inhomogeneous Poisson processes is introduced. Poisson intensity estimation is a vital task in various applications including medical imaging, astrophysics and network traffic analysis. The new approach enables full posterior inference of the intensity in a non-parametric regression setting. The performance of the novel scheme is demonstrated through simulation studies on synthetic and real datasets up to five dimensions, and the new scheme is compared with alternative approaches.
Bayesian additive regression trees (BART) is a semi-parametric regression model offering state-of-the-art performance on out-of-sample prediction. Despite this success, standard implementations of BART typically provide inaccurate prediction and overly narrow prediction intervals at points outside the range of the training data. This paper proposes a novel extrapolation strategy that grafts Gaussian processes to the leaf nodes in BART for predicting points outside the range of the observed data. The new method is compared to standard BART implementations and recent frequentist resampling-based…
Few methods in Bayesian non-parametric statistics/ machine learning have received as much attention as Bayesian Additive Regression Trees (BART). While BART is now routinely performed for prediction tasks, its theoretical properties began to be understood only very recently. In this work, we continue the theoretical investigation of BART initiated by Rockova and van der Pas (2017). In particular, we study the Bernstein-von Mises (BvM) phenomenon (i.e. asymptotic normality) for smooth linear functionals of the regression surface within the framework of non-parametric regression with fixed covar…
UC Berkeley（加州大学伯克利分校）；National University of Singapore（新加坡国立大学）；Microsoft Research（微软研究院）
Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression algorithm. The posterior is a distribution over sums of decision trees, and predictions are made by averaging approximate samples from the posterior. The combination of strong predictive performance and the ability to provide uncertainty measures has led BART to be commonly used in the social sciences, biostatistics, and causal inference. BART uses Markov Chain Monte Carlo (MCMC) to obtain approximate posterior samples over a parameterized space of sums of trees, but it has often been observed that the ch…
Johns Hopkins University（约翰斯·霍普金斯大学）；Brown University（布朗大学）；University of Florida（佛罗里达大学）；Moi University（莫伊大学）
The multinomial probit Bayesian additive regression trees (MPBART) framework was proposed by Kindo et al. (KD), approximating the latent utilities in the multinomial probit (MNP) model with BART (Chipman et al. 2010). Compared to multinomial logistic models, MNP does not assume independent alternatives and the correlation structure among alternatives can be specified through multivariate Gaussian distributed latent utilities. We introduce two new algorithms for fitting the MPBART and show that the theoretical mixing rates of our proposals are equal or superior to the existing algorithm in KD. …
We develop a Bayesian "sum-of-trees" model where each tree is constrained by a regularization prior to be a weak learner, and fitting and inference are accomplished via an iterative Bayesian backfitting MCMC algorithm that generates samples from a posterior. Effectively, BART is a nonparametric Bayesian regression approach which uses dimensionally adaptive random basis elements. Motivated by ensemble methods in general, and boosting algorithms in particular, BART is defined by a statistical model: a prior and a likelihood. This approach enables full posterior inference including point and inte…
University of Pennsylvania（宾夕法尼亚大学）；Medical College of Wisconsin（威斯康星医学院）；Arizona State University（亚利桑那州立大学）
Bayesian Additive Regression Trees (BART) is a fully Bayesian approach to modeling with ensembles of trees. BART can uncover complex regression functions with high dimensional regressors in a fairly automatic way and provide Bayesian quantification of the uncertainty through the posterior. However, BART assumes IID normal errors. This strong parametric assumption can lead to misleading inference and uncertainty quantification. In this paper, we use the classic Dirichlet process mixture (DPM) mechanism to nonparametrically model the error distribution. A key strength of BART is that default pri…
Bayesian additive regression tree (BART) models have seen increased attention in recent years as a general-purpose nonparametric modeling technique. BART combines the flexibility of modern machine learning techniques with the principled uncertainty quantification of Bayesian inference, and it has been shown to be uniquely appropriate for addressing the high-noise problems that occur commonly in many areas of science, including medicine and the social sciences. This paper introduces the SoftBart package for fitting the Soft BART algorithm of Linero and Yang (2018). In addition to improving upon…
On the Computational Efficiency of Bayesian Additive Regression Trees: An Asymptotic Analysis
Department of Statistics and Data Science, National University of Singapore（统计与数据科学系，新加坡国立大学）；Department of Statistics, UC Berkeley（统计系，加州大学伯克利分校）；Department of Electrical Engineering and Computer Sciences, UC Berkeley（电气工程与计算机科学系，加州大学伯克利分校）；Center for Computational Biology, UC Berkeley（计算生物学中心，加州大学伯克利分校）
In recent years, infinite-dimensional methods have been introduced for the Gaussian channels estimation. The aim of this paper is to study the application of similar methods to Poisson channels. In particular we compute the Bayesian estimator of a Poisson channel using the likelihood ratio and the discrete Malliavin gradient. This algorithm is suitable for numerical implementation via the Monte-Carlo scheme. As an application we provide an new proof of the formula obtained recently by Guo, Shamai and Verduú relating some derivatives of the input-output mutual information of a time-continuous P…
Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity $λ_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,λ_0)$ at these rates. To the best of our knowledge, construction of nonparametric density est…
We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and…
Functional BART with Shape Priors: A Bayesian Tree Approach to Constrained Functional Regression
The University of Texas Health Science Center at Houston（德克萨斯大学健康科学中心休斯顿分校）；Beijing Technology and Business University（北京工商大学）；Beijing Normal-Hong Kong Baptist University（香港中文大学（深圳））
Bayesian Additive Regression Trees (BART) is a Bayesian approach to flexible non-linear regression which has been shown to be competitive with the best modern predictive methods such as those based on bagging and boosting. BART offers some advantages. For example, the stochastic search Markov Chain Monte Carlo (MCMC) algorithm can provide a more complete search of the model space and variation across MCMC draws can capture the level of uncertainty in the usual Bayesian way. The BART prior is robust in that reasonable results are typically obtained with a default prior specification. However, t…
In this paper we present Poisson mixture approaches for origin-destination (OD) modeling in transportation analysis. We introduce covariate-based models which incorporate different transport modeling phases and also allow for direct probabilistic inference on link traffic based on Bayesian predictions. Emphasis is placed on the Poisson-inverse Gaussian as an alternative to the commonly-used Poisson-gamma and Poisson-lognormal models. We present a first full Bayesian formulation and demonstrate that the Poisson-inverse Gaussian is particularly suited for OD analysis due to desirable marginal an…
Bayesian inference for graphical models has received much attention in the literature in recent years. It is well known that when the graph G is decomposable, Bayesian inference is significantly more tractable than in the general non-decomposable setting. Penalized likelihood inference on the other hand has made tremendous gains in the past few years in terms of scalability and tractability. Bayesian inference, however, has not had the same level of success, though a scalable Bayesian approach has its respective strengths, especially in terms of quantifying uncertainty. To address this gap, we…
Bayesian Additive Regression Trees(BART) is a Bayesian nonparametric approach which has been shown to be competitive with the best modern predictive methods such as random forest and Gradient Boosting Decision Tree.The sum of trees structure combined with a Bayesian inferential framework provide a accurate and robust statistic method.BART variant named SBART using randomized decision trees has been developed and show practical benefits compared to BART. The primary bottleneck of SBART is the speed to compute the sufficient statistics and the publicly avaiable implementation of the SBART algori…
Bayesian inference for models with intractable likelihood functions represents a challenging suite of problems in modern statistics. In this work we analyse the Conway-Maxwell-Poisson (COM-Poisson) distribution, a two parameter generalisation of the Poisson distribution. COM-Poisson regression modelling allows the flexibility to model dispersed count data as part of a generalised linear model (GLM) with a COM-Poisson response, where exogenous covariates control the mean and dispersion level of the response. The major difficulty with COM-Poisson regression is that the likelihood function contai…
We present the first fully variational Bayesian inference scheme for continuous Gaussian-process-modulated Poisson processes. Such point processes are used in a variety of domains, including neuroscience, geo-statistics and astronomy, but their use is hindered by the computational cost of existing inference schemes. Our scheme: requires no discretisation of the domain; scales linearly in the number of observed events; and is many orders of magnitude faster than previous sampling based approaches. The resulting algorithm is shown to outperform standard methods on synthetic examples, coal mining…
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…
University College London（伦敦大学学院）；University of Toronto（多伦多大学）；University of Oxford（牛津大学）
Additive regression trees are flexible non-parametric models and popular off-the-shelf tools for real-world non-linear regression. In application domains, such as bioinformatics, where there is also demand for probabilistic predictions with measures of uncertainty, the Bayesian additive regression trees (BART) model, introduced by Chipman et al. (2010), is increasingly popular. As data sets have grown in size, however, the standard Metropolis-Hastings algorithms used to perform inference in BART are proving inadequate. In particular, these Markov chains make local changes to the trees and suff…
Minimum message length is a general Bayesian principle for model selection and parameter estimation that is based on information theory. This paper applies the minimum message length principle to a small-sample model selection problem involving Poisson and geometric data models. Since MML is a Bayesian principle, it requires prior distributions for all model parameters. We introduce three candidate prior distributions for the unknown model parameters with both light- and heavy-tails. The performance of the MML methods is compared with objective Bayesian inference and minimum description length…
Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics, etc., to name but a few) and the bivariate Poisson distribution which is a generalization of the Poisson distribution plays an important role in modeling such data. In this article, we consider the inferential aspect of a bivariate Poisson conditionals distribution for which both the conditionals are Poisson but the marginals are typically non-Poisson. It has Poisson marginals only in the case of independence. It appears that a simple iterative procedure under the maximum likelihood method …
We provide a complete framework for performing infinite-dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data. In particular, we address the following issues to make the Bayesian framework applicable in practice. We first introduce a positivity-preserving reparametrization, and we prove that under the reparametrization and a hybrid prior, the posterior distribution is well-posed in the infinite dimensional setting. Second we provide a dimension-independent MCMC algorithm, based on the preconditioned Crank-Nicolson Langevin method, in which we …
A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more difficult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter $m$. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved …
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…
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…
Analyzing point patterns with linear structures has recently been of interest in e.g. neuroscience and geography. To detect anisotropy in such cases, we introduce a functional summary statistic, called the cylindrical $K$-function, since it is a directional $K$-function whose structuring element is a cylinder. Further we introduce a class of anisotropic Cox point processes, called Poisson line cluster point processes. The points of such a process are random displacements of Poisson point processes defined on the lines of a Poisson line process. Parameter estimation based on moment methods or B…
This paper introduces a new stochastic process with values in the set Z of integers with sign. The increments of process are Poisson differences and the dynamics has an autoregressive structure. We study the properties of the process and exploit the thinning representation to derive stationarity conditions and the stationary distribution of the process. We provide a Bayesian inference method and an efficient posterior approximation procedure based on Monte Carlo. Numerical illustrations on both simulated and real data show the effectiveness of the proposed inference.
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Raymond Corporation（雷蒙德公司）
Prediction of events such as part replacement and failure events plays a critical role in reliability engineering. Event stream data are commonly observed in manufacturing and teleservice systems. Designing predictive models for individual units based on such event streams is challenging and an under-explored problem. In this work, we propose a non-parametric prognostic framework for individualized event prediction based on the inhomogeneous Poisson processes with a multivariate Gaussian convolution process (MGCP) prior on the intensity functions. The MGCP prior on the intensity functions of t…
We develop nonparametric Bayesian modelling approaches for Poisson processes, using weighted combinations of structured beta densities to represent the point process intensity function. For a regular spatial domain, such as the unit square, the model construction implies a Bernstein-Dirichlet prior for the Poisson process density, which supports general inference for point process functionals. The key contribution of the methodology is two classes of flexible and computationally efficient models for spatial Poisson process intensities over irregular domains. We address the choice or estimation…
Questions about information encoded by the brain demand statistical frameworks for inferring relationships between neural firing and features of the world. The landmark discovery of grid cells demonstrates that neurons can represent spatial information through regularly repeating firing fields. However, the influence of covariates may be masked in current statistical models of grid cell activity, which by employing approaches such as discretizing, aggregating and smoothing, are computationally inefficient and do not account for the continuous nature of the physical world. These limitations mot…
We present a Bayesian reconstruction algorithm to generate unbiased samples of the underlying dark matter field from halo catalogues. Our new contribution consists of implementing a non-Poisson likelihood including a deterministic non-linear and scale-dependent bias. In particular we present the Hamiltonian equations of motions for the negative binomial (NB) probability distribution function. This permits us to efficiently sample the posterior distribution function of density fields given a sample of galaxies using the Hamiltonian Monte Carlo technique implemented in the Argo code. We have tes…
We present an approximate Bayesian inference approach for estimating the intensity of an inhomogeneous Poisson process, where the intensity function is modelled using a Gaussian process (GP) prior via a sigmoid link function. Augmenting the model using a latent marked Poisson process and Pólya--Gamma random variables we obtain a representation of the likelihood which is conjugate to the GP prior. We estimate the posterior using a variational free--form mean field optimisation together with the framework of sparse GPs. Furthermore, as alternative approximation we suggest a sparse Laplace's meth…
We develop a Bayesian Poisson matrix factorization model for forming recommendations from sparse user behavior data. These data are large user/item matrices where each user has provided feedback on only a small subset of items, either explicitly (e.g., through star ratings) or implicitly (e.g., through views or purchases). In contrast to traditional matrix factorization approaches, Poisson factorization implicitly models each user's limited attention to consume items. Moreover, because of the mathematical form of the Poisson likelihood, the model needs only to explicitly consider the observed …
We study the problem of non-parametric Bayesian estimation of the intensity function of a Poisson point process. The observations are $n$ independent realisations of a Poisson point process on the interval $[0,T]$. We propose two related approaches. In both approaches we model the intensity function as piecewise constant on $N$ bins forming a partition of the interval $[0,T]$. In the first approach the coefficients of the intensity function are assigned independent gamma priors, leading to a closed form posterior distribution. On the theoretical side, we prove that as $n\rightarrow\infty,$ the…
Statistical modeling of point patterns is an important and common problem in several areas. The Poisson process is the most common process used for this purpose, in particular, its generalization that considers the intensity function to be stochastic. This is called a Cox process and different choices to model the dynamics of the intensity gives rise to a wide range of models. We present a new class of unidimensional Cox process models in which the intensity function assumes parametric functional forms that switch among them according to a continuous-time Markov chain. A novel methodology is p…
We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministically from a parameter value and a PRM realisation. This forms the basis of our Data Augmented MCMC, which consists in augmenting parameter space with the unobserved PRM value. The resulting simple Metropolis-Hastings sampler acts as an efficient simu…
In this paper, a multivariate count distribution with Conway-Maxwell (COM)-Poisson marginals is proposed. To do this, we develop a modification of the Sarmanov method for constructing multivariate distributions. Our multivariate COM-Poisson (MultCOMP) model has desirable features such as (i) it admits a flexible covariance matrix allowing for both negative and positive non-diagonal entries; (ii) it overcomes the limitation of the existing bivariate COM-Poisson distributions in the literature that do not have COM-Poisson marginals; (iii) it allows for the analysis of multivariate counts and is …
Vector autoregressive (VAR) models assume linearity between the endogenous variables and their lags. This assumption might be overly restrictive and could have a deleterious impact on forecasting accuracy. As a solution, we propose combining VAR with Bayesian additive regression tree (BART) models. The resulting Bayesian additive vector autoregressive tree (BAVART) model is capable of capturing arbitrary non-linear relations between the endogenous variables and the covariates without much input from the researcher. Since controlling for heteroscedasticity is key for producing precise density f…
University of Cambridge（剑桥大学）；University of Bristol（布里斯托大学）
Recent work introduced deep kernel processes as an entirely kernel-based alternative to NNs (Aitchison et al. 2020). Deep kernel processes flexibly learn good top-layer representations by alternately sampling the kernel from a distribution over positive semi-definite matrices and performing nonlinear transformations. A particular deep kernel process, the deep Wishart process (DWP), is of particular interest because its prior can be made equivalent to deep Gaussian process (DGP) priors for kernels that can be expressed entirely in terms of Gram matrices. However, inference in DWPs has not yet b…
Analyzing the underlying structure of multiple time-sequences provides insights into the understanding of social networks and human activities. In this work, we present the \emph{Bayesian nonparametric Poisson process allocation} (BaNPPA), a latent-function model for time-sequences, which automatically infers the number of latent functions. We model the intensity of each sequence as an infinite mixture of latent functions, each of which is obtained using a function drawn from a Gaussian process. We show that a technical challenge for the inference of such mixture models is the unidentifiabilit…
Individualized treatment rules (ITR) can improve health outcomes by recognizing that patients may respond differently to treatment and assigning therapy with the most desirable predicted outcome for each individual. Flexible and efficient prediction models are desired as a basis for such ITRs to handle potentially complex interactions between patient factors and treatment. Modern Bayesian semiparametric and nonparametric regression models provide an attractive avenue in this regard as these allow natural posterior uncertainty quantification of patient specific treatment decisions as well as th…
We present a general framework, the coupled compound Poisson factorization (CCPF), to capture the missing-data mechanism in extremely sparse data sets by coupling a hierarchical Poisson factorization with an arbitrary data-generating model. We derive a stochastic variational inference algorithm for the resulting model and, as examples of our framework, implement three different data-generating models---a mixture model, linear regression, and factor analysis---to robustly model non-random missing data in the context of clustering, prediction, and matrix factorization. In all three cases, we tes…
While the hurdle Poisson regression is a popular class of models for count data with excessive zeros, the link function in the binary component may be unsuitable for highly imbalanced cases. Ordinary Poisson regression is unable to handle the presence of dispersion. In this paper, we introduce Conway-Maxwell-Poisson (CMP) distribution and integrate use of flexible skewed Weibull link functions as better alternative. We take a fully Bayesian approach to draw inference from the underlying models to better explain skewness and quantify dispersion, with Deviance Information Criteria (DIC) used for…
