We consider nonparametric Bayesian estimation of a probability density $p$ based on a random sample of size $n$ from this density using a hierarchical prior. The prior consists, for instance, of prior weights on the regularity of the unknown density combined with priors that are appropriate given that the density has this regularity. More generally, the hierarchy consists of prior weights on an abstract model index and a prior on a density model for each model index. We present a general theorem on the rate of contraction of the resulting posterior distribution as $n\to \infty$, which gives conditions under which the rate of contraction is the one attached to the model that best approximates the true density of the observations. This shows that, for instance, the posterior distribution can adapt to the smoothness of the underlying density. We also study the posterior distribution of the model index, and find that under the same conditions the posterior distribution gives negligible weight to models that are bigger than the optimal one, and thus selects the optimal model or smaller models that also approximate the true density well. We apply these result to log spline density models, where we show that the prior weights on the regularity index interact with the priors on the models, making the exact rates depend in a complicated way on the priors, but also that the rate is fairly robust to specification of the prior weights.
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…
A density estimation method in a Bayesian nonparametric framework is presented when recorded data are not coming directly from the distribution of interest, but from a length biased version. From a Bayesian perspective, efforts to computationally evaluate posterior quantities conditionally on length biased data were hindered by the inability to circumvent the problem of a normalizing constant. In this paper we present a novel Bayesian nonparametric approach to the length bias sampling problem which circumvents the issue of the normalizing constant. Numerical illustrations as well as a real dat…
We estimate the quantum state of a light beam from results of quantum homodyne tomography noisy measurements performed on identically prepared quantum systems. We propose two Bayesian nonparametric approaches. The first approach is based on mixture models and is illustrated through simulation examples. The second approach is based on random basis expansions. We study the theoretical performance of the second approach by quantifying the rate of contraction of the posterior distribution around the true quantum state in the $L^2$ metric.
This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional co…
We develop a set of scalable Bayesian inference procedures for a general class of nonparametric regression models. Specifically, nonparametric Bayesian inferences are separately performed on each subset randomly split from a massive dataset, and then the obtained local results are aggregated into global counterparts. This aggregation step is explicit without involving any additional computation cost. By a careful partition, we show that our aggregated inference results obtain an oracle rule in the sense that they are equivalent to those obtained directly from the entire data (which are computa…
We consider inference in the scalar diffusion model $dX_t=b(X_t)dt+σ(X_t)dW_t$ with discrete data $(X_{jΔ_n})_{0\leq j \leq n}$, $n\to \infty,~Δ_n\to 0$ and periodic coefficients. For $σ$ given, we prove a general theorem detailing conditions under which Bayesian posteriors will contract in $L^2$-distance around the true drift function $b_0$ at the frequentist minimax rate (up to logarithmic factors) over Besov smoothness classes. We exhibit natural nonparametric priors which satisfy our conditions. Our results show that the Bayesian method adapts both to an unknown sampling regime and to unkn…
Measure-valued Markov chains have raised interest in Bayesian nonparametrics since the seminal paper by (Math. Proc. Cambridge Philos. Soc. 105 (1989) 579--585) where a Markov chain having the law of the Dirichlet process as unique invariant measure has been introduced. In the present paper, we propose and investigate a new class of measure-valued Markov chains defined via exchangeable sequences of random variables. Asymptotic properties for this new class are derived and applications related to Bayesian nonparametric mixture modeling, and to a generalization of the Markov chain proposed by (M…
We propose a new, two-step empirical Bayes-type of approach for neural networks. We show in context of the nonparametric regression model that the procedure (up to a logarithmic factor) provides optimal recovery of the underlying functional parameter of interest and provides Bayesian credible sets with frequentist coverage guarantees. The approach requires fitting the neural network only once, hence it is substantially faster than Bootstrapping type approaches. We demonstrate the applicability of our method over synthetic data, observing good estimation properties and reliable uncertainty quan…
In this paper, we describe a Bayesian nonparametric approach to make inference for a bivariate spherically symmetric distribution. We consider a Dirichlet invariant process prior on the set of all bivariate spherically symmetric distributions and we derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of Dirichlet invariant process for the symmetric distributions on the real line to a bivariate spherically symmetric distribution where the underlying distribution is invariant under a finite group of rotations. Moreover, we obtain the Dirichlet invariant process…
We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.
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…
Bayesian nonparametric statistics is an area of considerable research interest. While recently there has been an extensive concentration in developing Bayesian nonparametric procedures for model checking, the use of the Dirichlet process, in its simplest form, along with the Kullback-Leibler divergence is still an open problem. This is mainly attributed to the discreteness property of the Dirichlet process and that the Kullback-Leibler divergence between any discrete distribution and any continuous distribution is infinity. The approach proposed in this paper, which is based on incorporating t…
Favaro, Lijoi, and Pruenster (2012, Biometrics, 68, 1188--1196) derive a novel Bayesian nonparametric estimator of the probability of detecting at the $(n+m+1)$th observation a species already observed with any given frequency in an enlarged sample of size $n+m$, conditionally on a basic sample of size $n$. Unfortunately the general result under Gibbs priors (Theorem 2), and consequently the explicit result under $(α, θ)$ Poisson-Dirichlet priors (Proposition 3), appear to be wrong. Here we provide the correct formulas for both the results, obtained by means of a new technique devised in Cerqu…
We develop a unifying framework for Bayesian nonparametric regression to study the rates of contraction with respect to the integrated $L_2$-distance without assuming the regression function space to be uniformly bounded. The framework is very flexible and can be applied to a wide class of nonparametric prior models. Three non-trivial applications of the proposed framework are provided: The finite random series regression of an $α$-Hölder function, with adaptive rates of contraction up to a logarithmic factor; The un-modified block prior regression of an $α$-Sobolev function, with adaptive-and…
In this work, nonparametric statistical inference is provided for the continuous-time M/G/1 queueing model from a Bayesian point of view. The inference is based on observations of the inter-arrival and service times. Beside other characteristics of the system, particular interest is in the waiting time distribution which is not accessible in closed form. Thus, we use an indirect statistical approach by exploiting the Pollaczek-Khinchine transform formula for the Laplace transform of the waiting time distribution. Due to this, an estimator is defined and its frequentist validation in terms of p…
We study location-scale mixture priors for nonparametric statistical problems, including multivariate regression, density estimation and classification. We show that a rate-adaptive procedure can be obtained if the prior is properly constructed. In particular, we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a Gaussian kernel, an inverse gamma bandwidth, and Gaussian mixing weights.
We introduce a three-parameter random walk with reinforcement, called the $(θ,α,β)$ scheme, which generalizes the linearly edge reinforced random walk to uncountable spaces. The parameter $β$ smoothly tunes the $(θ,α,β)$ scheme between this edge reinforced random walk and the classical exchangeable two-parameter Hoppe urn scheme, while the parameters $α$ and $θ$ modulate how many states are typically visited. Resorting to de Finetti's theorem for Markov chains, we use the $(θ,α,β)$ scheme to define a nonparametric prior for Bayesian analysis of reversible Markov chains. The prior is applied in…
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).
We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law.
Assume we observe a finite number of inspection times together with information on whether a specific event has occurred before each of these times. Suppose replicated measurements are available on multiple event times. The set of inspection times, including the number of inspections, may be different for each event. This is known as mixed case interval censored data. We consider Bayesian estimation of the distribution function of the event time while assuming it is concave. We provide sufficient conditions on the prior such that the resulting procedure is consistent from the Bayesian point of…
We propose a kernel mixture of polynomials prior for Bayesian nonparametric regression. The regression function is modeled by local averages of polynomials with kernel mixture weights. We obtain the minimax-optimal rate of contraction of the full posterior distribution up to a logarithmic factor that adapts to the smoothness level of the true function by estimating metric entropies of certain function classes. We also provide a frequentist sieve maximum likelihood estimator with a near-optimal convergence rate. We further investigate the application of the kernel mixture of polynomials to the …
Discrete Bayesian nonparametric models whose expectation is a convex linear combination of a point mass at some point of the support and a diffuse probability distribution allow to incorporate strong prior information, while still being extremely flexible. Recent contributions in the statistical literature have successfully implemented such a modelling strategy in a variety of applications, including density estimation, nonparametric regression and model-based clustering. We provide a thorough study of a large class of nonparametric models we call inner spike and slab hNRMI models, which are o…
The problem of deciding whether two samples arise from the same distribution is often the question of interest in many research investigations. Numerous statistical methods have been devoted to this issue, but only few of them have considered a Bayesian nonparametric approach. We propose a nonparametric Bayesian index (WIKS) which has the goal of quantifying the difference between two populations $P_1$ and $P_2$ based on samples from them. The WIKS index is defined by a weighted posterior expectation of the Kolmogorov-Smirnov distance between $P_1$ and $P_2$ and, differently from most existing…
A Bayesian nonparametric approach to the study of species diversity based on choosing a random discrete distribution as a prior model for the unknown relative abundances of species has been recently introduced in Lijoi et al. (2007, 2008). Explicit posterior predictive estimation of {\it species richness} has been obtained under priors belonging to the $α$-Gibbs class (Gnedin & Pitman, 2006). Here we focus on posterior estimation of {\it species evenness} which accounts for diversity in terms of the proximity to the situation of uniform distribution of the population into different species. We…
Posterior contractions rates (PCRs) strengthen the notion of Bayesian consistency, quantifying the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the true model, with probability tending to 1 or almost surely, as the sample size goes to infinity. Under the Bayesian nonparametric framework, a common assumption in the study of PCRs is that the model is dominated for the observations; that is, it is assumed that the posterior can be written through the Bayes formula. In this paper, we consider the problem of establishing PCRs in Bayesian nonparametric…
We introduce verifiable criteria for weak posterior consistency of identifiable Bayesian nonparametric inference for jump diffusions with unit diffusion coefficient and uniformly Lipschitz drift and jump coefficients in arbitrary dimension. The criteria are expressed in terms of coefficients of the SDEs describing the process, and do not depend on intractable quantities such as transition densities. We also show that products of discrete net and Dirichlet mixture model priors satisfy our conditions, again under an identifiability assumption. This generalises known results by incorporating jump…
This paper studies the problem of testing whether a function is monotone from a nonparametric Bayesian perspective. Two new families of tests are constructed. The first uses constrained smoothing splines, together with a hierarchical stochastic-process prior that explicitly controls the prior probability of monotonicity. The second uses regression splines, together with two proposals for the prior over the regression coefficients. The finite-sample performance of the tests is shown via simulation to improve upon existing frequentist and Bayesian methods. The asymptotic properties of the Bayes …
This paper deals with the estimation of the unknown distribution of hidden random variables from the observation of pairwise comparisons between these variables. This problem is inspired by recent developments on Bradley-Terry models in random environment since this framework happens to be relevant to predict for instance the issue of a championship from the observation of a few contests per team. This paper provides three contributions on a Bayesian nonparametric approach to solve this problem. First, we establish contraction rates of the posterior distribution. We also propose a Markov Chain…
We present the Gaussian process density sampler (GPDS), an exchangeable generative model for use in nonparametric Bayesian density estimation. Samples drawn from the GPDS are consistent with exact, independent samples from a distribution defined by a density that is a transformation of a function drawn from a Gaussian process prior. Our formulation allows us to infer an unknown density from data using Markov chain Monte Carlo, which gives samples from the posterior distribution over density functions and from the predictive distribution on data space. We describe two such MCMC methods. Both me…
Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical res…
Given discrete time observations over a fixed time interval, we study a nonparametric Bayesian approach to estimation of the volatility coefficient of a stochastic differential equation. We postulate a histogram-type prior on the volatility with piecewise constant realisations on bins forming a partition of the time interval. The values on the bins are assigned an inverse Gamma Markov chain (IGMC) prior. Posterior inference is straightforward to implement via Gibbs sampling, as the full conditional distributions are available explicitly and turn out to be inverse Gamma. We also discuss in deta…
Models phrased though moment conditions are central to much of modern inference. Here these moment conditions are embedded within a nonparametric Bayesian setup. Handling such a model is not probabilistically straightforward as the posterior has support on a manifold. We solve the relevant issues, building new probability and computational tools using Hausdorff measures to analyze them on real and simulated data. These new methods which involve simulating on a manifold can be applied widely, including providing Bayesian analysis of quasi-likelihoods, linear and nonlinear regression, missing da…
We consider nonparametric Bayesian inference in a reflected diffusion model $dX_t = b (X_t)dt + σ(X_t) dW_t,$ with discretely sampled observations $X_0, X_Δ, \dots, X_{nΔ}$. We analyse the nonlinear inverse problem corresponding to the `low frequency sampling' regime where $Δ>0$ is fixed and $n \to \infty$. A general theorem is proved that gives conditions for prior distributions $Π$ on the diffusion coefficient $σ$ and the drift function $b$ that ensure minimax optimal contraction rates of the posterior distribution over Hölder-Sobolev smoothness classes. These conditions are verified for nat…
We consider the problem of the Bayesian inference of drift and diffusion coefficient functions in a stochastic differential equation given discrete observations of a realisation of its solution. We give conditions for the well-posedness and stable approximations of the posterior measure. These conditions in particular allow for priors with unbounded support. Our proof relies on the explicit construction of transition probability densities using the parametrix method for general parabolic equations. We then study an application of these results in inferring the rates of Birth-and-Death processe…
In this paper we review recently developed methods for nonparametric Bayesian inference for one-dimensional diffusion models. We discuss different possible prior distributions, computational issues, and asymptotic results.
Mixture models are one of the most widely used statistical tools when dealing with data from heterogeneous populations. This paper considers the long-standing debate over finite mixture and infinite mixtures and brings the two modelling strategies together, by showing that a finite mixture is simply a realization of a point process. Following a Bayesian nonparametric perspective, we introduce a new class of prior: the Normalized Independent Point Processes. We investigate the probabilistic properties of this new class. Moreover, we design a conditional algorithm for finite mixture models with …
In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. {We revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give a…
This paper is a note on the use of Bayesian nonparametric mixture models for continuous time series. We identify a key requirement for such models, and then establish that there is a single type of model which meets this requirement. As it turns out, the model is well known in multiple change-point problems.
Due to their great flexibility, nonparametric Bayes methods have proven to be a valuable tool for discovering complicated patterns in data. The term "nonparametric Bayes" suggests that these methods inherit model-free operating characteristics of classical nonparametric methods, as well as coherent uncertainty assessments provided by Bayesian procedures. However, as the authors say in the conclusion to their article, nonparametric Bayesian methods may be more aptly described as "massively parametric." Furthermore, I argue that many of the default nonparametric Bayes procedures are only Bayesia…
This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Polya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type…
A key problem in statistical modeling is model selection, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number ofclusters in mixture models or the number of factors in factor analysis. In this tutorial we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduction to Bayesian nonparametric methods and contains several examples of their application.
Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regression can be derived this way from Bayesian linear regression. Despite the success of the Gaussian process framework, the kernel trick is rarely explicitly considered in the Bayesian literature. In this paper, we aim to fill this gap and demonstrate the potential of applying the …
The vast majority of models for the spread of communicable diseases are parametric in nature and involve underlying assumptions about how the disease spreads through a population. In this article we consider the use of Bayesian nonparametric approaches to analysing data from disease outbreaks. Specifically we focus on methods for estimating the infection process in simple models under the assumption that this process has an explicit time-dependence.
