We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion. We also consider regression with Gaussian errors and smooth classification under a logistic or probit link function combined with various priors.
We derive the posteror contraction rate for non-parametric Bayesian estimation of a deterministic dispersion coefficient of a linear stochastic differential equation.
We study posterior rates of contraction in Gaussian process regression with unbounded covariate domain. Our argument relies on developing a Gaussian approximation to the posterior of the leading coefficients of a Karhunen--Loéve expansion of the Gaussian process. The salient feature of our result is deriving such an approximation in the $L^2$ Wasserstein distance and relating the speed of the approximation to the posterior contraction rate using a coupling argument. Specific illustrations are provided for the Gaussian or squared-exponential covariance kernel.
We study nonparametric Bayesian inference with location mixtures of the Laplace density and a Dirichlet process prior on the mixing distribution. We derive a contraction rate of the corresponding posterior distribution, both for the mixing distribution relative to the Wasserstein metric and for the mixed density relative to the Hellinger and $L_q$ metrics.
We study posterior contraction rates for a class of deep Gaussian process priors applied to the nonparametric regression problem under a general composition assumption on the regression function. It is shown that the contraction rates can achieve the minimax convergence rate (up to $\log n$ factors), while being adaptive to the underlying structure and smoothness of the target function. The proposed framework extends the Bayesian nonparametric theory for Gaussian process priors.
Given a sample of a Poisson point process with intensity $λ_f(x,y) = n \mathbf{1}(f(x) \leq y),$ we study recovery of the boundary function $f$ from a nonparametric Bayes perspective. Because of the irregularity of this model, the analysis is non-standard. We establish a general result for the posterior contraction rate with respect to the $L^1$-norm based on entropy and one-sided small probability bounds. From this, specific posterior contraction results are derived for Gaussian process priors and priors based on random wavelet series.
This paper studies posterior contraction rates in multi-category logit models with priors incorporating group sparse structures. We consider a general class of logit models that includes the well-known multinomial logit models as a special case. Group sparsity is useful when predictor variables are naturally clustered and particularly useful for variable selection in the multinomial logit models. We provide a unified platform for posterior contraction rates of group-sparse logit models that include binary logistic regression under individual sparsity. No size restriction is directly imposed on…
We analyze the posterior contraction rates of parameters in Bayesian models via the Langevin diffusion process, in particular by controlling moments of the stochastic process and taking limits. Analogous to the non-asymptotic analysis of statistical M-estimators and stochastic optimization algorithms, our contraction rates depend on the structure of the population log-likelihood function, and stochastic perturbation bounds between the population and sample log-likelihood functions. Convergence rates are determined by a non-linear equation that relates the population-level structure to stochast…
We study posterior contraction behaviors for parameters of interest in the context of Bayesian mixture modeling, where the number of mixing components is unknown while the model itself may or may not be correctly specified. Two representative types of prior specification will be considered: one requires explicitly a prior distribution on the number of mixture components, while the other places a nonparametric prior on the space of mixing distributions. The former is shown to yield an optimal rate of posterior contraction on the model parameters under minimal conditions, while the latter can be…
We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman-Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upo…
We consider a family of infinite dimensional product measures with tails between Gaussian and exponential, which we call $p$-exponential measures. We study their measure-theoretic properties and in particular their concentration. Our findings are used to develop a general contraction theory of posterior distributions on nonparametric models with $p$-exponential priors in separable Banach parameter spaces. Our approach builds on the general contraction theory for Gaussian process priors in van der Vaart and van Zanten 2008, namely we use prior concentration to verify prior mass and entropy cond…
We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance…
In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows us to derive con- traction rates for the parameter of interest from contraction rates of the related direct problem of estimating transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive con- traction rates for priors that are not related to the singular value decomposition of the operator. We apply our result to several examples of linear inverse pro…
We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. If the observational noise is assumed to be Gaussian then this prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for give…
In recent years, shrinkage priors have received much attention in high-dimensional data analysis from a Bayesian perspective. Compared with widely used spike-and-slab priors, shrinkage priors have better computational efficiency. But the theoretical properties, especially posterior contraction rate, which is important in uncertainty quantification, are not established in many cases. In this paper, we apply global-local shrinkage priors to high-dimensional multivariate linear regression with unknown covariance matrix. We show that when the prior is highly concentrated near zero and has heavy ta…
Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz's consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new po…
The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking $L^r$-norm neighborhoods, $1\le r\le\infty$, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for $1\le r\le2$, but deteriorate as $r$ increases beyond 2. In the cas…
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…
We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributi…
This paper considers the posterior contraction of non-parametric Bayesian inference on non-homogeneous Poisson processes. We consider the quality of inference on a rate function $λ$, given non-identically distributed realisations, whose rates are transformations of $λ$. Such data arises frequently in practice due, for instance, to the challenges of making observations with limited resources or the effects of weather on detectability of events. We derive contraction rates for the posterior estimates arising from the Sigmoidal Gaussian Cox Process and Quadratic Gaussian Cox Process models. These…
We investigate the frequentist properties of Bayesian procedures for estimation based on the horseshoe prior in the sparse multivariate normal means model. Previous theoretical results assumed that the sparsity level, that is, the number of signals, was known. We drop this assumption and characterize the behavior of the maximum marginal likelihood estimator (MMLE) of a key parameter of the horseshoe prior. We prove that the MMLE is an effective estimator of the sparsity level, in the sense that it leads to (near) minimax optimal estimation of the underlying mean vector generating the data. Bes…
Supremum norm loss is intuitively more meaningful to quantify function estimation error in statistics. In the context of multivariate nonparametric regression with unknown error, we propose a Bayesian procedure based on spike-and-slab prior and wavelet projections to estimate the regression function and all its mixed partial derivatives. We show that their posterior distributions contract to the truth optimally and adaptively under supremum-norm loss. The master theorem through tests with exponential errors used in Bayesian nonparametrics was not adequate to deal with this problem, and we deve…
We study the posterior contraction behavior of the latent population structure that arises in admixture models as the amount of data increases. We adopt the geometric view of admixture models - alternatively known as topic models - as a data generating mechanism for points randomly sampled from the interior of a (convex) population polytope, whose extreme points correspond to the population structure variables of interest. Rates of posterior contraction are established with respect to Hausdorff metric and a minimum matching Euclidean metric defined on polytopes. Tools developed include posteri…
We study the contraction properties of a quasi-posterior distribution $\checkΠ_{n,d}$ obtained by combining a quasi-likelihood function and a sparsity inducing prior distribution on $\rset^d$, as both $n$ (the sample size), and $d$ (the dimension of the parameter) increase. We derive some general results that highlight a set of sufficient conditions under which $\checkΠ_{n,d}$ puts increasingly high probability on sparse subsets of $\rset^d$, and contracts towards the true value of the parameter. We apply these results to the analysis of logistic regression models, and binary graphical models,…
In this paper, we consider the so-called Shape Invariant Model which stands for the estimation of a function f0 submitted to a random translation of law g0 in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process P(f0,g0) as well as f0 and g0. We first provide some identifiability result on this model and then adopt a Bayesian point of view. In this view, we find some prior on f and g such that the posterior distribution concentrates around the functions f0 and g0 when n goes to infinity, we then obtain a co…
In this paper, we consider the so-called Shape Invariant Model which stands for the estimation of a function f0 submitted to a random translation of law g0 in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process P(f0,g0). In this perspective, we adopt a Bayesian point of view and find prior on f and g such that the posterior distribution concentrates at a polynomial rate around P(f0,g0) when n goes to infinity. We intensively use some Bayesian non parametric tools coupled with mixture models and believe tha…
We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge…
We offer a general Bayes theoretic framework to derive posterior contraction rates under a hierarchical prior design: the first-step prior serves to assess the model selection uncertainty, and the second-step prior quantifies the prior belief on the strength of the signals within the model chosen from the first step. In particular, we establish non-asymptotic oracle posterior contraction rates under (i) a local Gaussianity condition on the log likelihood ratio of the statistical experiment, (ii) a local entropy condition on the dimensionality of the models, and (iii) a sufficient mass conditio…
The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a large number of continuous shrinkage priors has been proposed. Many of these shrinkage priors can be written as a scale mixture of normals, which makes them particularly easy to implement. We propose general conditions on the prior on the local variance in scale mixtures of normals, such that posterior contraction at the minimax rate is assured. The conditions require tails at least as heavy as La…
We consider the nonparametric regression problem with multiple predictors and an additive error, where the regression function is assumed to be coordinatewise nondecreasing. We propose a Bayesian approach to make an inference on the multivariate monotone regression function, obtain the posterior contraction rate, and construct a universally consistent Bayesian testing procedure for multivariate monotonicity. To facilitate posterior analysis, we set aside the shape restrictions temporarily, and endow a prior on blockwise constant regression functions with heights independently normally distribu…
In the setting of nonparametric multivariate regression with unknown error variance, we study asymptotic properties of a Bayesian method for estimating a regression function f and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for f, and the error variance is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, L2 and supremum norm posterior contraction rates for f and its mixed partial derivatives, and show that they coinc…
Due to their conjugate posteriors, Gaussian process priors are attractive for estimating the drift of stochastic differential equations with continuous time observations. However, their performance strongly depends on the choice of the hyper-parameters. We employ the marginal maximum likelihood estimator to estimate the scaling and/or smoothness parameter(s) of the prior and show that the corresponding posterior has optimal rates of convergence. General theorems do not apply directly to this model as the usual test functions are with respect to a random Hellinger-type metric. We allow for cont…
Shrinkage prior are becoming more and more popular in Bayesian modeling for high dimensional sparse problems due to its computational efficiency. Recent works show that a polynomially decaying prior leads to satisfactory posterior asymptotics under regression models. In the literature, statisticians have investigated how the global shrinkage parameter, i.e., the scale parameter, in a heavy tail prior affects the posterior contraction. In this work, we explore how the shape of the prior, or more specifically, the polynomial order of the prior tail affects the posterior. We discover that, under …
We consider the fractional posterior distribution that is obtained by updating a prior distribution via Bayes theorem with a fractional likelihood function, a usual likelihood function raised to a fractional power. First, we analyze the contraction property of the fractional posterior in a general misspecified framework. Our contraction results only require a prior mass condition on certain Kullback-Leibler (KL) neighborhood of the true parameter (or the KL divergence minimizer in the misspecified case), and obviate constructions of test functions and sieves commonly used in the literature for…
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…
Principal component analysis (PCA) is possibly one of the most widely used statistical tools to recover a low-rank structure of the data. In the high-dimensional settings, the leading eigenvector of the sample covariance can be nearly orthogonal to the true eigenvector. A sparse structure is then commonly assumed along with a low rank structure. Recently, minimax estimation rates of sparse PCA were established under various interesting settings. On the other side, Bayesian methods are becoming more and more popular in high-dimensional estimation, but there is little work to connect frequentist…
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…
Sparse Bayesian factor models are routinely implemented for parsimonious dependence modeling and dimensionality reduction in high-dimensional applications. We provide theoretical understanding of such Bayesian procedures in terms of posterior convergence rates in inferring high-dimensional covariance matrices where the dimension can be larger than the sample size. Under relevant sparsity assumptions on the true covariance matrix, we show that commonly-used point mass mixture priors on the factor loadings lead to consistent estimation in the operator norm even when $p\gg n$. One of our major co…
The Indian buffet process (IBP) and phylogenetic Indian buffet process (pIBP) can be used as prior models to infer latent features in a data set. The theoretical properties of these models are under-explored, however, especially in high dimensional settings. In this paper, we show that under mild sparsity condition, the posterior distribution of the latent feature matrix, generated via IBP or pIBP priors, converges to the true latent feature matrix asymptotically. We derive the posterior convergence rate, referred to as the contraction rate. We show that the convergence holds even when the dim…
Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces
Department of Statistics, Seoul National University（首尔国立大学统计系）；Department of Statistics, Inha University（inha大学统计系）；Department of Mathematics, University of Maryland（马里兰大学数学系）
By expressing prior distributions as general stochastic processes, nonparametric Bayesian methods provide a flexible way to incorporate prior knowledge and constrain the latent structure in statistical inference. The Indian buffet process (IBP) is such an example that can be used to define a prior distribution on infinite binary features, where the exchangeability among subjects is assumed. The phylogenetic Indian buffet process (pIBP), a derivative of IBP, enables the modeling of non-exchangeability among subjects through a stochastic process on a rooted tree, which is similar to that used in…
We investigate the problem of deriving adaptive posterior rates of contraction on $\mathbb{L}^{\infty}$ balls in density estimation. Although it is known that log-density priors can achieve optimal rates when the true density is sufficiently smooth, adaptive rates were still to be proven. Here we establish that the so-called spike-and-slab prior can achieve adaptive and optimal posterior contraction rates. Along the way, we prove a generic $\mathbb{L}^{\infty}$ contraction result for log-density priors with independent wavelet coefficients. Interestingly, our approach is different from previou…
We consider the problem of estimation in Hidden Markov models with finite state space and nonparametric emission distributions. Efficient estimators for the transition matrix are exhibited, and a semiparametric Bernstein-von Mises result is deduced. Following from this, we propose a modular approach using the cut posterior to jointly estimate the transition matrix and the emission densities. We derive a general theorem on contraction rates for this approach. We then show how this result may be applied to obtain a contraction rate result for the emission densities in our setting; a key intermed…
We study full Bayesian procedures for high-dimensional linear regression. We adopt data-dependent empirical priors introduced in [1]. In their paper, these priors have nice posterior contraction properties and are easy to compute. Our paper extend their theoretical results to the case of unknown error variance . Under proper sparsity assumption, we achieve model selection consistency, posterior contraction rates as well as Bernstein von-Mises theorem by analyzing multivariate t-distribution.
Non-linear latent variable models have become increasingly popular in a variety of applications. However, there has been little study on theoretical properties of these models. In this article, we study rates of posterior contraction in univariate density estimation for a class of non-linear latent variable models where unobserved U(0,1) latent variables are related to the response variables via a random non-linear regression with an additive error. Our approach relies on characterizing the space of densities induced by the above model as kernel convolutions with a general class of continuous …
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…
This paper presents a new approach to the classical problem of quantifying posterior contraction rates (PCRs) in Bayesian statistics. Our approach relies on Wasserstein distance, and it leads to two main contributions which improve on the existing literature of PCRs. The first contribution exploits the dynamic formulation of Wasserstein distance, for short referred to as Wasserstein dynamics, in order to establish PCRs under dominated Bayesian statistical models. As a novelty with respect to existing approaches to PCRs, Wasserstein dynamics allows us to circumvent the use of sieves in both sta…
Non-linear latent variable models have become increasingly popular in a variety of applications. However, there has been little study on theoretical properties of these models. In this article, we study rates of posterior contraction in univariate density estimation for a class of non-linear latent variable models where unobserved U(0,1) latent variables are related to the response variables via a random non-linear regression with an additive error. Our approach relies on characterizing the space of densities induced by the above model as kernel convolutions with a general class of continuous …
We study the behavior of the posterior distribution in high-dimensional Bayesian Gaussian linear regression models having $p\gg n$, with $p$ the number of predictors and $n$ the sample size. Our focus is on obtaining quantitative finite sample bounds ensuring sufficient posterior probability assigned in neighborhoods of the true regression coefficient vector, $β^0$, with high probability. We assume that $β^0$ is approximately $S$-sparse and obtain universal bounds, which provide insight into the role of the prior in controlling concentration of the posterior. Based on these finite sample bound…
