Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"
Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets" by Szabó, van der Vaart and van Zanten [arXiv:1310.4489v5].
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arXiv subjects
Subhashis Ghosal
Publications and source records attributed to Subhashis Ghosal.
Bayesian inference for higher order ordinary differential equation models
Often the regression function appearing in fields like economics, engineering, biomedical sciences obeys a system of higher order ordinary differential equations (ODEs). The equations are usually not analytically solvable. We are interested in inferring on the unknown parameters appearing in the equations. Significant amount of work has been done on parameter estimation in first order ODE models. Bhaumik and Ghosal (2014a) considered a two-step Bayesian approach by putting a finite random series prior on the regression function using B-spline basis. The posterior distribution of the parameter vector is induced from that of the regression function. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. Bhaumik and Ghosal (2014b) remedied this by directly considering the distance between the function in the nonparametric model and a Runge-Kutta (RK$4$) approximate solution of the ODE while inducing the posterior distribution on the parameter. They also studied the direct Bayesian method obtained from the approximate likelihood obtained by the RK4 method. In this paper we extend these ideas for the higher order ODE model and establish Bernstein-von Mises theorems for the posterior distribution of the parameter vector for each method with $n^{-1/2}$ contraction rate.
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MCMC-free adaptive Bayesian procedures using random series prior
We consider priors for several nonparametric Bayesian models which use finite random series with a random number of terms. The prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on the construction of an appropriate sieve and obtain adaptive posterior contraction rates for all smoothness levels of the function in the true model. We apply this general result on several statistical problems such as signal processing, density estimation, nonparametric additive regression, classification, spectral density estimation, functional regression etc. The prior can be viewed as an alternative to commonly used Gaussian process prior, but can be analyzed by relatively simpler techniques and in many cases allows a simpler approach to computation without using Markov chain Monte-Carlo (MCMC) methods. A simulation study was conducted to show that the performance of the random series prior is comparable to that of a Gaussian process prior.
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Adaptive Bayesian procedures using random series priors
We consider a prior for nonparametric Bayesian estimation which uses finite random series with a random number of terms. The prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior convergence rates for all smoothness levels of the function in the true model by constructing an appropriate "sieve" and applying the general theory of posterior convergence rates. We apply this general result on several statistical problems such as signal processing, density estimation, various nonparametric regressions, classification, spectral density estimation, functional regression etc. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analyzed by relatively simpler techniques and in many cases allows a simpler approach to computation without using Markov chain Monte-Carlo (MCMC) methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on two interesting data sets on functional regression.
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Posterior convergence rates for estimating large precision matrices using graphical models
We consider Bayesian estimation of a $p\times p$ precision matrix, when $p$ can be much larger than the available sample size $n$. It is well known that consistent estimation in such ultra-high dimensional situations requires regularization such as banding, tapering or thresholding. We consider a banding structure in the model and induce a prior distribution on a banded precision matrix through a Gaussian graphical model, where an edge is present only when two vertices are within a given distance. For a proper choice of the order of graph, we obtain the convergence rate of the posterior distribution and Bayes estimators based on the graphical model in the $L_{\infty}$-operator norm uniformly over a class of precision matrices, even if the true precision matrix may not have a banded structure. Along the way to the proof, we also compute the convergence rate of the maximum likelihood estimator (MLE) under the same set of condition, which is of independent interest. The graphical model based MLE and Bayes estimators are automatically positive definite, which is a desirable property not possessed by some other estimators in the literature. We also conduct a simulation study to compare finite sample performance of the Bayes estimators and the MLE based on the graphical model with that obtained by using a Cholesky decomposition of the precision matrix. Finally, we discuss a practical method of choosing the order of the graphical model using the marginal likelihood function.
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Bayesian two-step estimation in differential equation models
Ordinary differential equations (ODEs) are used to model dynamic systems appearing in engineering, physics, biomedical sciences and many other fields. These equations contain unknown parameters, say $\bmθ$ of physical significance which have to be estimated from the noisy data. Often there is no closed form analytic solution of the equations and hence we cannot use the usual non-linear least squares technique to estimate the unknown parameters. There is a two-step approach to solve this problem, where the first step involves fitting the data nonparametrically. In the second step the parameter is estimated by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the system of ODEs. The statistical aspects of this approach have been studied under the frequentist framework. We consider this two-step estimation under the Bayesian framework. The response variable is allowed to be multidimensional and the true mean function of it is not assumed to be in the model. We induce a prior on the regression function using a random series based on the B-spline basis functions. We establish the Bernstein-von Mises theorem for the posterior distribution of the parameter of interest. Interestingly, even though the posterior distribution of the regression function based on splines converges at a rate slower than $n^{-1/2}$, the parameter vector $\bmθ$ is nevertheless estimated at $n^{-1/2}$ rate.
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Bayesian estimation of a sparse precision matrix
We consider the problem of estimating a sparse precision matrix of a multivariate Gaussian distribution, including the case where the dimension $p$ is large. Gaussian graphical models provide an important tool in describing conditional independence through presence or absence of the edges in the underlying graph. A popular non-Bayesian method of estimating a graphical structure is given by the graphical lasso. In this paper, we consider a Bayesian approach to the problem. We use priors which put a mixture of a point mass at zero and certain absolutely continuous distribution on off-diagonal elements of the precision matrix. Hence the resulting posterior distribution can be used for graphical structure learning. The posterior convergence rate of the precision matrix is obtained. The posterior distribution on the model space is extremely cumbersome to compute. We propose a fast computational method for approximating the posterior probabilities of various graphs using the Laplace approximation approach by expanding the posterior density around the posterior mode, which is the graphical lasso by our choice of the prior distribution. We also provide estimates of the accuracy in the approximation.
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Bayesian estimation in differential equation models
Ordinary differential equations (ODEs) are used to model dynamic systems appearing in engineering, physics, biomedical sciences and many other fields. These equations contain unknown parameters, say $θ$ of physical significance which have to be estimated from the noisy data. Often there is no closed form analytic solution of the equations and hence we cannot use the usual non-linear least squares technique to estimate the unknown parameters. There is a two step approach to solve this problem, where the first step involves fitting the data nonparametrically. In the second step the parameter is estimated by minimizing the distance between the nonparametrically estimated derivative and the derivative suggested by the system of ODEs. The statistical aspects of this approach have been studied under the frequentist framework. We consider this two step estimation under the Bayesian framework. The response variable is allowed to be multidimensional and the true mean function of it is not assumed to be in the model. We induce a prior on the regression function using a random series based on the B-spline basis functions. We establish the Bernstein-von Mises theorem for the posterior distribution of the parameter of interest. Interestingly, even though the posterior distribution of the regression function based on splines converges at a rate slower than $n^{-1/2}$, the parameter vector $θ$ is nevertheless estimated at $n^{-1/2}$ rate.
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Adaptive Bayesian multivariate density estimation with Dirichlet mixtures
We show that rate-adaptive multivariate density estimation can be performed using Bayesian methods based on Dirichlet mixtures of normal kernels with a prior distribution on the kernel's covariance matrix parameter. We derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is optimal minimax for the smoothness class to which the true density belongs. No prior knowledge of smoothness is assumed. The sufficient conditions are shown to hold for the Dirichlet location mixture of normals prior with a Gaussian base measure and an inverse-Wishart prior on the covariance matrix parameter. Locally Hölder smoothness classes and their anisotropic extensions are considered. Our study involves several technical novelties, including sharp approximation of finitely differentiable multivariate densities by normal mixtures and a new sieve on the space of such densities.
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Optimal two-stage procedures for estimating location and size of the maximum of a multivariate regression function
We propose a two-stage procedure for estimating the location $\boldsμ$ and size M of the maximum of a smooth d-variate regression function f. In the first stage, a preliminary estimator of $\boldsμ$ obtained from a standard nonparametric smoothing method is used. At the second stage, we "zoom-in" near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity. We fit an appropriate polynomial regression model to estimate the location and size of the maximum. We establish that, under suitable smoothness conditions and appropriate choice of the zooming, the second stage estimators have better convergence rates than the corresponding first stage estimators of $\boldsμ$ and M. More specifically, for $α$-smooth regression functions, the optimal nonparametric rates $n^{-(α-1)/(2α+d)}$ and $n^{-α/(2α+d)}$ at the first stage can be improved to $n^{-(α-1)/(2α)}$ and $n^{-1/2}$, respectively, for $α>1+\sqrt{1+d/2}$. These rates are optimal in the class of all possible sequential estimators. Interestingly, the two-stage procedure resolves "the curse of the dimensionality" problem to some extent, as the dimension d does not control the second stage convergence rates, provided that the function class is sufficiently smooth. We consider a multi-stage generalization of our procedure that attains the optimal rate for any smoothness level $α>2$ starting with a preliminary estimator with any power-law rate at the first stage.
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Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh
Jayanta Kumar Ghosh is one of the most extraordinary professors in the field of Statistics. His research in numerous areas, especially asymptotics, has been groundbreaking, influential throughout the world, and widely recognized through awards and other honors. His leadership in Statistics as Director of the Indian Statistical Institute and President of the International Statistical Institute, among other eminent positions, has been likewise outstanding. In recognition of Jayanta's enormous impact, this volume is an effort to honor him by drawing together contributions to the main areas in which he has worked and continues to work. The papers naturally fall into five categories. First, sequential estimation was Jayanta's starting point. Thus, beginning with that topic, there are two papers, one classical by Hall and Ding leading to a variant on p-values, and one Bayesian by Berger and Sun extending reference priors to stopping time problems. Second, there are five papers in the general area of prior specification. Much of Jayanta's earlier work involved group families as does Sweeting's paper here for instance. There are also two papers dwelling on the link between fuzzy sets and priors, by Meeden and by Delampady and Angers. Equally daring is the work by Mukerjee with data dependent priors and the pleasing confluence of several prior selection criteria found by Ghosh, Santra and Kim. Jayanta himself studied a variety of prior selection criteria including probability matching priors and reference priors.
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J. K. Ghosh's contribution to statistics: A brief outline
Professor Jayanta Kumar Ghosh has contributed massively to various areas of Statistics over the last five decades. Here, we survey some of his most important contributions. In roughly chronological order, we discuss his major results in the areas of sequential analysis, foundations, asymptotics, and Bayesian inference. It is seen that he progressed from thinking about data points, to thinking about data summarization, to the limiting cases of data summarization in as they relate to parameter estimation, and then to more general aspects of modeling including prior and model selection.
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Posterior consistency of Dirichlet mixtures of beta densities in estimating positive false discovery rates
In recent years, multiple hypothesis testing has come to the forefront of statistical research, ostensibly in relation to applications in genomics and some other emerging fields. The false discovery rate (FDR) and its variants provide very important notions of errors in this context comparable to the role of error probabilities in classical testing problems. Accurate estimation of positive FDR (pFDR), a variant of the FDR, is essential in assessing and controlling this measure. In a recent paper, the authors proposed a model-based nonparametric Bayesian method of estimation of the pFDR function. In particular, the density of p-values was modeled as a mixture of decreasing beta densities and an appropriate Dirichlet process was considered as a prior on the mixing measure. The resulting procedure was shown to work well in simulations. In this paper, we provide some theoretical results in support of the beta mixture model for the density of p-values, and show that, under appropriate conditions, the resulting posterior is consistent as the number of hypotheses grows to infinity.
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Kullback Leibler property of kernel mixture priors in Bayesian density estimation
Positivity of the prior probability of Kullback-Leibler neighborhood around the true density, commonly known as the Kullback-Leibler property, plays a fundamental role in posterior consistency. A popular prior for Bayesian estimation is given by a Dirichlet mixture, where the kernels are chosen depending on the sample space and the class of densities to be estimated. The Kullback-Leibler property of the Dirichlet mixture prior has been shown for some special kernels like the normal density or Bernstein polynomial, under appropriate conditions. In this paper, we obtain easily verifiable sufficient conditions, under which a prior obtained by mixing a general kernel possesses the Kullback-Leibler property. We study a wide variety of kernel used in practice, including the normal, $t$, histogram, gamma, Weibull densities and so on, and show that the Kullback-Leibler property holds if some easily verifiable conditions are satisfied at the true density. This gives a catalog of conditions required for the Kullback-Leibler property, which can be readily used in applications.
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Nonparametric Bayesian model selection and averaging
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.
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Posterior convergence rates of Dirichlet mixtures at smooth densities
We study the rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the Hellinger metric entropy. We apply this new general theorem on posterior convergence rates by computing bounds for Hellinger (bracketing) entropy numbers for the involved class of densities, the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior. The best obtainable rate of convergence of the posterior turns out to be equivalent to the well-known frequentist rate for integrated mean squared error $n^{-2/5}$ up to a logarithmic factor.
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Convergence rates of posterior distributions for noniid observations
We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinite-dimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model.
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Posterior consistency of Gaussian process prior for nonparametric binary regression
Consider binary observations whose response probability is an unknown smooth function of a set of covariates. Suppose that a prior on the response probability function is induced by a Gaussian process mapped to the unit interval through a link function. In this paper we study consistency of the resulting posterior distribution. If the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values, we show that the posterior distribution is consistent in the $L_1$-distance. As an auxiliary result to our proofs, we show that, under certain conditions, a Gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function. This result may be of independent interest in the literature for small ball probabilities of Gaussian processes.
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