Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets"

Discussion of "Frequentist coverage of adaptive nonparametric Bayesian credible sets" by Szab\'o, van der Vaart and van Zanten [arXiv:1310.4489v5].Comment: Published at http://dx.doi.org/10.1214/15-AOS1270E in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multivariate Gaussian Network Structure Learning

We consider a graphical model where a multivariate normal vector is associated with each node of the underlying graph and estimate the graphical structure. We minimize a loss function obtained by regressing the vector at each node on those at the remaining ones under a group penalty. We show that the proposed estimator can be computed by a fast convex optimization algorithm. We show that as the sample size increases, the estimated regression coefficients and the correct graphical structure are correctly estimated with probability tending to one. By extensive simulations, we show the superiority of the proposed method over comparable procedures. We apply the technique on two real datasets. The first one is to identify gene and protein networks showing up in cancer cell lines, and the second one is to reveal the connections among different industries in the US.Comment: 30 pages, 17 figures, 3 table

Bayesian ROC surface estimation under verification bias

The Receiver Operating Characteristic (ROC) surface is a generalization of ROC curve and is widely used for assessment of the accuracy of diagnostic tests on three categories. A complication called the verification bias, meaning that not all subjects have their true disease status verified often occur in real application of ROC analysis. This is a common problem since the gold standard test, which is used to generate true disease status, can be invasive and expensive. In this paper, we will propose a Bayesian approach for estimating the ROC surface based on continuous data under a semi-parametric trinormality assumption. Our proposed method often adopted in ROC analysis can also be extended to situation in the presence of verification bias. We compute the posterior distribution of the parameters under trinormality assumption by using a rank-based likelihood. Consistency of the posterior under mild conditions is also established. We compare our method with the existing methods for estimating ROC surface and conclude that our method performs well in terms of accuracy

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.Comment: Published in at http://dx.doi.org/10.1214/07-EJS130 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficient Bayesian estimation and uncertainty quantification in ordinary differential equation models

Often the regression function is specified by a system of ordinary differential equations (ODEs) involving some unknown parameters. Typically analytical solution of the ODEs is not available, and hence likelihood evaluation at many parameter values by numerical solution of equations may be computationally prohibitive. Bhaumik and Ghosal (2015) considered a Bayesian two-step approach by embedding the model in a larger nonparametric regression model, where a prior is put through a random series based on B-spline basis functions. A posterior on the parameter is induced from the regression function by minimizing an integrated weighted squared distance between the derivative of the regression function and the derivative suggested by the ODEs. Although this approach is computationally fast, the Bayes estimator is not asymptotically efficient. In this paper we suggest a modification of the two-step method by directly considering the distance between the function in the nonparametric model and that obtained from a four stage Runge-Kutta (RK4) method. We also study the asymptotic behavior of the posterior distribution of the parameter based on an approximate likelihood obtained from an RK4 numerical solution of the ODEs. We establish a Bernstein-von Mises theorem for both methods which assures that Bayesian uncertainty quantification matches with the frequentist one and the Bayes estimator is asymptotically efficient

Adaptive Bayesian density regression for high-dimensional data

Density regression provides a flexible strategy for modeling the distribution of a response variable $Y$ given predictors $\mathbf{X}=(X_1,\ldots,X_p)$ by letting that the conditional density of $Y$ given $\mathbf{X}$ as a completely unknown function and allowing its shape to change with the value of $\mathbf{X}$. The number of predictors $p$ may be very large, possibly much larger than the number of observations $n$, but the conditional density is assumed to depend only on a much smaller number of predictors, which are unknown. In addition to estimation, the goal is also to select the important predictors which actually affect the true conditional density. We consider a nonparametric Bayesian approach to density regression by constructing a random series prior based on tensor products of spline functions. The proposed prior also incorporates the issue of variable selection. We show that the posterior distribution of the conditional density contracts adaptively at the truth nearly at the optimal oracle rate, determined by the unknown sparsity and smoothness levels, even in the ultra high-dimensional settings where $p$ increases exponentially with $n$. The result is also extended to the anisotropic case where the degree of smoothness can vary in different directions, and both random and deterministic predictors are considered. We also propose a technique to calculate posterior moments of the conditional density function without requiring Markov chain Monte Carlo methods.Comment: Published at http://dx.doi.org/10.3150/14-BEJ663 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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 $\theta$ 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 $\theta$ is nevertheless estimated at $n^{-1/2}$ rate

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.Comment: arXiv admin note: substantial text overlap with arXiv:1411.116

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.Comment: arXiv admin note: substantial text overlap with arXiv:1204.423

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.Comment: Published at http://dx.doi.org/10.1214/009053606000000795 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org