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Register a new userComplexity analysis of Bayesian learning of high-dimensional DAG models and their equivalence classes
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Added by
Quan Zhou
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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We consider MCMC methods for learning equivalence classes of sparse Gaussian DAG models when $p = e^{o(n)}$. The main contribution of this work is a rapid mixing result for a random walk Metropolis-Hastings algorithm, which we prove using a canonical path method. It reveals that the complexity of Bayesian learning of sparse equivalence classes grows only polynomially in $n$ and $p$, under some common high-dimensional assumptions. Further, a series of high-dimensional consistency results is obtained by the path method, including the strong selection consistency of an empirical Bayes model for structure learning and the consistency of a greedy local search on the restricted search space. Rapid mixing and slow mixing results for other structure-learning MCMC methods are also derived. Our path method and mixing time results yield crucial insights into the computational aspects of high-dimensional structure learning, which may be used to develop more efficient MCMC algorithms.
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The emergence of big data has led to a growing interest in so-called convergence complexity analysis, which is the study of how the convergence rate of a Monte Carlo Markov chain (for an intractable Bayesian posterior distribution) scales as the underlying data set grows in size. Convergence complexity analysis of practical Monte Carlo Markov chains on continuous state spaces is quite challenging, and there have been very few successful analyses of such chains. One fruitful analysis was recently presented by Qin and Hobert (2021b), who studied a Gibbs sampler for a simple Bayesian random effects model. These authors showed that, under regularity conditions, the geometric convergence rate of this Gibbs sampler converges to zero as the data set grows in size. It is shown herein that similar behavior is exhibited by Gibbs samplers for more general Bayesian models that possess both random effects and traditional continuous covariates, the so-called mixed models. The analysis employs the Wasserstein-based techniques introduced by Qin and Hobert (2021b).
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The problem of estimating ARMA models is computationally interesting due to the nonconcavity of the log-likelihood function. Recent results were based on the convex minimization. Joint model selection using penalization by a convex norm, e.g. the nuclear norm of a certain matrix related to the state space formulation was extensively studied from a computational viewpoint. The goal of the present short note is to present a theoretical study of a nuclear norm penalization based variant of the method of cite{Bauer:Automatica05,Bauer:EconTh05} under the assumption of a Gaussian noise process.
Following the seminal approach by Talagrand, the concept of Rademacher complexity for independent sequences of random variables is extended to Markov chains. The proposed notion of block Rademacher complexity (of a class of functions) follows from renewal theory and allows to control the expected values of suprema (over the class of functions) of empirical processes based on Harris Markov chains as well as the excess probability. For classes of Vapnik-Chervonenkis type, bounds on the block Rademacher complexity are established. These bounds depend essentially on the sample size and the probability tails of the regeneration times. The proposed approach is employed to obtain convergence rates for the kernel density estimator of the stationary measure and to derive concentration inequalities for the Metropolis-Hasting algorithm.
We consider the problem of constructing nonparametric undirected graphical models for high-dimensional functional data. Most existing statistical methods in this context assume either a Gaussian distribution on the vertices or linear conditional means. In this article we provide a more flexible model which relaxes the linearity assumption by replacing it by an arbitrary additive form. The use of functional principal components offers an estimation strategy that uses a group lasso penalty to estimate the relevant edges of the graph. We establish statistical guarantees for the resulting estimators, which can be used to prove consistency if the dimension and the number of functional principal components diverge to infinity with the sample size. We also investigate the empirical performance of our method through simulation studies and a real data application.
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