Bernstein - von Mises theorem and misspecified models: a review

This is a review of asymptotic and non-asymptotic behaviour of Bayesian methods under model specification. In particular we focus on consistency, i.e. convergence of the posterior distribution to the point mass at the best parametric approximation to the true model, and conditions for it to be locally Gaussian around this point. For well specified regular models, variance of the Gaussian approximation coincides with the Fisher information, making Bayesian inference asymptotically efficient. In this review, we discuss how this is affected by model misspecification. We also discuss approaches to adjust Bayesian inference to make it asymptotically efficient under model misspecification

Bayesian Inverse Problems with Heterogeneous Variance

We consider inverse problems in Hilbert spaces contaminated by Gaussian noise, and use a Bayesian approach to find its regularised smooth solution. We consider the so called conjugate diagonal setting where the covariance operators of the noise and of the prior are diagnolisable in the orthogonal bases associated with the forward operator of the inverse problem. Firstly, we derive the minimax rate of convergence in such problems with known covariance operator of the noise, showing that in the case of heterogeneous variance the ill posed inverse problem can become self regularised in some cases when the eigenvalues of the variance operator decay to zero, achieving parametric rate of convergence; as far as we are aware, this is a striking novel result that have not been observed before in nonparametric problems. Secondly, we give a general expression of the rate of contraction of the posterior distribution in case of known noise covariance operator in case the noise level is small, for a given prior distribution. We also investigate when this contraction rate coincides with the optimal rate in the minimax sense which is typically used as a benchmark for studying the posterior contraction rates. We apply our results to known variance operators with polynomially decreasing or increasing eigenvalues as an example. We also discuss when the plug in estimator of the eigenvalues of the covariance operator of the noise does not affect the rate of the contraction of the posterior distribution of the signal. We show that plugging in the maximum marginal likelihood estimator of the prior scaling parameter leads to the optimal posterior contraction rate, adaptively. Effect of the choice of the prior parameters on the contraction in such models is illustrated on simulated data with Volterra operator

Shared Differential Clustering across Single-cell RNA Sequencing Datasets with the Hierarchical Dirichlet Process

Single-cell RNA sequencing (scRNA-seq) is powerful technology that allows researchers to understand gene expression patterns at the single-cell level. However, analysing scRNA-seq data is challenging due to issues and biases in data collection. In this work, we construct an integrated Bayesian model that simultaneously addresses normalization, imputation and batch effects and also nonparametrically clusters cells into groups across multiple datasets. A Gibbs sampler based on a finite-dimensional approximation of the HDP is developed for posterior inference

Testing for equal correlation matrices with application to paired gene expression data

We present a novel method for testing the hypothesis of equality of two correlation matrices using paired high-dimensional datasets. We consider test statistics based on the average of squares, maximum and sum of exceedances of Fisher transform sample correlations and we derive approximate null distributions using asymptotic and non-parametric distributions. Theoretical results on the power of the tests are presented and backed up by a range of simulation experiments. We apply the methodology to a case study of colorectal tumour gene expression data with the aim of discovering biological pathway lists of genes that present significantly different correlation matrices on healthy and tumour samples. We find strong evidence for a large part of the pathway lists correlation matrices to change among the two medical conditions.Comment: 31 pages, 3 figure

Use of didactic terminology by teachers at various stages of professional communication

The article is based on the study that involved 115 students acquiring professional pedagogical education and 115 practicing teachers. The article describes the process of emergence of individual conceptual and terminological frameworks during various stages of professional communication, i.e. training and pedagogical activity. Individual frameworks of concepts have been studied through comparison of interpretations of definitions of basic didactic concepts by respondents (a total of 3487 definitions have been processed), 353 concept maps, as well as wordings of professionally significant problems in which didactic terms were used as well (a total of 400 statements have been analyzed). Factors that influence the nature of how teachers use didactic terms in various instances of professional communication have been described

Adaptive density estimation based on a mixture of Gammas

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local H\"older densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness the rates are adaptive to the smoothness

The Bernstein-von Mises theorem and non-regular models

We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the "true" solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein-von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary and have a faster rate of convergence. We also demonstrate a remarkable property of Bayesian inference, that for some models, there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. We illustrate the results on a problem from emission tomography.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1239 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org