Natalia Bochkina

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 the 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 that depend on the behaviour of the prior distribution on the boundary and have a faster rate of convergence. We also show 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.

Oscillation source location using wavelet transforms and generalized linear models

Recent developments in measurement and processing techniques have been used to monitor the stability of oscillations directly. Combined with fast, real-time data processing, this technology allows real-time monitoring of the margin of safety maintained against potential instability. This technology has been used to detect and warn of the system approaching instability. The use of real-time continuous dynamics monitoring often indicates dynamic behaviour that was anticipated by the model-based studies. In such cases it can be difficult to track down the sources of problems using conventional tools. This paper details the possibility of diagnosing the causes of problems related to oscillatory stability using measurement-based techniques. A dynamic model based on a real system is used to simulate periods of instability so that the methodology can be applied to the system to determine significant variables that contribute to mode dynamics. The aim of this paper is to develop a statistical model to encompass the linear and non-linear relationships in the system, which can then be used to identify causes of unstable system dynamics. To this end the discrete wavelet transform is used in conjunction with generalized linear models to fit the model data and to predict the system response with minimum deviance. This robust model could then be used in real time with real system variables to determine the correct course of action in rectifying a range of dynamics problems.

Consistency of the posterior distribution in generalized linear inverse problems

For ill-posed inverse problems, a regularized solution can be interpreted as a mode of the posterior distribution in a Bayesian framework. This framework enriches the set of possible solutions, as other posterior estimates can be used as a solution to the inverse problem, such as the posterior mean which can be easier to compute in practice. In this paper we prove consistency of Bayesian solutions of an ill-posed linear inverse problem in the Ky Fan metric for a general class of likelihoods and prior distributions in a finite-dimensional setting. This result can be applied to study infinite dimensional problems by letting the dimension of the unknown parameter grow to infinity which can be viewed as discretization on a grid or spectral approximation of an infinite dimensional problem. The likelihood and the prior distribution are assumed to be in an exponential form that includes distributions from the exponential family, and to be differentiable. The prior distribution can be non-conjugate or improper. The observations can be dependent, and no assumption of finite moments of observations, such as expected value or the variance, is necessary thus allowing for possibly non-regular likelihoods. If the variance exists, it may be heteroscedastic, namely, it may depend on the unknown function. We observe quite a surprising phenomenon when applying our result to the spectral approximation framework where it is possible to achieve the parametric rate of convergence, i.e. the problem becomes self-regularized. We also consider a particular case of the unknown parameter being on the boundary of the parameter set, and show that the rate of convergence in this case is faster than for an interior point parameter.

On the posterior median estimators of possibly sparse sequences

We adopt the Bayesian paradigm and discuss certain properties of posterior median estimators of possibly sparse sequences. The prior distribution considered is a mixture of an atom of probability at zero and a symmetric unimodal distribution, and the noise distribution is taken as another symmetric unimodal distribution. We derive an explicit form of the corresponding posterior median and show that it is an antisymmetric function and, under some conditions, a shrinkage and a thresholding rule. Furthermore we show that, as long as the tails of the nonzero part of the prior distribution are heavier than the tails of the noise distribution, the posterior median, under some constraints on the involved parameters, has the bounded shrinkage property, extending thus recent results to larger families of prior and noise distributions. Expressions of posterior distributions and posterior medians in particular cases of interest are obtained. The asymptotes of the derived posterior medians, which provide valuable information of how the corresponding estimators treat large coefficients, are also given. These results could be particularly useful for studying frequentist optimality properties and developing statistical techniques of the resulting posterior median estimators of possibly sparse sequences for a wider set of prior and noise distributions.

Oscillation source location in power systems using logic regression

Various attempts at solving the oscillation source location problem have been detailed in the literature[1][2][3] and each has had their own shortcomings. The dynamic behaviour of a power system is such that periods of instability may arise that are not solely due to large generators sitting beside each other in large plants. While these large machines operating at near full capacity certainly have an effect on modes in the system, the trigger may be something more inconspicuous like a smaller generator or load, or group thereof, that can produce instability in a number of system modes. The use of real-time continuous dynamics monitoring often indicates dynamic behaviour that was not anticipated by the modelbased studies. In such cases it can be difficult to track down the sources of problems using conventional tools. This paper details the possibility of diagnosing the causes of problems related to oscillatory stability using measurement-based techniques, with measurements derived from dynamic power system models. A dynamic model based on a real system is used to simulate periods of instability, so that the methodology can be applied to the data to determine the interaction of significant variables that contribute to poor mode damping. To this end, the discrete wavelet transform is used in conjunction with general linear models and logic regression to fit the model data and to predict the system response with minimum statistical deviance. This measurement-based modelling technique could then be used in real time with real system variables to determine the best course of action to rectify a range of dynamics problems.

Selection of the Regularization Parameter in Graphical Models using Network Characteristics

ABSTRACT Gaussian graphical models represent the underlying graph structure of conditional dependence between random variables, which can be determined using their partial correlation or precision matrix. In a high-dimensional setting, the precision matrix is estimated using penalized likelihood by adding a penalization term, which controls the amount of sparsity in the precision matrix and totally characterizes the complexity and structure of the graph. The most commonly used penalization term is the L1 norm of the precision matrix scaled by the regularization parameter, which determines the trade-off between sparsity of the graph and fit to the data. In this article, we propose several procedures to select the regularization parameter in the estimation of graphical models that focus on recovering reliably the appropriate network structure of the graph. We conduct an extensive simulation study to show that the proposed methods produce useful results for different network topologies. The approaches are also applied in a high-dimensional case study of gene expression data with the aim to discover the genes relevant to colon cancer. Using these data, we find graph structures, which are verified to display significant biological gene associations. Supplementary material is available online.

Bayesian truncated beta nonlinear mixed-effects models

ABSTRACT Truncated regression models arise in many applications where it is not possible to observe values of the response variable that are above or below certain thresholds. We propose a Bayesian truncated beta nonlinear mixed-effects model by considering the truncated variable to follow a truncated beta distribution. The mean parameter of the distribution is modeled by a nonlinear function of unknown fixed parameters and covariates and by random effects. The proposed model is suitable for response variables, y, bounded to an interval without the need to consider a transformed variable to apply the well-known beta regression model and its extensions, which are primarily appropriate for responses in the interval . Bayesian estimates and credible intervals are computed based on draws from the posterior distribution of parameters generated using an MCMC procedure. Posterior predictive checks, Bayesian standardized residuals and a Bayesian influence measures are considered for model diagnostics. Model selection is performed using the sum of log-CPO metric and a Bayesian model selection criterion based on Bayesian mixture modeling. Simulated datasets are used for prior sensitivity analysis and to evaluate finite sample properties of Bayesian estimates. The model is applied to a real dataset on soil–water retention.

CONSISTENCY BETWEEN PEDAGOGICAL ASSESSMENT AND SELF-ASSESSMENT OF EDUCATIONAL RESULTS IN THE CONTEXT OF PROFESSIONAL AND PERSONAL DEVELOPMENT OF STUDENTS

This article substantiates the role of assessment in professional and personal development of a specialist and explores the reasons behind inconsistency between pedagogical assessment and selfassessment of students. It describes results of studies of consistency of pedagogical assessment and self-assessment as a stimulus for students to develop professionally and personally (using psychology and pedagogy curriculum of a Russian higher education institution as an example). The sample included 185 students from the Saint Petersburg State University, both looking to pursue a career in pedagogy (92) and those not looking to pursue a career in pedagogy (93). It has been established that adequate self-assessment by teachers-to-be is caused by and, to a greater extent, contributes to further personal development of the future professional. Overassessment, especially in terms of completeness and consistency in apprehending didactic terminology is the basis for further targeted work towards increasing terminological literacy of a pedagogical student, focusing on dedicated professional development. Overassessment of own knowledge and proficiency in didactic terminology demonstrated by students of non-pedagogical lines of studies shows predominant focus on professional development and the strive to apprehend all types of knowledge within ones curriculum so as to receive quality education not only in fields of specialization, but also in general academic sense. This allows to conclude that adequate self-assessment in a specialized academic area makes future specialists focus on further individual personal development, while inflated self-assessment, regardless of specialization, makes them focus on dedicated professional development.

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.

Bayesian Perspectives on Sparse Empirical Bayes Analysis (SEBA)

We consider a joint processing of n independent similar sparse regression problems. Each is based on a sample \((y_{i1}, x_{i1})\ldots, (y_{im},x_{im})\) of m i.i.d. observations from \(y_{i1}=x_{i1}^T\;\beta_i+\epsilon_{i1},y_{i1}\in \mathbb{R},x_{i1}\in \mathbb{R}^p,\ {\rm and}\ \epsilon_{i1} \sim N(0, \sigma^2)\), say. The dimension p is large enough so that the empirical risk minimizer is not feasible. We consider, from a Bayesian point of view, three possible extensions of the lasso. Each of the three estimators, the lassoes, the group lasso, and the RING lasso, utilizes different assumptions on the relation between the n vectors \(\beta_1, \ldots, \beta_n\).