Consistency of the posterior distribution in generalized linear inverse problems

For ill-posed inverse problems, a regularised solution can be interpreted as a mode of the posterior distribution in a Bayesian framework. This framework enriches the set the solutions, as other posterior estimates can be used as a solution to the inverse problem, such as the posterior mean that 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 discretisation on a grid or spectral approximation of an infinite dimensional problem. 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 observations can be dependent. No assumption of finite moments of observations, such as expected value or the variance, is necessary thus allowing for possibly non-regular likelihoods, and allowing for non-conjugate and improper priors. 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-regularised. 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.Comment: arXiv admin note: substantial text overlap with arXiv:1110.301

Validation of differential gene expression algorithms: Application comparing fold-change estimation to hypothesis testing

<p>Abstract</p> <p>Background</p> <p>Sustained research on the problem of determining which genes are differentially expressed on the basis of microarray data has yielded a plethora of statistical algorithms, each justified by theory, simulation, or ad hoc validation and yet differing in practical results from equally justified algorithms. Recently, a concordance method that measures agreement among gene lists have been introduced to assess various aspects of differential gene expression detection. This method has the advantage of basing its assessment solely on the results of real data analyses, but as it requires examining gene lists of given sizes, it may be unstable.</p> <p>Results</p> <p>Two methodologies for assessing predictive error are described: a cross-validation method and a posterior predictive method. As a nonparametric method of estimating prediction error from observed expression levels, cross validation provides an empirical approach to assessing algorithms for detecting differential gene expression that is fully justified for large numbers of biological replicates. Because it leverages the knowledge that only a small portion of genes are differentially expressed, the posterior predictive method is expected to provide more reliable estimates of algorithm performance, allaying concerns about limited biological replication. In practice, the posterior predictive method can assess when its approximations are valid and when they are inaccurate. Under conditions in which its approximations are valid, it corroborates the results of cross validation. Both comparison methodologies are applicable to both single-channel and dual-channel microarrays. For the data sets considered, estimating prediction error by cross validation demonstrates that empirical Bayes methods based on hierarchical models tend to outperform algorithms based on selecting genes by their fold changes or by non-hierarchical model-selection criteria. (The latter two approaches have comparable performance.) The posterior predictive assessment corroborates these findings.</p> <p>Conclusions</p> <p>Algorithms for detecting differential gene expression may be compared by estimating each algorithm's error in predicting expression ratios, whether such ratios are defined across microarray channels or between two independent groups.</p> <p>According to two distinct estimators of prediction error, algorithms using hierarchical models outperform the other algorithms of the study. The fact that fold-change shrinkage performed as well as conventional model selection criteria calls for investigating algorithms that combine the strengths of significance testing and fold-change estimation.</p

Beyond comparisons of means: understanding changes in gene expression at the single-cell level

Traditional differential expression tools are limited to detecting changes in overall expression, and fail to uncover the rich information provided by single-cell level data sets. We present a Bayesian hierarchical model that builds upon BASiCS to study changes that lie beyond comparisons of means, incorporating built-in normalization and quantifying technical artifacts by borrowing information from spike-in genes. Using a probabilistic approach, we highlight genes undergoing changes in cell-to-cell heterogeneity but whose overall expression remains unchanged. Control experiments validate our method’s performance and a case study suggests that novel biological insights can be revealed. Our method is implemented in R and available at https://github.com/catavallejos/BASiCS

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 Hölder 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. One of the novel feature of the paper is that these results hold for densities with polynomial tails. Similar results are obtained using a hierarchical Bayesian model based on the mixture of inverse gamma densities which can be used to estimate adaptively densities with very heavy tails, including Cauchy density.</p

Educational Technologies in Modern Higher Education Institutions (Analysis of Russian and Foreign Research and Practice)

Introduction. Globalisation, openness to innovation, implementation of digital content, realisation of competency-based approach in modern education generate a consistent tendency towards strengthening of the role of educational technologies (ET) in different types of professional pedagogical activity, reinforcing pertinence of their application at all levels of education. Mastery and efficient use of a wide range of educational technologies have expanded the opportunities for educators to use productively traditional pedagogical and digital information technologies in order to obtain the guaranteed intended (expected) educational outcome. The interest of researchers and practitioners in selection, design, implementation and evaluation of effectiveness of educational technologies in higher educational institutions is evidenced by the increase in the number of publications suggesting various points of view on the use of such technologies within the educational process, in educational communications and in professional and personal development of students. The search for the most effective means of training of the new generation of specialists to achieve academic and technological breakthrough in Russia has increased the significance of systematic scientific information on modern educational technologies used in higher education institutions.The aim of the current research is to present the results of review and systematisation of research information on theoretical foundations of development and application of educational technologies in Russian and foreign higher educational institutions.Methodology and research methods. Analytical review has been prepared based on systematisation and generalisation of foreign and Russian scientific publications dedicated to the issues of design and use of educational technologies in the higher educational practice. 319 sources published since 2010 and included in Web of Science and Scopus research and citation databases as well as in the Russian Science Citation Index have been analysed. In order to achieve the set objective, authors have used such methods as content analysis, systems analysis, comparative analysis and benchmarking, terminological analysis, context analysis, generalisation and systematisation, analytical grouping, as well as methods of mathematical statistics.Results and scientific novelty. The article contains the results of analysis of Russian and foreign publications based on the following: 1) levels of presentation of research information (theoretical, theoretical and experimental, empirical); 2) attributes and properties of educational technologies; 3) approaches to selection of classification principles of educational technologies; 4) capabilities of educational technologies to shape academic, communicative and professional competency of a specialist; 5) criteria for evaluation of effectiveness of application of ET in the higher educational practice; 6) preparedness of teacher of higher educational institutions to develop and apply educational technologies. It has been established that Russian authors are still more oriented towards describing the characteristics and algorithms of predominantly traditional educational technologies and their applicability, evaluating their effectiveness through influencing students’ professional and personal development and competent communication. Foreign authors prefer to assess how widely applicable digital educational technologies are (within the boundaries of the educational organisation and beyond), measure students’ acceptance of educational technologies and their effectiveness through the impact on improvement of educational quality. Statistically significant differences were identified for all considered functional characteristics of educational technology in Russian and foreign publications, with the exception of “Student Attitude to Educational Technology”. Both Russian and foreign authors in the last 5 years have increased their interest in mixed educational technologies and in the problems of integrating traditional and digital technologies in higher education. In Russian journals there is an increase of the number of publications about the digital technologies, and in foreign publications there is an increase in the number of publications on traditional technologies, resources and conditions for improvement of effectiveness of educational technologies.Practical significance. The materials of the article can be used in the higher educational practice and in further research on development and application of educational technologies in Russian and foreign higher education

Educational Technologies in Modern Higher Education Institutions (Analysis of Russian and Foreign Research and Practice)

Introduction. Globalisation, openness to innovation, implementation of digital content, realisation of competency-based approach in modern education generate a consistent tendency towards strengthening of the role of educational technologies (ET) in different types of professional pedagogical activity, reinforcing pertinence of their application at all levels of education. Mastery and efficient use of a wide range of educational technologies have expanded the opportunities for educators to use productively traditional pedagogical and digital information technologies in order to obtain the guaranteed intended (expected) educational outcome. The interest of researchers and practitioners in selection, design, implementation and evaluation of effectiveness of educational technologies in higher educational institutions is evidenced by the increase in the number of publications suggesting various points of view on the use of such technologies within the educational process, in educational communications and in professional and personal development of students. The search for the most effective means of training of the new generation of specialists to achieve academic and technological breakthrough in Russia has increased the significance of systematic scientific information on modern educational technologies used in higher education institutions. The aim of the current research is to present the results of review and systematisation of research information on theoretical foundations of development and application of educational technologies in Russian and foreign higher educational institutions. Methodology and research methods. Analytical review has been prepared based on systematisation and generalisation of foreign and Russian scientific publications dedicated to the issues of design and use of educational technologies in the higher educational practice. 319 sources published since 2010 and included in Web of Science and Scopus research and citation databases as well as in the Russian Science Citation Index have been analysed. In order to achieve the set objective, authors have used such methods as content analysis, systems analysis, comparative analysis and benchmarking, terminological analysis, context analysis, generalisation and systematisation, analytical grouping, as well as methods of mathematical statistics. Results and scientific novelty. The article contains the results of analysis of Russian and foreign publications based on the following: 1) levels of presentation of research information (theoretical, theoretical and experimental, empirical); 2) attributes and properties of educational technologies; 3) approaches to selection of classification principles of educational technologies; 4) capabilities of educational technologies to shape academic, communicative and professional competency of a specialist; 5) criteria for evaluation of effectiveness of application of ET in the higher educational practice; 6) preparedness of teacher of higher educational institutions to develop and apply educational technologies. It has been established that Russian authors are still more oriented towards describing the characteristics and algorithms of predominantly traditional educational technologies and their applicability, evaluating their effectiveness through influencing students’ professional and personal development and competent communication. Foreign authors prefer to assess how widely applicable digital educational technologies are (within the boundaries of the educational organisation and beyond), measure students’ acceptance of educational technologies and their effectiveness through the impact on improvement of educational quality. Statistically significant differences were identified for all considered functional characteristics of educational technology in Russian and foreign publications, with the exception of “Student Attitude to Educational Technology”. Both Russian and foreign authors in the last 5 years have increased their interest in mixed educational technologies and in the problems of integrating traditional and digital technologies in higher education. In Russian journals there is an increase of the number of publications about the digital technologies, and in foreign publications there is an increase in the number of publications on traditional technologies, resources and conditions for improvement of effectiveness of educational technologies. Practical significance. The materials of the article can be used in the higher educational practice and in further research on development and application of educational technologies in Russian and foreign higher education.Введение. Глобализация, открытость инновациям, внедрение цифрового контента, реализация компетентностного подхода в современном образовании порождают устойчивую тенденцию к повышению роли образовательных технологий в различных видах профессиональной деятельности педагога, усиливая актуальность их применения на всех уровнях образования. Владение и эффективное использование широкого спектра образовательных технологий расширили возможности педагогов продуктивно использовать гуманитарно-педагогические и информационно-технические средства для получения гарантированно заданного (ожидаемого) образовательного результата. Интерес ученых и практиков к отбору, проектированию, внедрению и оценке эффективности образовательных технологий в вузе подтверждается увеличением числа публикаций, в которых предлагаются самые разнообразные точки зрения на их использование в образовательном процессе, образовательной коммуникации и профессиональноличностном развитии студентов. Поиск самых эффективных средств подготовки специалистов нового поколения для научно-технологического прорыва России усилил значимость системной научной информации о современных образовательных технологиях, применяемых в высшей школе. Цель статьи. Изложение результатов обзора и систематизации научной информации о теоретических основах разработки и применения образовательных технологий в отечественных и зарубежных вузах. Методология и методики исследования. Аналитический обзор подготовлен на основе систематизации и обобщения зарубежных и отечественных научных публикаций, посвященных проблеме проектирования и применения образовательных технологий в практике высшего образования. Проанализировано 319 источников, изданных с 2010 г. и включенных в наукометрические базы Web of science, Scopus и Российский индекс научного цитирования. Для реализации поставленной цели авторы применили методы контент-анализа, системного, сравнительно-сопоставительного, терминологического и контекстологического анализа, обобщения и систематизации, аналитической группировки, а также методы математической статистики. Результаты и научная новизна. В статье отражены результаты анализа отечественных и зарубежных публикаций по следующим основаниям: 1) уровни представления научной информации (теоретический, теоретико-экспериментальный, эмпирический); 2) признаки и свойства образовательных технологий; 3) подходы к выбору оснований для классификаций образовательных технологий; 4) возможности образовательных технологий в формировании академической, коммуникативной и профессиональной компетентности специалиста; 5) критерии для оценки эффективности их применения в вузовской образовательной практике; 6) готовность вузовских преподавателей к разработке и внедрению образовательных технологий. Установлено, что российские авторы в большей мере ориентированы на описание характеристик и алгоритмов преимущественно гуманитарных образовательных технологий, условий применения и оценку их эффективности через влияние на профессионально-личностное развитие и формирование информационно-коммуникативной компетентности студентов. Иностранные авторы отдают предпочтение оценке масштабов распространения цифровых образовательных технологий (в границах образовательной организации и за ее пределами), мере принятия студентами образовательных технологий и их эффективности через влияние на повышение качества образовательных результатов. Зафиксированы статистически значимые различия по всем рассмотренным функциональным характеристикам образовательной технологии между отечественными и зарубежными публикациями, за исключением «Отношение студента к образовательной технологии». И российские, и иностранные авторы в последние 5 лет усилили интерес к смешанным образовательным технологиям и к проблемам интеграции гуманитарных и цифровых технологий в высшем образовании. В российских журналах наметилась тенденция к увеличению публикаций о цифровых технологиях, а в зарубежных – о гуманитарных технологиях, а также проблемах изучения ресурсов и условий повышения эффективности образовательных технологий. Практическая значимость. Материалы статьи могут быть использованы в вузовской образовательной практике и при проведении исследований по проблемам разработки и эффективности применения образовательных технологий в отечественном и зарубежном высшем образовании.The present research was carried out with the financial support of the Russian Foundation for Basic Research in the framework of the research project No. 19-113-50056 (competition “Expansion”). The authors are deeply grateful to the reviewers for their expert evaluation of the manuscript.Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта № 19-113-50056 (конкурс «Экспансия»). Авторы выражают глубокую признательность рецензентам за экспертную оценку рукописи

Glycomics investigation into insulin action.

Defects in glycosylation are becoming increasingly associated with a range of human diseases. In some cases, the disease is caused by the glycosylation defect, whereas in others, the aberrant glycosylation may be a consequence of the disease. The implementation of highly sensitive and rapid mass spectrometric screening strategies for profiling the glycans present in model biological systems is revealing valuable insights into disease phenotypes. In addition, glycan screening is proving useful in the analysis of knock-out mice where it is possible to assess the role of glycosyltransferases and glycosidases and what function they have at the cellular and whole organism level. In this study, we analysed the effect of insulin on the glycosylation of 3T3-L1 cells and the effect of insulin resistance on glycosylation in a mouse model. Transcription profiling of 3T3-L1 cells treated with and without insulin revealed expression changes of several glycogenes. In contrast, mass spectrometric screening analysis of the glycans from these cells revealed very similar profiles suggesting that any changes in glycosylation were most likely on specific proteins rather than a global phenomenon. A fat-fed versus carbohydrate-fed mouse insulin resistant model was analysed to test the consequences of chronic insulin resistance. Muscle and liver N-glycosylation profiles from these mice are reported

MAP estimators and their consistency in Bayesian nonparametric inverse problems

We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map $\mathcal {G}$ applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager–Machlup functional defined on the Cameron–Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of $\mathcal {G}(u)$ can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier–Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics

Reporting FDR analogous confidence intervals for the log fold change of differentially expressed genes

<p>Abstract</p> <p>Background</p> <p>Gene expression experiments are common in molecular biology, for example in order to identify genes which play a certain role in a specified biological framework. For that purpose expression levels of several thousand genes are measured simultaneously using DNA microarrays. Comparing two distinct groups of tissue samples to detect those genes which are differentially expressed one statistical test per gene is performed, and resulting <it>p</it>-values are adjusted to control the false discovery rate. In addition, the expression change of each gene is quantified by some effect measure, typically the log fold change. In certain cases, however, a gene with a significant <it>p</it>-value can have a rather small fold change while in other cases a non-significant gene can have a rather large fold change. The biological relevance of the change of gene expression can be more intuitively judged by a fold change then merely by a <it>p</it>-value. Therefore, confidence intervals for the log fold change which accompany the adjusted <it>p</it>-values are desirable.</p> <p>Results</p> <p>In a new approach, we employ an existing algorithm for adjusting confidence intervals in the case of high-dimensional data and apply it to a widely used linear model for microarray data. Furthermore, we adopt a concept of different relevance categories for effects in clinical trials to assess biological relevance of genes in microarray experiments. In a brief simulation study the properties of the adjusting algorithm are maintained when being combined with the linear model for microarray data. In two cancer data sets the adjusted confidence intervals can indicate significance of large fold changes and distinguish them from other large but non-significant fold changes. Adjusting of confidence intervals also corrects the assessment of biological relevance.</p> <p>Conclusions</p> <p>Our new combination approach and the categorization of fold changes facilitates the selection of genes in microarray experiments and helps to interpret their biological relevance.</p