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Register a new userCompositional Active Inference I: Bayesian Lenses. Statistical Games
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Toby St. Clere Smithe
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We introduce the concepts of Bayesian lens, characterizing the bidirectional structure of exact Bayesian inference, and statistical game, formalizing the optimization objectives of approximate inference problems. We prove that Bayesian
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This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-free to distribution-dependent, from sub-Gaussian to sub-exponential, sub-Gamma, and sub-Weibull random variables, and from the mean to the maximum concentration. This review provides results in these settings with some fresh new results. Given the increasing popularity of high-dimensional data and inference, results in the context of high-dimensional linear and Poisson regressions are also provided. We aim to illustrate the concentration inequalities with known constants and to improve existing bounds with sharper constants.
In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Frechet mean of some distribution $mathbb{P}$ supported on a subspace of positive semi-definite Hermitian operators $mathbb{H}_{+}(d)$. We allow a barycenter to be restricted to some affine subspace of $mathbb{H}_{+}(d)$ and provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of $Q_*$ in both Frobenius norm and Bures-Wasserstein distance, and explain, how obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.
This paper considers distributed statistical inference for general symmetric statistics %that encompasses the U-statistics and the M-estimators in the context of massive data where the data can be stored at multiple platforms in different locations. In order to facilitate effective computation and to avoid expensive communication among different platforms, we formulate distributed statistics which can be conducted over smaller data blocks. The statistical properties of the distributed statistics are investigated in terms of the mean square error of estimation and asymptotic distributions with respect to the number of data blocks. In addition, we propose two distributed bootstrap algorithms which are computationally effective and are able to capture the underlying distribution of the distributed statistics. Numerical simulation and real data applications of the proposed approaches are provided to demonstrate the empirical performance.
Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoretical results justifying the Bayesian approach. Recent papers, see e.g. cite{Nickl2017,lu2017bernsteinvon} and references therein, illustrate the main difficulties and challenges in studying the properties of the posterior distribution in the nonparametric setup. This paper offers a new approach for study the frequentist properties of the nonparametric Bayes procedures. The idea of the approach is to relax the nonlinear structural equation by introducing an auxiliary functional parameter and replacing the structural equation with a penalty and by imposing a prior on the auxiliary parameter. For the such extended model, we state sharp bounds on posterior concentration and on the accuracy of the penalized MLE and on Gaussian approximation of the posterior, and a number of further results. All the bounds are given in terms of effective dimension, and we show that the proposed calming device does not significantly affect this value.
Microbes can affect processes from food production to human health. Such microbes are not isolated, but rather interact with each other and establish connections with their living environments. Understanding these interactions is essential to an understanding of the organization and complex interplay of microbial communities, as well as the structure and dynamics of various ecosystems. A common and essential approach toward this objective involves the inference of microbiome interaction networks. Although network inference methods in other fields have been studied before, applying these methods to estimate microbiome associations based on compositional data will not yield valid results. On the one hand, features of microbiome data such as compositionality, sparsity and high-dimensionality challenge the data normalization and the design of computational methods. On the other hand, several issues like microbial community heterogeneity, external environmental interference and biological concerns also make it more difficult to deal with the network inference. In this paper, we provide a comprehensive review of emerging microbiome interaction network inference methods. According to various assumptions and research targets, estimated networks are divided into four main categories: correlation networks, conditional correlation networks, mixture networks and differential networks. Their scope of applications, advantages and limitations are presented in this review. Since real microbial interactions can be complex and dynamic, no unifying method has captured all the aspects of interest to date. In addition, we discuss the challenges now confronting current microbial associations study and future prospects. Finally, we highlight that the research in microbial network inference requires the joint promotion of statistical computation methods and experimental techniques.
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