M. Grendar
Posterior distribution over a countable set M of continuous data-sampling distributions piles up at L-projection of the true distribution r on M, provided that the L-projection is unique. If there are several L-projections of r on M, then the posterior probability splits among them equally.
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Consistency of variational approximations under bounded Kullback--Leibler divergence
研究变分近似在贝叶斯推断中继承后验一致性的条件，证明在一般度量空间上，若近似测度与目标测度序列的KL散度一致有界且目标后验弱收敛到真参数处的狄拉克测度，则变分序列也一致。
We introduce verifiable criteria for weak posterior consistency of identifiable Bayesian nonparametric inference for jump diffusions with unit diffusion coefficient and uniformly Lipschitz drift and jump coefficients in arbitrary dimension. The criteria are expressed in terms of coefficients of the SDEs describing the process, and do not depend on intractable quantities such as transition densities. We also show that products of discrete net and Dirichlet mixture model priors satisfy our conditions, again under an identifiability assumption. This generalises known results by incorporating jump…
Bayesian nonparametric statistics is an area of considerable research interest. While recently there has been an extensive concentration in developing Bayesian nonparametric procedures for model checking, the use of the Dirichlet process, in its simplest form, along with the Kullback-Leibler divergence is still an open problem. This is mainly attributed to the discreteness property of the Dirichlet process and that the Kullback-Leibler divergence between any discrete distribution and any continuous distribution is infinity. The approach proposed in this paper, which is based on incorporating t…
We study the asymptotic consistency properties of $α$-Rényi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the $α$-Rényi divergence from the true posterior. Unique to our work is that we consider settings with $α> 1$, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result i…
In this paper we study posterior consistency for different topologies on the parameters for hidden Markov models with finite state space. We first obtain weak and strong posterior consistency for the marginal density function of finitely many consecutive observations. We deduce posterior consistency for the different components of the parameter. We also obtain posterior consistency for marginal smoothing distributions in the discrete case. We finally apply our results to independent emission probabilities, translated emission probabilities and discrete HMMs, under various types of priors.
Large Deviation Asymptotics and Bayesian Posterior Consistency on Stochastic Processes and Dynamical Systems
本文结合 Varadhan 引理、淬火/退火大偏差与指数逼近，建立长程相关随机过程及动力系统的广义贝叶斯后验收敛性框架，并给出混合过程与 Markov 过程的新结果。
Robust estimation under multivariate normal (MVN) mixture model is always a computational challenge. A recently proposed maximum pseudo \b{eta}-likelihood estimator aims to estimate the unknown parameters of a MVN mixture model in the spirit of minimum density power divergence (DPD) methodology but with a relatively simpler and tractable computational algorithm even for larger dimensions. In this letter, we will rigorously derive the existence and weak consistency of the maximum pseudo \b{eta}-likelihood estimator in case of MVN mixture models under a reasonable set of assumptions.
The variational framework for learning inducing variables (Titsias, 2009a) has had a large impact on the Gaussian process literature. The framework may be interpreted as minimizing a rigorously defined Kullback-Leibler divergence between the approximating and posterior processes. To our knowledge this connection has thus far gone unremarked in the literature. In this paper we give a substantial generalization of the literature on this topic. We give a new proof of the result for infinite index sets which allows inducing points that are not data points and likelihoods that depend on all functio…
In this paper we introduce objective proper prior distributions for hypothesis testing and model selection based on measures of divergence between the competing models; we call them divergence based (DB) priors. DB priors have simple forms and desirable properties, like information (finite sample) consistency; often, they are similar to other existing proposals like the intrinsic priors; moreover, in normal linear models scenarios, they exactly reproduce Jeffreys-Zellner-Siow priors. Most importantly, in challenging scenarios such as irregular models and mixture models, the DB priors are well …
A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy and $f$-divergences. Several properties of divergence and their duality with law invariant risk measures are developed, most notably relating their chain rules or additivity properties with certain notions of time consistency for dynamic law invariant risk measures known as accept…
Contraction of Rényi Divergences for Discrete Channels: Properties and Applications
本文研究了离散信道中Rényi分歧的收缩性质，揭示了其与φ-分歧的区别，并将其应用于马尔可夫链收敛速度的界分析。
Concentration Bounds for Discrete Distribution Estimation in KL Divergence
该研究针对KL散度下离散分布估计问题，推导Laplace估计器的集中界，将偏差量级从k/n改进为√k/n（n≥k时），并给出匹配下界证明其紧性。
In this work we give a comprehensive overview of the time consistency property of dynamic risk and performance measures, focusing on a the discrete time setup. The two key operational concepts used throughout are the notion of the LM-measure and the notion of the update rule that, we believe, are the key tools for studying time consistency in a unified framework.
Statistical Consistency of Discrete-to-Continuous Limits of Determinantal Point Processes
该研究探讨了离散DPPs向连续DPPs的极限行为，提出弱一致性概念，并展示了在不可访问核和空间情况下如何利用离散多项式族实现更高效的采样方法。
In this paper we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures (also known as acceptability indices). The proposed framework not only integrates existing forms of time consistency, but also provides a comprehensive toolbox for analysis and synthesis of the concept of time consistency in decision making. In particular, it allows for in depth comparative analysis of (most of) the existing types of time consistency -- a feat that has not be possible before and which is done in the companion paper [BCP2016] to this one. In o…
Contraction of Rényi Divergences for Discrete Channels
本文研究有限空间上Rényi散度的SDPI常数，证明其单调性与凸性，给出支撑集限制、闭式表达式及张量化界，并应用于本地差分隐私和马尔可夫链，获得尖锐收缩保证与改进的收敛界。
We study Gaussian approximations to the distribution of a diffusion. The approximations are easy to compute: they are defined by two simple ordinary differential equations for the mean and the covariance. Time correlations can also be computed via solution of a linear stochastic differential equation. We show, using the Kullback-Leibler divergence, that the approximations are accurate in the small noise regime. An analogous discrete time setting is also studied. The results provide both theoretical support for the use of Gaussian processes in the approximation of diffusions, and methodological…
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