Stationary and Transition Probabilities in Slow Mixing, Long Memory Markov Processes

Asadi, Meysam; Torghabeh, Ramezan Paravi; Santhanam, Narayana

doi:10.1109/tit.2014.2334594

Cited by 12 publications

(3 citation statements)

References 40 publications

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“…Likewise, the nucleotides in a DNA sequence do not form a random sample. Recently, there has been interest in the estimation of the missing mass and coverage probabilities when observable samples are modeled as Markov chains [Asadi et al, 2014;Falahatgar et al, 2016;Hao et al, 2018;Wolfer and Kontorovich, 2019;Cha et al, 2021]. In BNPs, Bacallado et al [2013] considered SSPs for observable samples modeled as reversible Markov chains.…”

Section: Some Generalizations Of Sspsmentioning

confidence: 99%

Bayesian Nonparametric Inference for "Species-sampling" Problems

Balocchi¹,

Favaro²,

Naulet³

2022

Preprint

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Species-sampling" problems (SSPs) refer to a broad class of statistical problems that, given an observable sample from an unknown population of individuals belonging to some species, call for estimating features of the unknown species composition of additional unobservable samples from the same population. Among SSPs, the problems of estimating coverage probabilities, the number of unseen species and coverages of prevalences have emerged over the past three decades for being the objects of numerous studies, both in methods and applications, mostly within the field of biological sciences but also in machine learning, electrical engineering, theoretical computer science and information theory. In this paper, we present an overview of Bayesian nonparametric (BNP) inference for such three SSPs under the popular Pitman-Yor process (PYP) prior: i) we introduce each SSP in the classical (frequentist) nonparametric framework, and review its posterior analyses in the BNP framework; ii) we improve on computation and interpretability of existing posterior distributions, typically expressed through complicated combinatorial numbers, by establishing novel posterior representations in terms of simple compound Binomial and Hypergeometric distributions. The critical question of estimating the discount and scale parameters of the PYP prior is also considered and investigated, establishing a general property of Bayesian consistency with respect to the hierarchical Bayes and empirical Bayes approaches, that is: the discount parameter can be always estimated consistently, whereas the scale parameter cannot be estimated consistently, thus advising caution in posterior inference. We conclude our work by discussing other SSPs, and presenting some emerging generalizations of SSPs, mostly in biological sciences, which deal with "feature-sampling" problems, multiple populations of individuals sharing species and classes of Markov chains.

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Section: Some Generalizations Of Sspsmentioning

confidence: 99%

Bayesian Nonparametric Inference for "Species-sampling" Problems

Balocchi¹,

Favaro²,

Naulet³

2022

Preprint

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“…Therefore, it is interesting to analytically understand and study Good-Turing estimators for missing mass in Markov chains. Missing mass has been studied extensively in the iid case [3], [5]- [10], and estimation in Markov chains has been studied as well [11]- [14]. Recently, there has been interest in studying concentration and estimation of missing mass in Markov chains [15] [16].…”

Section: Prior Work and Problem Settingmentioning

confidence: 99%

Missing Mass of Rank-2 Markov Chains

Chandra¹,

Thangaraj²,

Rajaraman³

2021

Preprint

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Estimation of missing mass with the popular Good-Turing (GT) estimator is well-understood in the case where samples are independent and identically distributed (iid). In this article, we consider the same problem when the samples come from a stationary Markov chain with a rank-2 transition matrix, which is one of the simplest extensions of the iid case. We develop an upper bound on the absolute bias of the GT estimator in terms of the spectral gap of the chain and a tail bound on the occupancy of states. Borrowing tail bounds from known concentration results for Markov chains, we evaluate the bound using other parameters of the chain. The analysis, supported by simulations, suggests that, for rank-2 irreducible chains, the GT estimator has bias and mean-squared error falling with number of samples at a rate that depends loosely on the connectivity of the states in the chain.

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“…Naturally then, to build clustering and community detection algorithms on graphs, we build Markov random walks such that the restriction of the random walk to within any one cluster mixes much faster than the overall walk itself. Once we obtain a walk like that, a careful interpretation of samples from the random walk along the lines of the theorems we obtain in [27] should reveal the community structure of the graph.…”

Section: Core Ideamentioning

confidence: 99%

Modeling community detection using slow mixing random walks

Torghabeh

Santhanam

2015

2015 IEEE International Conference on Big Data (Big Data)

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The task of community detection in a graph formalizes the intuitive task of grouping together subsets of vertices such that vertices within clusters are connected tighter than those in disparate clusters. This paper approaches community detection in graphs by constructing Markov random walks on the graphs. The mixing properties of the random walk are then used to identify communities. We use coupling from the past as an algorithmic primitive to translate the mixing properties of the walk into revealing the community structure of the graph. We analyze the performance of our algorithms on specific graph structures, including the stochastic block models (SBM) and LFR random graphs.

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Stationary and Transition Probabilities in Slow Mixing, Long Memory Markov Processes

Cited by 12 publications

References 40 publications

Bayesian Nonparametric Inference for "Species-sampling" Problems

Bayesian Nonparametric Inference for "Species-sampling" Problems

Missing Mass of Rank-2 Markov Chains

Modeling community detection using slow mixing random walks

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