On the representation of finite multiple Markov chains by weighted circuits

In this paper circuit chains of superior order are defined as multiple Markov chains for which transition probabilities are expressed in terms of the weights of a finite class of circuits in a finite set, in connection with kinetic properties along the circuits. Conversely, it is proved that if we join any finite doubly infinite strictly stationary Markov chain of order r for which transitions hold cyclically with a second chain with the same transitions for the inverse time-sense, then they may be represented as circuit chains of order r.     

Markov approximation and consistent estimation of unbounded probabilistic suffix trees

We consider infinite order chains whose transition probabilities depend on a finite suffix of the past. These suffixes are of variable length and the set of the lengths of all suffix is unbounded. We assume that the probability transitions for each of these suffixes are continuous with exponential decay rate. For these chains, we prove the weak consistency of a modification of Rissanen's algorithm Context which estimates the length of the suffix needed to predict the next symbol, given a finite sample. This generalizes to the unbounded case the original result proved for variable length Markov chains in the seminal paper Rissanen (1983). Our basic tool is the canonical Markov approximation which enables to approximate the chain of infinite order by a sequence of variable length Markov chains of increasing order. Our proof is constructive and we present an explicit decreasing upper bound for the probability of wrong estimation of the length of the current suffix.     

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??An absorbing Markov chain is an important statistic model and widely used in algorithm modeling for many disciplines, such as digital image processing, network analysis and so on. In order to get the stationary distribution for such model, the inverse of the transition matrix usually needs to be calculated. However, it is still difficult and costly for large matrices. In this paper, for absorbing Markov chains with two absorbing states, we propose a simple method to compute the stationary distribution for models with diagonalizable transition matrices. With this approach, only an eigenvector with eigenvalue 1 needs to be calculated. We also use this method to derive probabilities of the gambler's ruin problem from a matrix perspective. And, it is able to handle expansions of this problem. In fact, this approach is a variant of the general method for absorbing Markov chains. Similar techniques can be used to avoid calculating the inverse matrix in the general method.     

Random walks on graphs with interval weights and precise marginals

We propose a model of random walks on weighted graphs where the weights are interval valued, and connect it to reversible imprecise Markov chains. While the theory of imprecise Markov chains is now well established, this is a first attempt to model reversible chains. In contrast with the existing theory, the probability models that have to be considered are now non-convex. This presents a difficulty in computational sense, since convexity is critical for the existence of efficient optimization algorithms used in the existing models. The second part of the paper therefore addresses the computational issues of the model. The goal is finding sets of weights which maximize or minimize expectations corresponding to multiple steps transition probabilities. In particular, we present a local optimization algorithm and numerically test its efficiency. We show that its application allows finding close approximations of the globally best solutions in reasonable time.     

Stability and monotonicity for interactive Markov chains

In a Markov chain model of a social process, interest often centers on the distribution of the population by state. One question, the stability question, is whether this distribution converges to an equilibrium value. For an ordinary Markov chain (a chain with constant transition probabilities), complete answers are available. For an interactive Markov chain (a chain which allows the transition probabilities governing each individual to depend on the locations by state of the rest of the population), few stability results are available. This paper presents new results. Roughly, the main result is that an interactive Markov chain with unique equilibrium will be stable if the chain satisfies a certain monotonicity property. The property is a generalization to interactive Markov chains of the standard definition of monotonicity for ordinary Markov chains.     

Discrete time Markov chains with interval probabilities

The parameters of Markov chain models are often not known precisely. Instead of ignoring this problem, a better way to cope with it is to incorporate the imprecision into the models. This has become possible with the development of models of imprecise probabilities, such as the interval probability model. In this paper we discuss some modelling approaches which range from simple probability intervals to the general interval probability models and further to the models allowing completely general convex sets of probabilities. The basic idea is that precisely known initial distributions and transition matrices are replaced by imprecise ones, which effectively means that sets of possible candidates are considered. Consequently, sets of possible results are obtained and represented using similar imprecise probability models.We first set up the model and then show how to perform calculations of the distributions corresponding to the consecutive steps of a Markov chain. We present several approaches to such calculations and compare them with respect to the accuracy of the results. Next we consider a generalisation of the concept of regularity and study the convergence of regular imprecise Markov chains. We also give some numerical examples to compare different approaches to calculations of the sets of probabilities.     

Probabilities for Queueing Systems with Embedded Markov Chains

Abstract

Transition probabilities of embedded Markov chain for single-server queues are considered when the distribution of the inter-arrival time or that of the service time is specified. A comprehensive collection of formulas is derived for the transition probabilities, covering some seventeen flexible families. The corresponding estimation procedures are also derived by the method of moments. It is expected that this work could serve as a useful reference for the modeling of queuing systems with embedded Markov chains.     

Computing the Stationary Distribution of Absorbing Markov Chains with One Eigenvector of Diagonalizable Transition Matrices

An absorbing Markov chain is an important statistic model and widely used in algorithm modeling for many disciplines, such as digital image processing, network analysis and so on. In order to get the stationary distribution for such model, the inverse of the transition matrix usually needs to be calculated. However, it is still difficult and costly for large matrices. In this paper, for absorbing Markov chains with two absorbing states, we propose a simple method to compute the stationary distribution for models with diagonalizable transition matrices. With this approach, only an eigenvector with eigenvalue 1 needs to be calculated. We also use this method to derive probabilities of the gambler's ruin problem from a matrix perspective. And, it is able to handle expansions of this problem. In fact, this approach is a variant of the general method for absorbing Markov chains. Similar techniques can be used to avoid calculating the inverse matrix in the general method.     

马氏双链函数的强大数定律及其应用

本文研究了马氏随机环境中马氏双链函数的强大数定律.利用将双链函数进行分段研究的方法,获得了马氏环境中马氏双链函数强大数定律成立的一个充分条件.运用该定律,推导出马氏双链从一个状态到另一个状态转移概率的极限性质,进而推广了马氏双链的极限性质.     

Bayesian Prediction via Partitioning

This article proposes a new Bayesian approach to prediction on continuous covariates. The Bayesian partition model constructs arbitrarily complex regression and classification surfaces by splitting the covariate space into an unknown number of disjoint regions. Within each region the data are assumed to be exchangeable and come from some simple distribution. Using conjugate priors, the marginal likelihoods of the models can be obtained analytically for any proposed partitioning of the space where the number and location of the regions is assumed unknown a priori. Markov chain Monte Carlo simulation techniques are used to obtain predictive distributions at the design points by averaging across posterior samples of partitions.     

Exponential bounds for discrete-time singularly perturbed Markov chains

This paper develops exponential type upper bounds for scaled occupation measures of singularly perturbed Markov chains in discrete time. By considering two-time scale in the Markov chains, asymptotic analysis is carried out. The cases of the fast changing transition probability matrix is irreducible and that are divisible into l ergodic classes are examined first; the upper bounds of a sequence of scaled occupation measures are derived. Then extensions to Markov chains involving transient states and/or nonhomogeneous transition probabilities are dealt with. The results enable us to further our understanding of the underlying Markov chains and related dynamic systems, which is essential for solving many control and optimization problems.     

Computing the nearest reversible Markov chain

Reversible Markov chains are the basis of many applications. However, computing transition probabilities by a finite sampling of a Markov chain can lead to truncation errors. Even if the original Markov chain is reversible, the approximated Markov chain might be non‐reversible and will lose important properties, like the real‐valued spectrum. In this paper, we show how to find the closest reversible Markov chain to a given transition matrix. It turns out that this matrix can be computed by solving a convex minimization problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Potential theory for finitely additive Markov chains

In this paper potential theory is developed for finitely additive Markov chains and this is used to obtain various characterization theorems for discrete time Markov chains with an arbitrary state space, with finitely additive stationary transition probabilities and a finitely additive initial distribution.     

Perturbed Markov Chains with Damping Component

The paper is devoted to studies of regularly and singularly perturbed Markov chains with damping component. In such models, a matrix of transition probabilities is regularised by adding a special damping matrix multiplied by a small damping (perturbation) parameter ε. We perform a detailed perturbation analysis for such Markov chains, particularly, give effective upper bounds for the rate of approximation for stationary distributions of unperturbed Markov chains by stationary distributions of perturbed Markov chains with regularised matrices of transition probabilities, asymptotic expansions for approximating stationary distributions with respect to damping parameter, explicit coupling type upper bounds for the rate of convergence in ergodic theorems for n-step transition probabilities, as well as ergodic theorems in triangular array mode.

     

Importance Link Function Estimation for Markov Chain Monte Carlo Methods

Abstract

This article focuses on improving estimation for Markov chain Monte Carlo simulation. The proposed methodology is based upon the use of importance link functions. With the help of appropriate importance sampling weights, effective estimates of functionals are developed. The method is most easily applied to irreducible Markov chains, where application is typically immediate. An important conceptual point is the applicability of the method to reducible Markov chains through the use of many-to-many importance link functions. Applications discussed include estimation of marginal genotypic probabilities for pedigree data, estimation for models with and without influential observations, and importance sampling for a target distribution with thick tails.     

A structural model of the higher‐order Markov process incorporating reversion effects

The paper presents a formal approach which may increase the realism and parsimony of higher‐order Markov models applied to certain human behaviors. Often in behavioral applications, any improvements in fit available from increasing the order of a Markov model would be more than offset by interpretive problems caused by the very rapid increase in the number of independent parameters. The model proposed here for the higher‐order process greatly reduces the number of independent parameters, replacing them with sociologically relevant effects of persistence in and reversion to previous conditions.

The general model is called the “reversion model.” In it, individuals are allowed to carry along some information about their pasts, for a number of periods corresponding to the order of the model. The parameters describing residence histories are constructed to give each individual an underlying set of first‐order transition probabilities, which are modified by experience of the various states of the system. When an individual occupies a particular state, his relative probability of future residence there (vis‐a‐vis the other states as a group) is permitted to change. But occupation of a particular state is not permitted to affect the relative chances of residence among the other states. With suitable constraints, the number of parameters of this higher‐order process no longer increases geometrically with the order, but only arithmetically.

Maximum likelihood estimation formulas are derived for the reversion model, which is then applied to longitudinal data on the work activities of U.S. Ph.D. physicists and chemists in 1960–1966, and is found to fit well using likelihood ratio tests.     

Two classes of time-inhomogeneous Markov chains: Analysis of the periodic case

We consider the M/G/1 and GI/M/1 types of Markov chains for which their one step transitions depend on the times of the transitions. These types of Markov chains are encountered in several stochastic models, including queueing systems, dams, inventory systems, insurance risk models, etc. We show that for the cases when the time parameters are periodic the systems can be analyzed using some extensions of known results in the matrix-analytic methods literature. We have limited our examples to those relating to queueing systems to allow us a focus. An example application of the model to a real life problem is presented.     

Spanning Forests of Digraphs and Limiting Probabilities of Markov Chains

A maximum out forest of a digraph is its spanning subgraph that consists of disjoint diverging trees and has the maximum possible number of arcs. For an arbitrary weighted digraph, we consider a matrix of specific weights of maximum out forests and demonstrate how this matrix can be used to get a graph-theoretic interpretation for the limiting probabilities of Markov chains. For a special (nonclassical) correspondence between Markov chains and weighted digraphs, the matrix of Cesáro limiting transition probabilities of any finite homogeneous Markov chain coincides with the normalized matrix of maximum out forests of the corresponding digraphs. This provides a finite (combinatorial) method to calculate the limiting probabilities of Markov chains and thus their stationary distributions. On the other hand, the Markov chain technique provides the proofs to some statements about digraphs.