Introduction

Symbolic dynamics is the study of shift (and other) transformations on spaces of infinite sequences or arrays of symbols and maps between such systems. A symbolic dynamical system, with a shift-invariant measure, corresponds to a stationary stochastic process. In the setting of information theory, such a system amounts to a collection of messages. Markov measures and hidden Markov measures, also called sofic measures, on symbolic dynamical systems have the desirable property of being determined by a finite set of data. But not all of their properties, for example the entropy, can be determined by finite algorithms. This article surveys some of the known and unknown properties of hidden Markov measures that are of special interest from the viewpoint of symbolic dynamics. To keep the article self contained, necessary background and related concepts are reviewed briefly. More can be found in [47, 56, 55, 71].

We discuss methods and tools that have been useful in the study of symbolic systems, measures supported on them, and maps between them.

This volume is a collection of papers on hidden Markov processes (HMPs) involving connections with symbolic dynamics and statistical mechanics. The subject was the focus of a five-day workshop held at the Banff International Research Station (BIRS) in October 2007, which brought together thirty mathematicians, computer scientists, and electrical engineers from institutions throughout the world. Most of the papers in this volume are based either on work presented at the workshop or on problems posed at the workshop.

From one point of view, an HMP is a stochastic process obtained as the noisy observation process of a finite-state Markov chain; a simple example is a binary Markov chain observed in binary symmetric noise, i.e., each symbol (0 or 1) in a binary state sequence generated by a two-state Markov chain may be flipped with some small probability, independently from time instant to time instant. In another (essentially equivalent) viewpoint, an HMP is a process obtained from a finite-state Markov chain by partitioning its state set into groups and completely “hiding” the distinction among states within each group; more precisely, there is a deterministic function on the states of the Markov chain, and the HMP is the process obtained by observing the sequences of function values rather than sequences of states (and hence such a process is sometimes called a “function of a Markov chain”).

HMPs are encountered in an enormous variety of applications involving phenomena observed in the presence of noise.

Introduction

The hidden Markov process

A hidden Markov process (HMP) is a discrete-time finite-state Markov chain (FSMC) observed through a memoryless channel. The HMP has become ubiquitous in statistics, computer science, and electrical engineering because it approximates many processes well using a dependency structure that leads to many efficient algorithms. While the roots of the HMP lie in the “grouped Markov chains” of Harris [20] and the “functions of a finite-state Markov chain” of Blackwell [8], the HMP first appears (in full generality) as the output process of a finite-state channel (FSC) [9]. The statistical inference algorithm of Baum and Petrie [5], however, cemented the HMP's place in history and is responsible for great advances in fields such as speech recognition and biological sequence analysis [22, 24]. An exceptional survey of HMPs, by Ephraim and Merhav, gives a nice summary of what is known in this area [12].

Introduction

The purpose of this article is twofold. First, in Section 2, we introduce a new complex version of the Hilbert metric on the standard real simplex. This metric is defined on a complex neighborhood of the interior of the standard real simplex, within the standard complex simplex. We show that if the neighborhood is sufficiently small, then any sufficiently small complex perturbation of a strictly positive square matrix acts as a contraction, with respect to this metric. While this article was nearing completion, we were informed of a different complex Hilbert metric, which was recently introduced. We briefly discuss the relation between this metric [3] and our metric in Remark 2.7.

Secondly, we show how one can use a complex Hilbert metric to obtain lower estimates of the domain of analyticity of the entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.

Introduction

We want to describe an approach to studying entropy rates for certain hidden Markov processes. Our motivation for studying this problem comes from a previous approach to Lyapunov exponents for random matrix products, which we shall also briefly describe. We were first introduced to the connection between Lyapunov exponents and entropy rates by the article of Jacquet et al. [8]. However, there is a close analogy which probably dates back as far as the work of Furstenberg [3] and Blackwell [1]. They studied Lyapunov exponents and entropy rates, respectively, by considering associated stationary measures. For simplicity, we shall restrict ourselves to the specific case of binary symmetric channels with noise. However, there is scope for generalizing this method to more general settings.

Our aim is to present new explicit formulae for the entropy rates and, thus, by suitable approximations, give an algorithm for the explicit computation. The usual techniques for studying entropy rates tend to give algorithms which give exponential convergence (reflecting the use of positive matrices and associated transfer operators). The techniques we describe typically lead to a faster super-exponential convergence.

Game theory is the mathematical study of cooperation and conflict. It provides a distinct and interdisciplinary approach to the study of human behavior and can be applied to any situation in which the choice of each player influences other players' utilities, and in which players take this mutual influence into consideration in their decision making processes. Such strategic interaction is commonly used in the analysis of systems involving human beings, such as economy, sociology, politics, and anthropology. Game theory is a very powerful conceptual and procedural tool to investigate social interaction, such as the rules of the game, the informational structure of the interactions, and the payoffs associated with particular user decisions. Game theory can be applied to all behavioral disciplines in a unified analytical framework. In the later chapters of this book, game theory will be the main tool for modeling and analyzing human behavior in media-sharing social networks. In this chapter we introduce the basic and most important concepts of game theory that will be used extensively in this book.

The idea of game theory was first suggested by Emile Borel, who proposed a formal theory of games in 1921. Later in 1944, the mathematician, John von Neumann and the economist Oskar Morgenstern published Theory of Games and Economic Behavior, which provided most of the basic noncooperative game terminologies and problem setups that are still in use today, such as two-person zero-sum games.

In general, side information is the information other than the target signal that can help improve system performance. For instance, in digital communications, side information about channel conditions at the transmitter's side can help reduce the bit error rate, and in learning theory, the side information map can also improve the classification accuracy. In this chapter, we use multimedia fingerprinting as an example and discuss how side information affects user behavior in media-sharing social networks.

In the scalable fingerprinting system in Chapter 5, given a test copy, the fingerprint detector simply uses fingerprints extracted from all layers collectively to identify colluders, and does not use any other information in the detection process. Intuitively, if some information about collusion can be made available during the colluder identification process, using such side information can help improve the traitor-tracing performance. In this chapter, we investigate two important issues in multimedia fingerprinting social networks that are related to side information: which side information can help improve the traitor-tracing performance, and how it affects user behavior in multimedia fingerprinting systems.

In this chapter, we first examine which side information can help improve the traitor tracing performance; our analysis shows that information about the statistical means of the detection statistics can significantly improve the detection performance. We then explore possible techniques for the fingerprint detector to probe and use such side information, and analyze its performance.

In the past decade, we have witnessed the emergence of large-scale media-sharing social network communities such as Napster, Facebook, and YouTube, in which millions of users form a dynamically changing infrastructure to share multimedia content. This proliferation of multimedia data has created a technological revolution in the entertainment and media industries, bringing new experiences to users and introducing the new concept of web-based social networking communities. The massive production and use of multimedia also pose new challenges to the scalable and reliable sharing of multimedia over large and heterogeneous networks; demand effective management of enormous amounts of unstructured media objects that users create, share, link, and reuse; and raise critical issues of protecting the intellectual property of multimedia.

In large-scale media-sharing social networks, millions of users actively interact with one another; such user dynamics not only influence each individual user but also affect the system performance. An example is peer-to-peer (P2P) file sharing systems, in which users cooperate with one another to provide an inexpensive, scalable, and robust platform for distributed data sharing. Because of the voluntary and unregulated participation nature of these systems, user cooperation cannot be guaranteed in P2P networks, and recent studies showed that many users are free riders, sharing no files at all. To provide a predictable and satisfactory level of service, it is important to analyze the impact of human factors on media-sharing social networks, and to provide important guidelines for better design of multimedia systems.