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.

As shown in Chapters 5 and 6, cooperation enables users in a social network to access extra resources from others and thus to receive higher payoffs. Meanwhile, each user also contributes his or her own resources to help others. However, because the nature of participation nature in many media-sharing social networks is often voluntary and unregulated, users' full cooperation cannot be guaranteed, and a critical issue to be resolved first is to analyze when users will cooperate with each other and design cooperation strategies. In this chapter, we use colluder social networks in multimedia fingerprinting as an example, and analyze when users will collaborate with each other and how they reach agreements.

For a colluder in multimedia fingerprinting systems, the first issue to address is to decide whether he or she would like to participate in collusion and with whom he or she would like to collude. When colluders' goal is to minimize their probability of being detected, a collusion attack with more attackers reduces the energy of each contributing fingerprint by a larger ratio and, therefore, each colluder has a smaller chance of being caught. Thus, to minimize the risk, colluders are always willing to cooperate with one another because it reduces all colluders' risk, and a colluder should find as many fellow attackers as possible.

Nevertheless, colluding with more attackers also means sharing with more people the reward from illegal usage of multimedia and, therefore, colluders may not always want to cooperate.

With recent advances in networking, multimedia signal processing, and communication technologies, we have witnessed the emergence of large-scale video streaming social networks, in which millions of users form a distributed and dynamically changing infrastructure to share video streams. Statistics showed that more than 75 percent of the total US Internet audience have viewed online video, and the average online video viewer watched four hours of video per month. With the fast deployment of high-speed residential network access, video is expected to be the dominating traffic on the Internet in the near future.

The traditional method for video streaming over the Internet is the client-server service model. A client sets up a connection with a video source server and video content is directly streamed to the client from either the video source server or a nearby content delivery server. The most popular client-server video stream service nowadays is YouTube, which drew 5 billion US video views in July 2008. However, client-server–based video streaming methods incur expensive bandwidth provision cost on the server. For example, the streaming rate for a TV-quality video is about 400 kilobits per second (kbps), which makes the client-server video streaming solution very expensive when more users join the system.

P2P video streaming encourages users to upload their downloaded data to other users in the network; each user acts as a server and a client at the same time. The system relies on voluntary contributions of resources from individual users to achieve high scalability and robustness and to provide satisfactory performance.

Mobile phones are among the most popular consumer devices; the recent developments of 3G networks and smart phones enable users to watch video programs by subscribing to data plans from service providers. Because of the ubiquity of mobile phones and phone-to-phone communication technologies, subscribers can redistribute the video content to nonsubscribers. Such a redistribution mechanism is a potential competitor for the service provider and is very difficult to trace, given users' high mobility. The service provider must set a reasonable price for the data plan to prevent such unauthorized redistribution behavior and to protect the provider's own profit. In this chapter, we analyze the optimal price setting for the service provider by investigating the equilibrium between the subscribers and the secondary buyers in the content redistribution network. We model the behavior between the subscribers and the secondary buyers as a noncooperative game and find the optimal price and quantity for both groups of users. Such an analysis can help the service provider preserve the profit under the threat of the redistribution networks and can improve the quality of service for end users.

Introduction

The explosive advance of multimedia processing technologies is creating dramatic shifts in the ways that video content is delivered to and consumed by end users. Also, the increased popularity of wireless networks and mobile devices has drawn a great deal of attention in the past decade about ubiquitous multimedia access in the multimedia community.