# Hidden Markov Processes: Theory and Applications to Biology

M. Vidyasagar
Pages: 304
https://www.jstor.org/stable/j.ctt6wq0db

1. Front Matter
(pp. i-vi)
(pp. vii-x)
3. Preface
(pp. xi-xiv)
M. Vidyasagar
4. PART 1. PRELIMINARIES
• Chapter One Introduction to Probability and Random Variables
(pp. 3-44)

Probability theory is an attempt to formalize the notion of uncertainty in the outcome of an experiment. For instance, suppose an urn contains four balls, colored red, blue, white, and green respectively. Suppose we dip our hand in the urn and pull out one of the balls “at random.” What is the likelihood that the ball we pull out will be red? If we make multiple draws, replacing the drawn ball each time and shaking the urn thoroughly before the next draw, what is the likelihood that we have to make at least ten draws before we draw a red...

• Chapter Two Introduction to Information Theory
(pp. 45-70)

In this chapter we give a brief introduction to some elementary aspects of information theory, specifically entropy in its various forms. One can think of entropy as the level of uncertainty associated with a random variable (or more precisely, the probability distribution of the random variable). Entropy has several useful properties, and the ones relevant to subsequent chapters are brought out here. When there are two or more random variables, it is worthwhile to study the conditional entropy of one random variable with respect to another. The last concept is relative entropy, also known as the Kullback-Leibler divergence, named after...

• Chapter Three Nonnegative Matrices
(pp. 71-98)

In this chapter, the focus is on nonnegative matrices. They are relevant in the study of Markov processes (see Chapter 4), because the state transition matrix of a Markov process is a special kind of nonnegative matrix, known as a stochastic matrix.¹ However, it turns out that practically all of the useful properties of a stochastic matrix also hold for the more general class of nonnegative matrices. Hence it is desirable to present the theory in the more general setting, and then specialize to Markov processes. The reader may find the more general results useful in some other context.

Nonnegative...

5. PART 2. HIDDEN MARKOV PROCESSES
• Chapter Four Markov Processes
(pp. 101-128)

In this chapter, we begin our study of Markov processes, which in turn lead to “hidden” Markov processes, the core topic of the book. We define the “Markov property,” and show that all the relevant information about a Markov process assuming values in a finite set of cardinalityncan be captured by a nonnegativen×nmatrix called the “state transition matrix,” and ann-dimensional probability distribution of the initial state. Then we invoke the results of Chapter 3 on nonnegative matrices to analyze the temporal evolution of Markov processes. Then we proceed to a discussion of more advanced topics...

• Chapter Five Introduction to Large Deviation Theory
(pp. 129-163)

In this chapter, we take some baby steps in a very important part of probability theory, known as large deviation theory.¹ We begin by describing briefly the motivation for the problem under study. Suppose$\mathbb{A}=\{{{a}_{1}},\ldots ,{{a}_{n}}\}$is a finite set. Let${\mathcal M({\mathbb A})}$denote the set of all probability distributions on the set${\mathbb {A}}$. Clearly one can identify${\mathcal M({\mathbb A})}$with then-simplex${{\mathbb {S}}_{n}}$. Suppose$\mu \in {\mathcal {M}}(\mathbb{A})$is a fixed but possibly unknown probability distribution, andXis a random variable assuming values in${\mathbb {A}}$with the distributionμ. In order to estimateμ, we generate independent samplesx1, …,x1, …,...

• Chapter Six Hidden Markov Processes: Basic Properties
(pp. 164-176)

In this chapter, we study a special type of stochastic process that forms the main focus of this book, called a “hidden” Markov process (HMP). Some authors also use the expression “hidden Markov model (HMM).” In this book we prefer to say “a process {Yt}isa hidden Markov process” or “a process {Yt}hasa hidden Markov model.” We use the two expressions interchangeably.

The chapter is organized as follows: In Section 6.1 we present three distinct types of HMMs, and show that they are all equivalent from the standpoint of their expressive power or modeling ability. In Section...

• Chapter Seven Hidden Markov Processes: The Complete Realization Problem
(pp. 177-222)

In this chapter we continue our study of hidden Markov processes begun in Chapter 6. The focus of study here is the so-called complete realization problem, which was broached in Section 6.2. In that section, we discussed the Baum-Welch algorithm, which attempts to construct a HMM given an output observation sequence of finite length, once the cardinality of the underlying state space is specified. The problem discussed in this chapter is far deeper, and can be stated as follows: Suppose${\mathbb {M}}=\{1,\ldots ,m\}$is a finite set¹ and that {Yt}t≥0 is a stationary stochastic process assuming values in${\mathbb {M}}$. We wish...

6. PART 3. APPLICATIONS TO BIOLOGY
• Chapter Eight Some Applications to Computational Biology
(pp. 225-254)

In the preceding chapters, we have introduced a great many ideas regarding Markov processes and hidden Markov processes. In the last part of the book, we study ways in which some of these ideas can be applied to problems in computational biology. One of the first observations to be made is that, unlike problems in engineering, problems in computational biology do not always perfectly fit into a nice mathematical framework. Instead, one uses a mathematical framework as a starting point, and makes some modifications to suit the practicalities of biology. Usually, in the process, “rigor” is not always maintained. However,...

• Chapter Nine BLAST Theory
(pp. 255-272)

BLAST (Basic Local Alignment Search Tool) is a widely used statistical method for finding similarities between sequences of symbols from finite alphabets. While the theory is completely general, the most widely used applications are to comparing sequences of nucleotides and sequences of amino acids. Though the letter B in BLAST stands for “basic,” in fact the theory itself is anything but basic. The objective of this chapter therefore is to present an accessible treatment of the theory.

The theory of BLAST was developed through a series of papers coauthored by Samuel Karlin; see [77, 74, 75, 76, 39, 40]. The...

7. Bibliography
(pp. 273-284)
8. Index
(pp. 285-287)
9. Back Matter
(pp. 288-288)