We introduce probability calculus in this chapter as a tool for representing and reasoning with degrees of belief.

Introduction

We provide in this chapter a framework for representing and reasoning with uncertain beliefs. According to this framework, each event is assigned a degree of belief which is interpreted as a probability that quantifies the belief in that event. Our focus in this chapter is on the semantics of degrees of belief, where we discuss their properties and the methods for revising them in light of new evidence. Computational and practical considerations relating to degrees of belief are discussed at length in future chapters.

We start in Section 3.2 by introducing degrees of belief, their basic properties, and the way they can be used to quantify uncertainty. We discuss the updating of degrees of belief in Section 3.3, where we show how they can increase or decrease depending on the new evidence made available. We then turn to the notion of independence in Section 3.4, which will be fundamental when reasoning about uncertain beliefs. The properties of degrees of belief are studied further in Section 3.5, where we introduce some of the key laws for manipulating them. We finally treat the subject of soft evidence in Sections 3.6 and 3.7, where we provide some tools for updating degrees of belief in light of uncertain information.

Degrees of belief

We have seen in Chapter 2 that a propositional knowledge base Δ classifies sentences into one of three categories: sentences that are implied by Δ, sentences whose negations are implied by Δ, and all other sentences (see Figure 2.2).

We consider in this chapter the problem of finding variable instantiations that have maximal probability under some given evidence. We present two classes of exact algorithms for this problem, one based on variable elimination and the other based on systematic search. We also present approximate algorithms based on local search.

Introduction

Consider the Bayesian network in Figure 10.1, which concerns a population that is 55% male and 45% female. According to this network, members of this population can suffer from a medical condition C that is more likely to occur in males. Moreover, two diagnostic tests are available for detecting this condition, T1 and T2, with the second test being more effective on females. The CPTs of this network also reveal that the two tests are equally effective on males.

One can partition the members of this population into four different groups depending on whether they are male or female and whether they have the condition or not. Suppose that a person takes both tests and all we know is that the two tests yield the same result, leading to the evidence A=yes. We may then ask: What is the most likely group to which this individual belongs? This query is therefore asking for the most likely instantiation of variables S and C given evidence A=yes, which is technically known as a MAP instantiation. We have already discussed this class of queries in Chapter 5, where we referred to variables S and C as the MAP variables.

INTRODUCTION

Computational approaches such as those described in Chapters 6 through 10 analyze protein-protein interaction (PPI) networks on the basis of network properties only, with little integration of information from outside sources. Current conventional methods can predict only whether two proteins share a specific function but not the universe of functions that they share. Their effectiveness is hampered by their inability to take into consideration the full range of available information about protein functions. The discussion in Chapter 11 has demonstrated the effectiveness of integrating Gene Ontology (GO) annotations into such analysis. It has become increasingly apparent that the fusion of multiple strands of biological data regarding each gene or protein will produce a more comprehensive picture of the relations among the components of a genome [191], including proteins, and a more specific representation of each protein. The sophisticated data set, graph, or tree generated through these means can be subjected to advanced computational analysis by methods such as machine learning algorithms. Such approaches have become increasingly widespread and are expected to improve the accuracy of protein function prediction.

In this chapter, we present some of the more recent approaches that have been developed for incorporating diverse biological information into the explorative analysis of PPI networks.

INTEGRATION OF GENE EXPRESSION WITH PPI NETWORKS

Current research efforts have resulted in the generation of large quantities of data related to the functional properties of genomes; specifically, gene expression and protein interaction data. Gene expression profiles provide a snapshot of the simultaneous activity of all the genes in a genome under a given condition, thus eliminating the need to examine each gene separately.

Bayesian networks have received a lot of attention over the last few decades from both scientists and engineers, and across a number of fields, including artificial intelligence (AI), statistics, cognitive science, and philosophy.

Perhaps the largest impact that Bayesian networks have had is on the field of AI, where they were first introduced by Judea Pearl in the midst of a crisis that the field was undergoing in the late 1970s and early 1980s. This crisis was triggered by the surprising realization that a theory of plausible reasoning cannot be based solely on classical logic [McCarthy, 1977], as was strongly believed within the field for at least two decades [McCarthy, 1959]. This discovery has triggered a large number of responses by AI researchers, leading, for example, to the development of a new class of symbolic logics known as non-monotonic logics (e.g., [McCarthy, 1980; Reiter, 1980; McDermott and Doyle, 1980]). Pearl's introduction of Bayesian networks, which is best documented in his book [Pearl, 1988], was actually part of his larger response to these challenges, in which he advocated the use of probability theory as a basis for plausible reasoning and developed Bayesian networks as a practical tool for representing and computing probabilistic beliefs.

From a historical perspective, the earliest traces of using graphical representations of probabilistic information can be found in statistical physics [Gibbs, 1902] and genetics [Wright, 1921]. However, the current formulations of these representations are of a more recent origin and have been contributed by scientists from many fields.

We consider in this chapter the relationship between the values of parameters that quantify a Bayesian network and the values of probabilistic queries applied to these networks. In particular, we consider the impact of parameter changes on query values, and the amount of parameter change needed to enforce some constraints on these values.

Introduction

Consider a laboratory that administers three tests for detecting pregnancy: a blood test, a urine test, and a scanning test. Assume also that these tests relate to the state of pregnancy as given by the network of Figure 16.1 (we treated this network in Chapter 5). According to this network, the prior probability of pregnancy is 87% after an artificial insemination procedure. Moreover, the posterior probability of pregnancy given three negative tests is 10.21%. Suppose now that this level of accuracy is not acceptable: the laboratory is interested in improving the tests so the posterior probability is no greater than 5% given three negative tests. The problem now becomes one of finding a certain set of network parameters (corresponding to the tests' false positive and negative rates) that guarantee the required accuracy. This is a classic problem of sensitivity analysis that we address in Section 16.3 as it is concerned with controlling network parameters to enforce some constraints on the queries of interest.

Assume now that we replace one of the tests with a more accurate one, leading to a new Bayesian network that results from updating the parameters corresponding to that test.

I am pleased to offer the research community my second book-length contribution to the field of bioinformatics. My first book, Advanced Analysis of Gene Expression Microarray Data, was published in 2006 by World Scientific as part of its Science, Engineering, and Biology Informatics (SEBI) series. I first became involved in the study of bioinformatics in 1998 and, over the ensuing decade, have been struck by the enormous quantity of data being generated and the need for effective approaches to its analysis.

The analysis of protein-protein interactions (PPIs) is fundamental to the understanding of cellular organizations, processes, and functions. It has been observed that proteins seldom act as single isolated species in the performance of their functions; rather, proteins involved in the same cellular processes often interact with each other. Therefore, the functions of uncharacterized proteins can be predicted through comparison with the interactions of similar known proteins. A detailed examination of a PPI network can thus yield significant new insights into protein functions. These interactions have traditionally been examined via intensive small-scale investigations of a small set of proteins of interest, each yielding information about a limited number of PPIs. The existing databases of PPIs have been compiled from such small-scale screens, presented in individual research papers. Because these data were subject to stringent controls and evaluation in the peer-review process, they can be considered to be fairly reliable. However, each experiment observes only a few interactions and yields a data set of very limited size. Recent large-scale investigations of PPIs using such techniques as two-hybrid systems, mass spectrometry, and protein microarrays have enriched the available protein interaction data and facilitated the construction of integrated PPI networks.

INTRODUCTION

The previous three chapters have discussed in detail the analysis of protein-proteinelusive interactions, which often compromise the effectiveness of the approaches presented so far. In this chapter, we will examine flow-based approaches, another avenue for the analysis of PPI networks. These methods permit information from other sources to be integrated with PPI data to enhance the effectiveness of algorithms for protein function prediction and functional module detection. Flow-based approaches offer a novel strategy for assessing the degree of biological and topological influence of each protein over other proteins in a PPI network. Through simulation of biological or functional flows within these complex networks, these methods seek to model and predict network behavior under the influence of various realistic external stimuli.

This chapter will discuss several flow-based methods for the prediction of protein function. The first section will address the concept of functional flow introduced by Nabieva et al. [221] and the FunctionalFlow algorithm based on this model. In this approach, each protein with a known functional annotation is treated as a source of functional flow, which is then propagated to unannotated nodes, using the edges in the interaction graph as a conduit. This process is based on simple local rules. A distance effect is formulated that considers the impact of each annotated protein on any other protein, with the effect diminishing as the distance between the proteins increases.

The generation of protein-protein interaction (PPI) data is proceeding at a rapid and accelerating pace, heightening the demand for advances in the computational methods used to analyze patterns and relationships in these complex data sets. This book has offered a systematic presentation of a variety of advanced computational approaches that are available for the analysis of PPI networks. In particular, we have focused on those approaches that address the modularity analysis and functional prediction of proteins in PPI networks. These computational techniques have been presented as belonging to seven categories:

1. Basic representation and modularity analysis. Throughout this book, PPI networks have been represented through mathematical graphs, and we have provided a detailed discussion of the basic properties of such graphs. PPI networks have been identified as modular and hierarchical in nature, and modularity analysis is therefore of particular utility in understanding their structure. A range of approaches has been proposed for the detection of modules within these networks and to guide the prediction of protein function. We have broadly classified these methods as distance-based, graph-theoretic, topology-based, flow-based, statistical, and domain knowledge-based. Clustering a PPI network permits a better understanding of its structure and the interrelationship of its constituent components. The potential functions of unannotated proteins may be predicted by comparison with other members of the same functional module.

2. Distance-based analysis. Chapter 7 surveyed five categories of approaches to distance-based clustering. All these methods use classic clustering techniques and focus on the definition of the topological or biological distance or similarity between two proteins in a network.

3. […]

INTRODUCTION

The ability of the various approaches discussed throughout this book to accurately analyze protein-protein interactions (PPIs) is often compromised by the errors and gaps that characterize the data. Their accuracy would be enhanced by the integration of data from all available sources. Modern experimental and computational techniques have resulted in the accumulation of massive amounts of information about the functional behavior of biological components and systems. These diverse data sources have provided useful insights into the functional association between components. The following types of data have frequently been drawn upon for functional analysis and could be integrated with PPI data [276, 297, 304, 305]:

• Amino acid sequences

• Protein structures

• Genomic sequences

• Phylogenetic profiles

• Microarray expressions

• Gene Ontology (GO) annotations

The development of sequence similarity search algorithms such as FASTA [244], BLAST [13], and PSI-BLAST [14] has been a major breakthrough in the field of bioinformatics. The algorithms rest on the understanding that proteins with similar sequences are functionally consistent. Searching for sequential homologies among proteins can facilitate their classification and the accurate prediction of their functions.

The availability of complete genomes for various organisms has shifted such sequence comparisons from the level of the single gene to the genome level [48, 97]. As discussed in Chapter 3, several genome-scale approaches have been introduced on the basis of the correlated evolutionary mechanisms of genes. For example, the conservation of gene neighborhoods across different, distantly-related genomes reveals potential functional linkages [80, 235, 296]. Gene fusion analysis infers pairs of interacting proteins and their functional relatedness [98, 208].