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Reviewer: George R. Mayforth

This short book is a mathematically intense exploration of three models for information retrieval (IR): logical, probabilistic, and vector-space models. Van Rijsbergen shows how these models can be represented in Hilbert space, a multi- (or infinite) dimensional extension of Euclidean space. Hilbert space is named after the German mathematician David Hilbert who first defined and investigated it. It was quickly determined that Hilbert's abstract mathematics could be applied to the then-emerging field of quantum mechanics. As shown in this book, mathematical principles and constructs developed for quantum mechanics can be applied to information retrieval. The book begins by presenting its main ideas as a discussion among three people, followed by a more conventional introduction. Among other things, a parallel is drawn between the uncertainty in measurements made on a quantum system and the uncertainty in how a person interacts with queries in an IR system. In essence, in Hilbert space, documents and queries have similar properties to physical systems, hence the same mathematics can be used to analyze them. In particular, the quantum mechanical approach to measurement can be correlated to the notion of relevance in an IR system. Chapters 2 to 4 set the stage for the integration of the mathematics with IR concepts. Chapter 2 describes concepts of IR, such as mathematical representation of queries in terms of sets, evaluation of queries in terms of precision and relevance, and different types of logic that are appropriate to IR. Here also is the first discussion of the relationship between queries and the geometry of a vector space. Chapters 3 and 4 provide an overview of the mathematical concepts that are important to the geometrical treatment: vectors and Hilbert spaces, linear transformations, operators, and matrices. Dirac notation is introduced as a powerful and concise way to manage calculations. These chapters are purely mathematical, with no specific links made to IR concepts. Chapters 5 and 6 connect the math to IR. Chapter 5 deals with conditional logic and implication. The connection with IR is that if a document (a set of propositions represented in a Hilbert space) implies a query (a single proposition represented in the Hilbert space), then the query must be about the document (although "aboutness" is a probability, not a binary condition). One conclusion reached from the mathematical analysis is that the act of making an observation on a query/document affects the final determination of how closely the query is about the document. This is analogous to the quantum mechanical situation where observing a very small object causes an interaction that affects the state of the object. Van Rijsbergen states that this connection between conditional and quantum logic is formally made here for the first time. This connection is important because it implies that other results from quantum mechanics may be applied to IR and provide guidance toward further developments and areas of research. Chapter 6 unifies the set-theoretic, logical, and algebraic approaches into a language and notation. The language is expressed as a small set of operators and functions, using Dirac's notation originally developed for quantum mechanical calculations. Here, the relationship is shown between probability as discussed above and the geometry of the Hilbert space. Specifically, the author makes a formal link between the angle between the vectors representing a document and a query and the probability that the document is about the query. Other IR functions analyzed are coordination-level matching (counting the index terms shared between query and document), pseudo-relevance and relevance feedback (ways of using a user's ranking of the relevance of documents for refining a query), dynamic clustering (guiding the examination of new documents based on the user's observations about ones already seen), and ostensive retrieval (the user's decisions about relevance during a query session are used to calculate the probability of relevance of new documents). All chapters end with a discussion of further reading. This chapter also provides considerations for future research using the tools developed. The book concludes with appendices on linear algebra, quantum mechanics (including a derivation of Heisenberg's uncertainty principle), and probability, followed by an extensive annotated bibliography. It is clear that the author is enthusiastic about and fond of his field. He has devoted great care to this book. The writing is very clear and readable, and the mathematics are presented in a straightforward fashion. I must respectfully disagree, however, with the author's contention that anyone who has taken high school mathematics should be able to handle the mathematics presented here (unless high school math has changed significantly since I took it). Although the concepts are generally developed from first principles, the exposition develops rapidly and assumes a familiarity with linear algebra that takes college courses and hands-on practice to achieve. That said, anyone working in the theory of IR would do well to learn the mathematics and comprehend this text. The author states that those working in artificial intelligence and quantum computing would also benefit from the book; I agree. Online Computing Reviews Service