This book provides an introduction to algebraic statistics. It includes exercises and open problems that complement the material and stimulate further research.
How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.
Markov Bases.- Likelihood Inference.- Conditional Independence.- Hidden Variables.- Bayesian Integrals.- Exercises.- Open Problems.
From the reviews:
"The book provides a succinct overview of the state of the art, including a long section discussing interesting and open problems. ... The book certainly achieves the authors' stated aims of encouraging 'dialogue between algebra and statistics, to benefit both disciplines.' ... the book particularly effective as a tool for a reading group on algebraic statistics; it is an excellent resource, cuts straight to the subject's open questions, and should be an interesting read for any researcher of theoretical statistics." (Robin J. Evans, SIAM Review, Vol. 53 (1), 2011)
"The present monograph grew out of a series of the authors' lectures and its aim is to provide an introduction to some of the fundamental notions in algebraic statistics. ... an excellent source for running seminars on the subject. It could also be used as a textbook in advanced undergraduate or graduate courses ... . presents some open problems giving a few snapshots of current research in the area and may be a good source for problems for young mathematicians interested in the subject." (Luis David Garcia-Puente, Mathematical Reviews, Issue 2012 d)
