I'm taking a course in Bayesian inference and Information Theory with application to physics. I have taken courses in Probability, Statistics, Measure Theory and Statistical Mechanics.
I would like a book mathematically rigorous but illustrated with physical examples if possible. In case it is not, what is a good book for learning about information theory and bayesian inference in more depth with the forementioned background?
1. M. Jordan (ed.), Learning in Graphical Models , MIT Press, 1998
Overview of the book:
Graphical models, a marriage between probability theory and graph theory, provide a natural tool for dealing with two problems that occur throughout applied mathematics and engineering—uncertainty and complexity. In particular, they play an increasingly important role in the design and analysis of machine learning algorithms. Fundamental to the idea of a graphical model is the notion of modularity: a complex system is built by combining simpler parts. Probability theory serves as the glue whereby the parts are combined, ensuring that the system as a whole is consistent and providing ways to interface models to data. Graph theory provides both an intuitively appealing interface by which humans can model highly interacting sets of variables and a data structure that lends itself naturally to the design of efficient general-purpose algorithms.
This book presents an in-depth exploration of issues related to learning within the graphical model formalism. Four chapters are tutorial chapters—Robert Cowell on Inference for Bayesian Networks, David MacKay on Monte Carlo Methods, Michael I. Jordan et al. on Variational Methods, and David Heckerman on Learning with Bayesian Networks. The remaining chapters cover a wide range of topics of current research interest.
2. M. Mezard and A. Montanari, Information, Physics and Computation , Oxford University Press, 2009
Overview of this book:
This book presents a unified approach to a rich and rapidly evolving research domain at the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory. It is accessible to graduate students and researchers without a specific training in any of these fields. The selected topics include spin glasses, error correcting codes, satisfiability, and are central to each field. The approach focuses on large random instances and adopts a common probabilistic formulation in terms of graphical models. It presents message passing algorithms like belief propagation and survey propagation, and their use in decoding and constraint satisfaction solving. It also explains analysis techniques like density evolution and the cavity method, and uses them to study phase transitions.
3. C. Bishop, Pattern Recognition and Machine Learning (Information Science and Statistics), 2010
Overview of this book:
This is the first textbook on pattern recognition to present the Bayesian viewpoint. The book presents approximate inference algorithms that permit fast approximate answers in situations where exact answers are not feasible. It uses graphical models to describe probability distributions when no other books apply graphical models to machine learning. No previous knowledge of pattern recognition or machine learning concepts is assumed. Familiarity with multivariate calculus and basic linear algebra is required, and some experience in the use of probabilities would be helpful though not essential as the book includes a self-contained introduction to basic probability theory.
These books are self-contained and thus the prerequisites mentioned in the details are a add-on for research in these areas. Of course these are just the subset of many other content-rich books out there.
Hope this helps.