Very often the true dynamics of a physical system is hidden from us: we are able to observe only a certain measurement of the state of the underlying system. For example, air temperature and the speed of wind are easily observable functions of the climate dynamics, while electroencephalogram provides insight into the functioning of the brain. Observing only partial information naturally limits our ability to describe or model the underlying system, detect changes in dynamics, make predictions, etc. Nevertheless, these problems must be addressed in practical situations, and are extremely challenging from the mathematical point of view.
Remarkably, many problems have a common nature. For example, correcting signals corrupted by noisy channels during transmission, analyzing neuronal spikes, analyzing genomic sequences, and validating the so-called renormalization group methods of theoretical physics, all have the same underlying mathematical structure: a hidden Gibbsmodel. In this model, the observable process is a function (either deterministic or random) of a process whose underlying probabilistic law is Gibbs. Gibbs models have been introduced in Statistical Mechanics, and have been highly successful in modeling physical systems (ferromagnets, dilute gases, polymer chains). Nowadays Gibbs models are ubiquitous and are used by researchers from different fields, often implicitly, i.e., without knowledge that a given model belongs to a much wider class.
A well-established Gibbs theory provides an excellent starting point to develop Hidden Gibbstheory. A subclass of Hidden Gibbs models is formed by the well-known Hidden Markov Models. Several examples have been considered in Statistical Mechanics and the theory of Dynamical Systems. However, many basic questions are still unanswered. Moreover, the relevance of Hidden Gibbs models to other areas, such as Information Theory, Bioinformatics, and Neuronal Dynamics, has never been made explicit and exploited.
Lecture Notes and other materials will be made available
Final grade will be based on attendance, active participation, homework assignments and, possibly, a final presentation (topics will be provided).
03/09/15 NO LECTURE
17/09/15 Gibbs formalism
24/09/15 Gibbs formalism
01/10/15 NO LECTURE
08/10/15 Hidden Markov Models
15/10/15 Application 1: Denoising in Information Theory
22/10/15 Renormalisation of Gibbs states
29/10/15 Preservation of Gibbs Property
05/11/15 NO LECTURE
12/11/15 Application 2: Nonlinear Time-Series Analysis
26/11/15 Students presentations
03/12/15 Students presentations
10/12/15 Students presentations