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CMI Special Lecture Series on Information, Control and Statistical Mechanics

Sanjoy Mitter, MIT

The theory of Stochastic control with Partial Observations when extended to stochastic systems with unknown parameters provides a comprehensive view of control when systems can be described using Markov Processes. Conceptually, this theory can be extended to stochastic systems, which have both a temporal and spatial dimension by describing them in terms of Markov Random Fields. The statistical mechanics of Gibbs Measures, which has a rich theoretical underpinning (See for example, H. O. Georgii: Gibbs Measures and Phase Transitions: de Gruy, 1988) can be effectively used for mathematical modeling of such spatio-temporal stochastic systems. Viewed thus, Shannon's Information theory would be interpreted as the stochastic control with a non-classical information pattern of a statistical mechanical system on the integers in thermodynamic limit. The coding and decoding problem for the reliable transmission of messages over a noisy channel can be transported to the study of a Spin glass in an asymptotic regime. These ideas are discussed in the first lecture from a high level viewpoint. In the next two lectures, I discuss the interconnections that exist between Path estimation for non-linear stochastic systems, Information theory and Non-equilibrium Statistical Mechanics. The fundamental idea is to view Bayesian Inference as minimization of an appropriately defined Free Energy. This Free Energy minimization is then given an information-theoretic interpretation. To illustrate these ideas in a concrete setting, I discuss the Kalman Filter for Gaussian Processes as a Statistical Mechanical System.

(1) Friday January 14 2005, 4:00pm, Jorgensen 74
(2) Tuesday January 18 2005, 4:00pm, Moore 070
(3) Monday January 24 2005, 1:00pm, Moore 070
(4) Tuesday January 25 2005, 4:00pm, Moore 070

(1) Mitter, S.K. and Newton, N. A Variational Approach to Nonlinear Estimation SIAM Jrn. on Control, Volume 42, Number 5 (2004), pp. 1813-1833. (pdf)

(2) Mitter, S.K. and Newton, N.J. Information and Entropy Flow in the Kalman-Bucy Filter to appear in J. of Stat. Phys., January 2005. (pdf)

(3) Mitter, S.K., and Tatikonda, S., Control over Noisy Channels IEEE Trans. on Auto. Control, Volume: 49 , Issue: 7 , July 2004, Pages:1196 - 1201. (pdf)

(4) Mitter, S.K., and Tatikonda, S., Control under Communication Constraints IEEE Trans. on Auto. Control, Volume: 49 , Issue: 7 , July 2004, Pages:1056 - 1068. (pdf)

(5) Tatikonda, S., Sahai, A. and Mitter, S.K. Stochastic Linear Control Over a Communication Channel IEEE Trans. on Auto Control, Special Issue on Networked Control Systems, Volume: 49 , Issue: 9 , Sept. 2004, Pages:1549 - 1561. (pdf)

(6) Borkar, V.S., Mitter, S.K. and Tatikonda, S. Optimal Sequential Vector Quantization of Markov Sources SIAM J. Control Optim., {\bf 40}, 135-148; proceedings of the 1998 IEEE Int'l Sym on Inf Th. (pdf)

(7) Mitter, S.K. Control with Limited Information: the Role of Systems Theory and Information Theory ISIT 2000 Plenary Talk, IEEE Information Theory Society Newsletter, 50, pgs.1-23; Eur. Jrn. Control, Vol. 7, pp. 122-131. (pdf)

(8) Markov Control Problems Under Communication Constraints, by Borkar, V.S.,Tatikonda, S. and Mitter, S.K., Comm. Inf. Sys. Vol. 1., No. 1., 2001, pp. 15-32. (pdf)