Separability as a modeling paradigm in large probabilistic models

Many interesting stochastic models can be formulated as finite-state vector Markov processes, with a state characterized by the values of a collection of random variables. In general, such models suffer from the curse of dimensionality: the size of the state space grows exponentially with the number of underlying random variables, thereby precluding conventional modeling and analysis. A potential cure to this curse is to work with models that allow the propagation of partial information, e.g. marginal distributions, expectations, higher-moments, or cross-correlations, as derived from the joint distribution for the network state. This thesis develops and rigorously investigates the notion of separability, associated with structure in probabilistic models that permits exact propagation of partial information. We show that when partial information can be propagated exactly, it can be done so linearly. The matrices for propagating such partial information share many valuable spectral relationships with the underlying transition matrix of the Markov chain. Separability can be understood from the perspective of subspace invariance in linear systems, though it relates to invariance in a non-standard way. We analyze the asymptotic generality-- as the number of random variables becomes large-of some special cases of separability that permit the propagation of marginal distributions. Within this discussion of separability, we introduce the generalized influence model, which incorporates as special cases two prominent models permitting the propagation of marginal distributions: the influence model and Markov chains on permutations (the symmetric group). The thesis proposes a potentially tractable solution to learning informative model parameters, and illustrates many advantageous properties of the estimator under the assumption of separability. Lastly, we illustrate separability in the general setting without any notion of time-homogeneity, and discuss potential benefits for inference in special cases.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2011

Author dc:creator

• Richoux, William J., Jr. (William Joseph)

Advisor dc:contributor.advisor

• George C. Verghese.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/64586

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/64586