Belief Propagation and Revision in Networks with Loops

Local belief propagation rules of the sort proposed by Pearl(1988) are guaranteed to converge to the optimal beliefs for singly connected networks. Recently, a number of researchers have empirically demonstrated good performance of these same algorithms on networks with loops, but a theoretical understanding of this performance has yet to be achieved. Here we lay the foundation for an understanding of belief propagation in networks with loops. For networks with a single loop, we derive ananalytical relationship between the steady state beliefs in the loopy network and the true posterior probability. Using this relationship we show a category of networks for which the MAP estimate obtained by belief update and by belief revision can be proven to be optimal (although the beliefs will be incorrect). We show how nodes can use local information in the messages they receive in order to correct the steady state beliefs. Furthermore we prove that for all networks with a single loop, the MAP estimate obtained by belief revisionat convergence is guaranteed to give the globally optimal sequence of states. The result is independent of the length of the cycle and the size of the statespace. For networks with multiple loops, we introduce the concept of a "balanced network" and show simulati.

Justified Generalization: Acquiring Procedures from Examples

This thesis describes an implemented system called NODDY for acquiring procedures from examples presented by a teacher. Acquiring procedures form examples involves several different generalization tasks. Generalization is an underconstrained task, and the main issue of machine learning is how to deal with this underconstraint. The thesis presents two principles for constraining generalization on which NODDY is based. The first principle is to exploit domain based constraints. NODDY demonstrated how such constraints can be used both to reduce the space of possible generalizations to manageable size, and how to generate negative examples out of positive examples to further constrain the generalization. The second principle is to avoid spurious generalizations by requiring justification before adopting a generalization. NODDY demonstrates several different ways of justifying a generalization and proposes a way of ordering and searching a space of candidate generalizations based on how much evidence would be required to justify each generalization. Acquiring procedures also involves three types of constructive generalizations: inferring loops (a kind of group), inferring complex relations and state variables, and inferring predicates. NODDY demonstrates three constructive generalization methods for these kinds of generalization.