Princeton University Computer Science Department Technical Report 1996

Finite Growth Models

Eric Sven Ristad and Peter N. Yianilos

Abstract: Finite growth models (FGM) are nonnegative functionals that arise from parametrically-weighted directed acyclic graphs and a tuple observation that affects these weights. The weight of a source-sink path is the product of the weights along it. The functional's value is the sum of the weights of all such paths. The mathematical foundations of hidden Markov modeling (HMM) and expectation maximization (EM) are generalized to address the problem of functional maximization given an observation. Probability models such as HMMs and stochastic context free grammars are examples that satisfy a particular constraint: that of summing or integrating to one. The FGM framework, algorithms, and data structures describe these and other similar stochastic models while providing a unified and natural way for computer scientists to learn and reason about them and their many variations.

Restricted to probabilistic form, FGMs correspond to stochastic automata that allow observations to be processed in many orderings and groupings -- not just one-by-one in sequential order. As a result the parameters of a highly general form of stochastic transducer can be learned from examples, and the particular case of string edit distance is developed.

In the FGM framework one may direct learning by criteria beyond simple maximum-likelihood. The MAP (maximum a posteriori estimate ) and MDL (minimum description length) are discussed along with the application of FGMs to causal-context probability models and unnormalized noncausal models.

Keywords: Expectation Maximization (EM), Hidden Markov Model (HMM), Baum-Welch Reestimation, Stochastic: Model, Grammar, Transducer, Transduction, Metric Learning, Maximum Likelihood (ML), Maximum a Posteriori (MAP), Minimum Description Length (MDL), String Edit Distance, String Similarity.