Finite Growth Models
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
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.