IMSE lunchtime seminar: October 16, at noon

A probabilistic function of a finite state Markov process is a stochastic process, , that generates a sequence of discrete symbols, , that is neither stationary nor Markovian of any order. The likelihood function for H, , is the probability that generates given that is specified by the parameter . An important problem is to determine given . The classical solution is to find that for which takes its maximum value. This solution yields a consistent estimator of the true process from which was produced. Baum’s algorithm is a computationally efficient way of maximizing with respect to subject to a stochasticity constraint on , . In particular, Baum derives a transformation, such that if , then except if is a critical point of in which case is a fixed point of . Iterative application of to an initial value of the parameter will cause to grow to a local maximum and, asymptotically in the length of , the unique global maximum.
There are two standard proofs of this result, one based on the convexity of and another based on Hölder’s inequality. These proofs are long and rather opaque giving little insight into the remarkable properties of . We give a new, short, and intuitively appealing geometric interpretation and proof of Baum’s algorithm. This proof begins by noticing that has the form of a homogeneous polynomial in with positive coefficients and restricts to lie on a manifold defined as that part of the hyperplane bounded by the first orthant of the space . From these observations it follows that the line segment from to has a positive projection on the gradient of everywhere on the the segment. Thus is, as Baum shows, a growth transformation.

Using various techniques from harmonic analysis and number theory we demonstrate the dependence of the qualitative behavior of the solution of NLS on the algebraic properties of time. To obtain the results we use some recent advances in low regularity theory of dispersive PDE. This is joint work with M. B. Erdogan.