IMSE lunchtime seminar: October 16, at noon

The second meeting of the seminar is, as usual, in Grainger Library, room 335.

Steven Levinson, ECE, talks on A Geometric Interpretation and Proof of Baum's Algorithm for Estimation of the Parameters of a Probabilistic Function of a Markov Process,

A probabilistic function of a finite state Markov process is a stochastic process, H, that generates a sequence of discrete symbols, O, that is neither stationary nor Markovian of any order. The likelihood function for H, P(O|S), is the probability that H generates O given that H is specified by the parameter S. An important problem is to determine S given O. The classical solution is to find that S for which P(O|S) takes its maximum value. This solution yields a consistent estimator of the true process from which O was produced. Baum’s algorithm is a computationally efficient way of maximizing P(O|S) with respect to S subject to a stochasticity constraint on S, G(S). In particular, Baum derives a transformation, T:G(S) \to G(S) such that if S' = T(S), then P(O|S') > P(O|S) except if S is a critical point of P(O|S) in which case S is a fixed point of T. Iterative application of T to an initial value of the parameter will cause P(O|S) to grow to a local maximum and, asymptotically in the length of O, the unique global maximum.
There are two standard proofs of this result, one based on the convexity of \log[P(O|S)] and another based on Hölder’s inequality. These proofs are long and rather opaque giving little insight into the remarkable properties of T. We give a new, short, and intuitively appealing geometric interpretation and proof of Baum’s algorithm. This proof begins by noticing that P(O|S) has the form of a homogeneous polynomial in S with positive coefficients and G(S) restricts S to lie on a manifold defined as that part of the hyperplane s_1 + s_2 + ... + s_n - 1 = 0 bounded by the first orthant of the space \mathbb{R}^n. From these observations it follows that the line segment from S to S' has a positive projection on the gradient of P(O|S) everywhere on the the segment. Thus T is, as Baum shows, a growth transformation.
followed by Nikolaos Tzirakis , Math, reporting on Talbot effect for a nonlinear Schrödinger equation on the torus.
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.

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