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
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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.
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