1,601 research outputs found
Total positivity in exponential families with application to binary variables
We study exponential families of distributions that are multivariate totally
positive of order 2 (MTP2), show that these are convex exponential families,
and derive conditions for existence of the MLE. Quadratic exponential familes
of MTP2 distributions contain attractive Gaussian graphical models and
ferromagnetic Ising models as special examples. We show that these are defined
by intersecting the space of canonical parameters with a polyhedral cone whose
faces correspond to conditional independence relations. Hence MTP2 serves as an
implicit regularizer for quadratic exponential families and leads to sparsity
in the estimated graphical model. We prove that the maximum likelihood
estimator (MLE) in an MTP2 binary exponential family exists if and only if both
of the sign patterns and are represented in the sample for
every pair of variables; in particular, this implies that the MLE may exist
with observations, in stark contrast to unrestricted binary exponential
families where observations are required. Finally, we provide a novel and
globally convergent algorithm for computing the MLE for MTP2 Ising models
similar to iterative proportional scaling and apply it to the analysis of data
from two psychological disorders
On an involution of Christoffel words and Sturmian morphisms
There is a natural involution on Christoffel words, originally studied by the second author in [A. de Luca, Combinatorics of standard Sturmian words, Lecture Notes in Computer Science 1261 (1997) 249–267]. We show that it has several equivalent definitions: one of them uses the slope of the word, and changes the numerator and the denominator respectively in their inverses modulo the length; another one uses the cyclic graph allowing the construction of the word, by interpreting it in two ways (one as a permutation and its ascents and descents, coded by the two letters of the word, the other in the setting of the Fine and Wilf periodicity theorem); a third one uses central words and generation through iterated palindromic closure, by reversing the directive word. We show further that this involution extends to Sturmian morphisms, in the sense that it preserves conjugacy classes of these morphisms, which are in bijection with Christoffel words. The involution on morphisms is the restriction of some conjugation of the automorphisms of the free group. Finally, we show that, through the geometrical interpretation of substitutions of Arnoux and Ito, our involution is the same thing as duality of endomorphisms (modulo some conjugation)
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