Eigenstructure of Maximum Likelihood from Counts Data

TL;DR

This paper introduces the eigenstructure of maximum likelihood estimates for generalized multinomial models, providing algebraic, geometric, and algorithmic insights, including a new iterative, derivative-free method called Weaver algorithms.

Contribution

It derives the algebraic form of the eigenstructure, proves the stationary point property, and develops the Weaver algorithms for eigenestimate computation.

Findings

Eigenstructure characterized by algebraic and geometric representations.

Eigenestimate is a stationary point of the likelihood.

Weaver algorithms efficiently compute eigenestimates iteratively.

Abstract

The MLE (Maximum Likelihood Estimate) for a multinomial model is proportional to the data. We call such estimate an eigenestimate and the relationship of it to the data as the eigenstructure. When the multinomial model is generalized to deal with data arise from incomplete or censored categorical counts, we would naturally look for this eigenstructure between MLE and data. The paper finds the algebraic representation of the eigenstructure (put as Eqn (2.1)), with which the intuition is visualized geometrically (Figures 2.2 and 4.3) and elaborated in a theory (Section 4). The eigenestimate constructed from the eigenstructure must be a stationary point of the likelihood, a result proved in Theorem 4.42. On the bridge between the algebraic definition of Eqn (2.1) and the Proof of Theorem 4.42, we have exploited an elementary inequality (Lemma 3.1) that governs the primitive cases, defined the thick objects of fragment and slice which can be assembled like mechanical parts (Definition 4.1), proved a few intermediary results that help build up the intuition (Section 4), conjectured the universal existence of an eigenestimate (Conjecture 4.32), established a criterion for boundary regularity (Criterion 4.37), and paved way (the Trivial Slicing Algorithm (TSA)) for the derivation of the Weaver algorithms (Section 5) that finds the eigenestimate by using it to reconstruct the observed counts through the eigenstructure, the reconstruction is iterative but derivative-free and matrix-inversion-free. As new addition to the current body of algorithmic methods, the Weaver algorithms craftily tighten threads that are weaved on a rectangular grid (Figure 2.3), and is one incarnation of the TSA. Finally, we put our method in the context of some existing methods (Section 6). Softwares are pseudocoded and put online. Visit http://hku.hk/jdong/eigenstruct2013a.html for demonstrations and download.