Annals of Combinatorics 3 (1999) 103-113


Inverting Random Functions

Michael A. Steel and László A. Székely

Biomathematics Research Centre, University of Canterbury, Private Bag 4800, Christchurch, New Zeal
m.steel@math.canterbury.ac.nz

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
laszlo@math.sc.edu

Received September 23, 1998

AMS Subject Classification: 92D15, 62C20, 90C46, 90C47

Abstract. In this paper we study how to invert random functions under different criteria. The motivation for this study is phylogeny reconstruction, since the evolution of biomolecular sequences may be considered as a random function from the set of possible phylogenetic trees to the set of collections of biomolecular sequences of observed species. Our results may affect how we think about maximum likelihood estimation (MLE) in phylogeny. For inverting random functions, MLE is optimal under a first criterion, although it is not optimal under a second criterion which is at least equally natural but more conservative. Furthermore, MLE has to be used differently from the way it has been used in the phylogeny literature, if we have a prior distribution on trees and mutation mechanisms and want to keep MLE optimal under the same first criterion. Some of the results of this paper have been known in the setting of statistical decision theory, but have never been discussed in the context of phylogeny.

Keywords: random function, maximum likelihood estimation, minimax problem, phylogeny reconstruction


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