Random Functions Iii : Discrete Mle

This paper continues our earlier investigations into the inversion of random functions in a general (abstract) setting. In Section 2 we investigate a concept of invertibility and the invertibility of the composition of random functions. In Section 3 we resolve some questions concerning the number of samples required to ensure the accuracy of parametric maximum likelihood estimation (MLE). A direct application to phylogeny reconstruction is given. 1. Review of random functions This paper is a sequel of our earlier papers [11, 12]. We assume that the reader is familiar with those papers; however, we repeat the most important definitions. For two finite sets, A and U , let us be given a U-valued random variable ξ a for every a ∈ A. We call the vector of random variables (ξ a : a ∈ A) a random function Ξ : A → U. Ordinary functions are specific instances of random functions. Given another random function, Γ, from U to V , we can speak about the composition of Γ and Ξ, Γ • Ξ : A → V , which is the vector variable (γ ξa : a ∈ A). In this paper we are concerned with inverting random functions. In other words, we look for random functions Γ : U → A in order to obtain the best approximations of the identity function ι : A → A by Γ • Ξ. We always assume that Ξ and Γ are independent. This assumption holds for free if either Ξ or Γ is a deterministic function. Consider the probability of returning a from a by the composition of two random functions, that is, r a = P[γ ξa = a]. The assumption on the independence of Ξ and Γ immediately implies (1) r a = u∈U P[ξ a = u] · P[γ u = a]. A natural criterion is to find Γ for a given Ξ in order to maximize a r a. More generally, we may have a weight function w : A → R + and we may wish to maximize