Introduction

Let X be a discrete random variable with values in a finite set X . If X has m elements, then without loss of generality, we assume that X = {1, . . . ,m} and we identify the probability distribution of X with a point p = (p1, . . . , pm) ∈ Rm such that px ≥ 0 for every x ∈ X and

∑ x∈X px = 1. The probability

simplex is the set of all such points

∆X := {p ∈ Rm : px ≥ 0, ∑ x∈X

px = 1}. (1.1)

Any statistical model for X is by definition a family of probability distributions and hence a family of points in ∆X . This gives a basic identification of discrete statistical models with geometric objects. In this book we study only parametric models and hence we are always given a parameter space Θ ⊆ Rd and a map p : Θ → ∆X such that the model is equal to the image of Θ under p. The coordinates of this map are typically denoted by px(θ) or p(x; θ) for x ∈ X and θ ∈ Θ.