Chapter 2 - Statistical Models and SCR

Abstract

In this chapter, we present a brief overview of the basic statistical principles that are referenced throughout the remainder of this book. Emphasis is placed on the definition of a random variable, the common probability distributions used to model random variables, and how hierarchical models can be used to describe conditionally related random variables. We discuss what a probability distribution is, and for a set of common probability distributions, we describe the distribution, its associated properties, and provide examples of when the distribution could be used. The common probability distributions described are: Bernoulli, binomial, Poisson, multinomial, uniform, and normal (Gaussian). We briefly cover classical parameter estimation based on likelihood, describing how to carry out maximum likelihood for a set of examples. Moving to more complex examples of more than one random variable, we also provide background material on the joint, marginal, and conditional distributions. Then, after covering some basic concepts of hierarchical modeling, the chapter ends by describing spatial capture-recapture models using hierarchical modeling notation. This makes the concepts outlined in the previous chapter more precise, and it highlights the fact that SCR models include explicit models for the ecological processes of interest (e.g., spatial variation in density) and the observation process, which describes how individuals are encountered.