Bringing Bayesian Models to Life

Mevin B. Hooten, Trevor J. Hefley, Trevor Hefley

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9780429516801
• Topic: Biological Sciences
• Subtopic: Probability & Statistics
• Index: Biological Sciences

Bringing Bayesian Models to Life empowers the reader to extend, enhance, and implement statistical models for ecological and environmental data analysis. We open the black box and show the reader how to connect modern statistical models to computer algorithms. These algorithms allow the user to fit models that answer their scientific questions without needing to rely on automated Bayesian software. We show how to handcraft statistical models that are useful in ecological and environmental science including: linear and generalized linear models, spatial and time series models, occupancy and capture-recapture models, animal movement models, spatio-temporal models, and integrated population-models.

Features:

• R code implementing algorithms to fit Bayesian models using real and simulated data examples.
• A comprehensive review of statistical models commonly used in ecological and environmental science.
• Overview of Bayesian computational methods such as importance sampling, MCMC, and HMC.
• Derivations of the necessary components to construct statistical algorithms from scratch.

Bringing Bayesian Models to Life contains a comprehensive treatment of models and associated algorithms for fitting the models to data. We provide detailed and annotated R code in each chapter and apply it to fit each model we present to either real or simulated data for instructional purposes. Our code shows how to create every result and figure in the book so that readers can use and modify it for their own analyses. We provide all code and data in an organized set of directories available at the authors' websites.

Section IV

Advanced Models and Concepts

18	Quantile Regression

18.1 QUANTILE MODELS FOR CONTINUOUS DATA

We covered a variety of approaches for modeling mean (and covariance) structure of data in the preceding chapters. For example, the classical linear regression model allowed us to infer how the heterogeneous mean of the data varies with predictor variables of interest. To do so, we parameterized a statistical model as $y_i\sim N(x_i^{\prime }\mathit{\beta },{\sigma }^2)$, for i = 1, …, n, where $x_i^{\prime }\mathit{\beta }$ represents the mean of the data distribution at covariate values xi. However, while this regression model tells us something about the average relationship between xi and y_i, there could be other important aspects of the relationship that it does not provide inference about. For example, what if we are interested in learning about the relationship between xi and y_i associated with the median of the data, or even the upper or lower tail of the data?

In a seminal paper, Koenker and Basset (1978) described a way to generalize the regression concept to model relationships between xi and y_i at various quantiles of the data. The later expansions of these ideas have spawned a variety of generalizations of quantile regression approaches to study relationships in data. In ecology specifically, Cade and Noon (2003) advocated the use of quantile regression methods for ecological inference, but the details on how to implement and extend quantile regression models were beyond the scope of their paper. In what follows, we provide a short introduction to the concept of quantile regression and show how to extend this concept to the Bayesian setting. We also present two different and contrasting perspectives on quantile regression and how we can use it in ecological and environmental science.

In the classical linear regression model, we assumed that $E(y_i|\mathit{\beta },{\sigma }^2)=x_i^{\prime }\mathit{\beta }$. By contrast, in quantile regression, we assume that $q_{\tau }(y_i|\mathit{\beta },{\sigma }^2)=x_i^{\prime }\mathit{\beta }$, where q_τ is the τth (0 < τ < 1) ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Acknowledgments
• Authors
• SECTION I Background
• SECTION II Basic Models and Concepts
• SECTION III Intermediate Models and Concepts
• SECTION IV Advanced Models and Concepts
• SECTION V Expert Models and Concepts
• Tips and Tricks
• Glossary
• References
• Probability Distributions
• Index