Book Image

Bayesian Analysis with Python - Third Edition

By : Osvaldo Martin
Book Image

Bayesian Analysis with Python - Third Edition

By: Osvaldo Martin

Overview of this book

The third edition of Bayesian Analysis with Python serves as an introduction to the main concepts of applied Bayesian modeling using PyMC, a state-of-the-art probabilistic programming library, and other libraries that support and facilitate modeling like ArviZ, for exploratory analysis of Bayesian models; Bambi, for flexible and easy hierarchical linear modeling; PreliZ, for prior elicitation; PyMC-BART, for flexible non-parametric regression; and Kulprit, for variable selection. In this updated edition, a brief and conceptual introduction to probability theory enhances your learning journey by introducing new topics like Bayesian additive regression trees (BART), featuring updated examples. Refined explanations, informed by feedback and experience from previous editions, underscore the book's emphasis on Bayesian statistics. You will explore various models, including hierarchical models, generalized linear models for regression and classification, mixture models, Gaussian processes, and BART, using synthetic and real datasets. By the end of this book, you will possess a functional understanding of probabilistic modeling, enabling you to design and implement Bayesian models for your data science challenges. You'll be well-prepared to delve into more advanced material or specialized statistical modeling if the need arises.
Table of Contents (15 chapters)
Preface
12
Bibliography
13
Other Books You May Enjoy
14
Index

4.3 Generalizing the linear model

The linear model we have been using is a special case of a more general model, the Generalized Linear Model (GLM). The GLM is a generalization of the linear model that allows us to use different distributions for the likelihood. At a high level, we can write a Bayesian GLM like:

𝛼 ∼ a prior 𝛽 ∼ another prior θ ∼ some prior μ = 𝛼 + 𝛽X Y ∼ ϕ (f (μ ),θ)

is an arbitrary distribution; some common cases are Normal, Student’s t, Gamma, and NegativeBinomial. θ represents any auxiliary parameter the distribution may have, like σ for the Normal. We also have f, usually called the inverse link function. When is Normal, then f is the identity function. For distributions like Gamma and NegativeBinomial, f is usually the exponential function. Why do we need f? Because the linear model will generally be on the real line, but the μ parameter (or its equivalent) may be defined on a different domain. For instance, μ for the NegativeBinomial is defined for positive values, so we need to transform μ....