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Bayesian Analysis with Python - Third Edition

You're reading from  Bayesian Analysis with Python - Third Edition

Product type Book
Published in Jan 2024
Publisher Packt
ISBN-13 9781805127161
Pages 394 pages
Edition 3rd Edition
Languages
Author (1):
Osvaldo Martin Osvaldo Martin
Profile icon Osvaldo Martin

Table of Contents (15) Chapters

Preface
1. Chapter 1 Thinking Probabilistically 2. Chapter 2 Programming Probabilistically 3. Chapter 3 Hierarchical Models 4. Chapter 4 Modeling with Lines 5. Chapter 5 Comparing Models 6. Chapter 6 Modeling with Bambi 7. Chapter 7 Mixture Models 8. Chapter 8 Gaussian Processes 9. Chapter 9 Bayesian Additive Regression Trees 10. Chapter 10 Inference Engines 11. Chapter 11 Where to Go Next 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....

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