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

6.5 Distributional models

We saw earlier that we can use linear models for parameters other than the mean (or location parameter). For example, we can use a linear model for the mean and a linear model for the standard deviation of a Gaussian distribution. These models are usually called distributional models. The syntax for distributional models is very similar; we just need to add a line for the auxiliary parameters we want to model. For instance, σ for a Gaussian, or α for a NegativeBinomial.

Let’s now reproduce an example from Chapter 4, the babies example:

Code 6.16

formula = bmb.Formula( 
    "length ∼ np.sqrt(month)", 
    "sigma ∼ month" 

model_dis = bmb.Model(formula, babies) 
idata_dis = model_dis.fit()

Figure 6.9 shows the posterior distribution values of sigma for model_dis (varying sigma) and for a model with constant sigma. We can see...