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

8.2 Modeling functions

We will begin our discussion of Gaussian processes by first describing a way to represent functions as probabilistic objects. We may think of a function f as a mapping from a set of inputs X to a set of outputs Y . Thus, we can write:

Y = f(X )

One very crude way to represent functions is by listing for each xi value its corresponding yi value as in Table 8.1. You may remember this way of representing functions from elementary school.

x y
0.00 0.46
0.33 2.60
0.67 5.90
1.00 7.91

Table 8.1: A tabular representation of a function (sort of)

As a general case, the values of X and Y will live on the real line; thus, we can see a function as a (potentially) infinite and ordered list of paired values (xi,yi). The order is important because, if we shuffle the values, we will get different functions.

Following this description, we can represent, numerically, any specific function we want. But what if we want to represent functions probabilistically? Well,...