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

8.3 Multivariate Gaussians and functions

In Figure 8.1, we represented a function as a collection of samples from 1-dimensional Gaussian distributions. One alternative is to use an n-dimensional multivariate Gaussian distribution to get a sample vector of length n. Actually, you may want to try to reproduce Figure 8.1 but replacing np.random.normal(0, 1, len(x)) with np.random.multivariate_normal, with a mean of np.zeros_like(x) and a standard deviation of np.eye(len(x). The advantage of working with a Multivariate Normal is that we can use the covariance matrix to encode information about the function. For instance, by setting the covariance matrix to np.eye(len(x)), we are saying that each of the 10 points, where we are evaluating the function, has a variance of 1. We are also saying that the variance between them, that is, their covariances, is 0. In other words, they are independent. If we replace those zeros with other numbers, we could get covariances telling a different story...