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

8.10 Hilbert space GPs

Gaussian processes can be slow. The main reason is that their computation requires us to invert a matrix, whose size grows with the number of observations. This operation is computationally costly and does not scale very nicely. For that reason, a large portion of the research around GPs has been to find approximations to compute them faster and allow scaling them to large data.

We are going to discuss only one of those approximations, namely the Hilbert Space Gaussian Process (HSGP), without going into the details of how this approximation is achieved. Conceptually, we can think of it as a basis function expansion similar, in spirit, to how splines are constructed (see Chapter 6). The consequence of this approximation is that it turns the matrix inversion into just matrix multiplication, a much faster operation.

But When Will It Work?

We can only use HSGPs for low dimensions (1 to maybe 3 or 4), and only for some kernels like the exponential quadratic or Matern...