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

10.3 Quadratic method

The quadratic approximation, also known as the Laplace method or the normal approximation, consists of approximating the posterior with a Gaussian distribution. To do this, we first find the model of the posterior distribution; numerically, we can do this with an optimization method. Then we compute the Hessian matrix, from which we can then estimate the standard deviation. If you are wondering, the Hessian matrix is a square matrix of second-order partial derivatives. For what we care we can use it to obtain the standard deviation of in general a covariance matrix.

Bambi can solve Bayesian models using the quadratic method for us. In the following code block, we first define a model for the coin-flipping problem, the same one we already defined for the grid method, and then we fit it using the quadratic method, called laplace in Bambi:

Code 10.2

data = pd.DataFrame(data, columns=["w"]) 
priors = {"Intercept": bmb.Prior("Uniform...