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Bayesian Analysis with Python
Bayesian Analysis with Python

Bayesian Analysis with Python: Unleash the power and flexibility of the Bayesian framework

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Bayesian Analysis with Python

Chapter 2. Programming Probabilistically – A PyMC3 Primer

Now that we have a basic understanding of Bayesian statistics we are going to learn how to build probabilistic models using computational tools; specifically we are going to learn about probabilistic programming. The main idea is that we are going to use code to describe our models and make inferences from them. It is not that we are too lazy to learn the mathematical way, nor are we elitist hardcore hackers—I-dream-in-code. One important reason behind this choice is that many models do not lead to a closed-form analytic posterior, that is, we can only compute those posteriors using numerical techniques. Another reason to learn probabilistic programing is that modern Bayesian statistics is done mainly by writing code, and since we already know Python, why would we do it in another way?! Probabilistic programming offers an effective way to build complex models and allows us to focus more on model design,...

Probabilistic programming

Bayesian statistics is conceptually very simple: we have some data that is fixed, in the sense that we cannot change what we have measured, and we have parameters whose values are of interest to us and hence we explore their plausible values. All the uncertainties we have are modeled using probabilities. In other statistical paradigms, there are different types of unknown quantities; in the Bayesian framework everything that is unknown is treated the same. If we do not know a quantity we assign a probability distribution to it. Then, Bayes' theorem is used to transform the prior probability distribution Probabilistic programming (what we know about a given problem before observing the data), into a posterior distribution Probabilistic programming (what we know after observing data). In other words, Bayesian statistics is a form of learning.

Although conceptually simple, fully probabilistic models often lead to analytically intractable expressions. For many years, this was a real problem and was probably one...

PyMC3 introduction

PyMC3 is a Python library for probabilistic programming. The last version at the moment of writing is 3.0.rc2 released on October 4th, 2016. PyMC3 provides a very simple and intuitive syntax that is easy to read and that is close to the syntax used in the statistical literature to describe probabilistic models. PyMC3 is written using Python, where the computationally demanding parts are written using NumPy and Theano. Theano is a Python library originally developed for deep learning that allows us to define, optimize, and evaluate mathematical expressions involving multidimensional arrays efficiently. The main reason PyMC3 uses Theano is because some of the sampling methods, like NUTS, need gradients to be computed and Theano knows how to do automatic differentiation. Also, Theano compiles Python code to C code, and hence PyMC3 is really fast. This is all the information about Theano we need to have to use PyMC3. If you still want to learn more about it start reading...

Summarizing the posterior

As we have already seen, the result of a Bayesian analysis is a posterior distribution. This contains all the information about our parameters, according to the data and the model. One way to visually summarize the posterior is to use the plot_posterior function that comes with PyMC3. This function accepts a PyMC3 trace or a NumPy array as a main argument. By default, plot_posterior shows a histogram for the credible parameters together with the mean of the distribution and the 95% HPD as a thick black line at the bottom of the plot. Different interval values can be set for the HPD with the argument alpha_level. We are going to refer to this type of plot as Kruschke's plot, since John K. Kruschke introduced this type of plot in his great book Doing Bayesian Data Analysis:

pm.plot_posterior(chain, kde_plot=True)
Summarizing the posterior

Posterior-based decisions

Sometimes describing the posterior is not enough. Sometimes we need to make decisions based on our inferences. We have to reduce...

Summary

In this chapter, we learned about probabilistic programming and how inference engines leverage the power of Bayesian modeling. We discussed the main conceptual ideas behind MCMC methods and its central role in modern Bayesian data analysis. We encountered, for the first time, the powerful and easy-to-use PyMC3 library. We revisited the coin-flipping problem from the previous chapter, this time using PyMC3 to define it, solve it, and also perform model checks and diagnoses that are a very important part of the modeling process.

In the next chapter, we will keep building our Bayesian analytics skills by learning how to work with models having more than one parameter and how to make parameters talk to each other.

Keep reading

Probabilistic programming


Bayesian statistics is conceptually very simple: we have some data that is fixed, in the sense that we cannot change what we have measured, and we have parameters whose values are of interest to us and hence we explore their plausible values. All the uncertainties we have are modeled using probabilities. In other statistical paradigms, there are different types of unknown quantities; in the Bayesian framework everything that is unknown is treated the same. If we do not know a quantity we assign a probability distribution to it. Then, Bayes' theorem is used to transform the prior probability distribution (what we know about a given problem before observing the data), into a posterior distribution (what we know after observing data). In other words, Bayesian statistics is a form of learning.

Although conceptually simple, fully probabilistic models often lead to analytically intractable expressions. For many years, this was a real problem and was probably one of the...

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Key benefits

  • Simplify the Bayes process for solving complex statistical problems using Python;
  • Tutorial guide that will take the you through the journey of Bayesian analysis with the help of sample problems and practice exercises;
  • Learn how and when to use Bayesian analysis in your applications with this guide.

Description

The purpose of this book is to teach the main concepts of Bayesian data analysis. We will learn how to effectively use PyMC3, a Python library for probabilistic programming, to perform Bayesian parameter estimation, to check models and validate them. This book begins presenting the key concepts of the Bayesian framework and the main advantages of this approach from a practical point of view. Moving on, we will explore the power and flexibility of generalized linear models and how to adapt them to a wide array of problems, including regression and classification. We will also look into mixture models and clustering data, and we will finish with advanced topics like non-parametrics models and Gaussian processes. With the help of Python and PyMC3 you will learn to implement, check and expand Bayesian models to solve data analysis problems.

Who is this book for?

Students, researchers and data scientists who wish to learn Bayesian data analysis with Python and implement probabilistic models in their day to day projects. Programming experience with Python is essential. No previous statistical knowledge is assumed.

What you will learn

  • • Understand the essentials Bayesian concepts from a practical point of view
  • • Learn how to build probabilistic models using the Python library PyMC3
  • • Acquire the skills to sanity-check your models and modify them if necessary
  • • Add structure to your models and get the advantages of hierarchical models
  • • Find out how different models can be used to answer different data analysis questions
  • • When in doubt, learn to choose between alternative models.
  • • Predict continuous target outcomes using regression analysis or assign classes using logistic and softmax regression.
  • • Learn how to think probabilistically and unleash the power and flexibility of the Bayesian framework

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Table of Contents

9 Chapters
1. Thinking Probabilistically - A Bayesian Inference Primer Chevron down icon Chevron up icon
2. Programming Probabilistically – A PyMC3 Primer Chevron down icon Chevron up icon
3. Juggling with Multi-Parametric and Hierarchical Models Chevron down icon Chevron up icon
4. Understanding and Predicting Data with Linear Regression Models Chevron down icon Chevron up icon
5. Classifying Outcomes with Logistic Regression Chevron down icon Chevron up icon
6. Model Comparison Chevron down icon Chevron up icon
7. Mixture Models Chevron down icon Chevron up icon
8. Gaussian Processes Chevron down icon Chevron up icon
Index Chevron down icon Chevron up icon

Customer reviews

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Full star icon Full star icon Full star icon Half star icon Empty star icon 3.4
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5 star 60%
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1 star 40%
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Luis Junior Aug 21, 2024
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Excellent
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Federico Carrone Jun 23, 2019
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I loved this book. I recommend you read it after reading "Probabilistic Programming & Bayesian Methods for Hackers by Cam Davidson-Pilon.
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Jean-Michel Patrick May 16, 2019
Full star icon Full star icon Full star icon Full star icon Full star icon 5
A great book, very well written and that will introduce you to the main themes of Bayesian inference. Tackles a breadth of topics that will allow you to start doing amazing - and even complex - Bayesian models.Hands-on with Python and PyMC3 and end-of-chapter exercises, which is very pedagogical and lets you really understand the guts of your models. Definitely recommend it to beginners!
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Richard BJ Apr 20, 2018
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Really useful, worked examples and all.
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Charles Fribourg Aug 22, 2017
Full star icon Full star icon Full star icon Full star icon Full star icon 5
It is not easy to find materials for a short introductory course in Bayesian Statistics, especially if you want to use PyMC3, and this book gives you all that. The book is highly practical, and goes much more in-depth than "Bayesian Methods for Hackers" or "Think Bayes". All the codes are in Jupyter notebook on Github so that the students can follow quite easily even without much Python experience. It does not assume too much knowledge in Probability or Statistics, and the pages on this is a bit limit. However, there is a reading list at the end of each chapter for the motivated student. All in all, it is a great book to kick start Bayesian Statistics.
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