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Bayesian Analysis with Python
Bayesian Analysis with Python

Bayesian Analysis with Python: Introduction to statistical modeling and probabilistic programming using PyMC3 and ArviZ , Second Edition

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Bayesian Analysis with Python

Programming Probabilistically

"Our golems rarely have a physical form, but they too are often made of clay living in silicon as computer code."
- Richard McElreath

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 with PyMC3. The basic idea is to specify models using code and then solve them in a more or less automatic way. It is not that we are too lazy to learn the mathematical way, nor are we elitist-hardcore-hackers-in-code. One important reason behind this choice is that many models do not lead to an analytic closed form, and thus we can only solve those models using numerical techniques.

Another reason to learn probabilistic programming is that modern Bayesian statistics is mainly done by writing code...

Probabilistic programming

Bayesian statistics is conceptually very simple; we have the knows and the unknowns; we use Bayes' theorem to condition the latter on the former. If we are lucky, this process will reduce the uncertainty about the unknowns. Generally, we refer to the knowns as data and treat it like a constant, and the unknowns as parameters and treat them as probability distributions. In more formal terms, we assign probability distributions to unknown quantities. Then, we use Bayes' theorem to transform the prior probability distribution into a posterior distribution . 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 main issues that hindered the wide adoption of Bayesian methods.

The arrival of the computational era and the development...

PyMC3 primer

PyMC3 is a Python library for probabilistic programming. The last version at the moment of writing is 3.6. 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's base code is written using Python, and the computationally demanding parts are written using NumPy and Theano.

Theano is a Python library that was originally developed for deep learning and 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, such as NUTS, need gradients to be computed, and Theano knows how to compute gradients using what is known as automatic differentiation. Also, Theano compiles Python code to C code, and hence PyMC3 is really...

Summarizing the posterior

Generally, the first task we will perform after sampling from the posterior is check what the results look like. The plot_trace function from ArviZ is ideally suited to this task:

az.plot_trace(trace)
Figure 2.1

By using az.plot_trace, we get two subplots for each unobserved variable. The only unobserved variable in our model is . Notice that y is an observed variable representing the data; we do not need to sample that because we already know those values. Thus, in Figure 2.1, we have two subplots. On the left, we have a Kernel Density Estimation (KDE) plot; this is like the smooth version of the histogram. On the right, we get the individual sampled values at each step during the sampling. From the trace plot, we can visually get the plausible values from the posterior. You should compare this result using PyMC3 with those from the previous chapter...

Gaussians all the way down

We introduced the main Bayesian notions using the beta-binomial model mainly because of its simplicity. Another very simple model is the Gaussian or normal model. Gaussians are very appealing from a mathematical point of view because working with them is easy; for example, we know that the conjugate prior of the Gaussian mean is the Gaussian itself. Besides, there are many phenomena that can be nicely approximated using Gaussians; essentially, almost every time that we measure the average of something, using a big enough sample size, that average will be distributed as a Gaussian. The details of when this is true, when this is not true, and when this is more or less true, are elaborated in the central limit theorem (CLT); you may want to stop reading now and search about this really central statistical concept (very bad pun intended).

Well, we were saying...

Groups comparison

One pretty common statistical analysis is group comparison. We may be interested in how well patients respond to a certain drug, the reduction of car accidents by the introduction of a new traffic regulation, student performance under different teaching approaches, and so on.

Sometimes, this type of question is framed under the hypothesis testing scenario with the goal of declaring a result statistically significant. Relying only on statistical significance can be problematic for many reasons: on the one hand, statistical significance is not equivalent to practical significance; on the other hand, a really small effect can be declared significant just by collecting enough data. The idea of hypothesis testing is connected to the concept of p-values. This is not a fundamental connection but a cultural one; people are used to thinking that way mostly because that...

Hierarchical models

Suppose we want to analyze the quality of water in a city, so we take samples by dividing the city into neighborhoods. We may think we have two options to analyze this data:

  • Study each neighborhood as a separate entity
  • Pool all the data together and estimate the water quality of the city as a single big group

Both options could be reasonable, depending on what we want to know. We can justify the first option by saying we obtain a more detailed view of the problem, which otherwise could become invisible or less evident if we average the data. The second option can be justified by saying that if we pool the data, we obtain a bigger sample size and hence a more accurate estimation. Both are good reasons, but we can do something else, something in-between. We can build a model to estimate the water quality of each neighborhood and, at the same time, estimate...

Summary

Although Bayesian statistics is conceptually simple, fully probabilistic models often lead to analytically intractable expressions. For many years, this was a huge barrier, hindering the wide adoption of Bayesian methods. Fortunately, math, statistics, physics, and computer science came to the rescue in the form of numerical methods that are capable—at least in principle—of solving any inference problem. The possibility of automating the inference process has led to the development of probabilistic programming languages, allowing for a clear separation between model definition and inference.

PyMC3 is a Python library for probabilistic programming with a very simple, intuitive, and easy to read syntax that is also very close to the statistical syntax used to describe probabilistic models. We introduced the PyMC3 library by revisiting the coin-flip model from...

Exercises

  1. Using PyMC3, change the parameters of the prior beta distribution in our_first_model to match those of the previous chapter. Compare the results to the previous chapter. Replace the beta distribution with a uniform one in the interval [0,1]. Are the results equivalent to the ? Is the sampling slower, faster, or the same? What about using a larger interval such as [-1, 2]? Does the model run? What errors do you get?
  2. Read about the coal mining disaster model that is part of the PyMC3 documentation: http://pymc-devs.github.io/pymc3/notebooks/getting_started.html#Case-study-2:-Coal-mining-disasters. Try to implement and run this model by yourself.
  1. Modify model_g, change the prior for the mean to a Gaussian distribution centered at the empirical mean, and play with a couple of reasonable values for the standard deviation of this prior. How robust/sensitive are the inferences...
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Key benefits

  • A step-by-step guide to conduct Bayesian data analyses using PyMC3 and ArviZ
  • A modern, practical and computational approach to Bayesian statistical modeling
  • A tutorial for Bayesian analysis and best practices with the help of sample problems and practice exercises.

Description

The second edition of Bayesian Analysis with Python is an introduction to the main concepts of applied Bayesian inference and its practical implementation in Python using PyMC3, a state-of-the-art probabilistic programming library, and ArviZ, a new library for exploratory analysis of Bayesian models. The main concepts of Bayesian statistics are covered using a practical and computational approach. Synthetic and real data sets are used to introduce several types of models, such as generalized linear models for regression and classification, mixture models, hierarchical models, and Gaussian processes, among others. By the end of the book, you will have a working knowledge of probabilistic modeling and you will be able to design and implement Bayesian models for your own data science problems. After reading the book you will be better prepared to delve into more advanced material or specialized statistical modeling if you need to.

Who is this book for?

If you are a student, data scientist, researcher, or a developer looking to get started with Bayesian data analysis and probabilistic programming, this book is for you. The book is introductory so no previous statistical knowledge is required, although some experience in using Python and NumPy is expected.

What you will learn

  • Build probabilistic models using the Python library PyMC3
  • Analyze probabilistic models with the help of ArviZ
  • Acquire the skills required to sanity check models and modify them if necessary
  • Understand the advantages and caveats of hierarchical models
  • Find out how different models can be used to answer different data analysis questions
  • Compare models and choose between alternative ones
  • Discover how different models are unified from a probabilistic perspective
  • Think probabilistically and benefit from the flexibility of the Bayesian framework

Product Details

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Publication date : Dec 26, 2018
Length: 356 pages
Edition : 2nd
Language : English
ISBN-13 : 9781789341652
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Product Details

Publication date : Dec 26, 2018
Length: 356 pages
Edition : 2nd
Language : English
ISBN-13 : 9781789341652
Category :
Languages :
Concepts :

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Table of Contents

10 Chapters
Thinking Probabilistically Chevron down icon Chevron up icon
Programming Probabilistically Chevron down icon Chevron up icon
Modeling with Linear Regression Chevron down icon Chevron up icon
Generalizing Linear Models Chevron down icon Chevron up icon
Model Comparison Chevron down icon Chevron up icon
Mixture Models Chevron down icon Chevron up icon
Gaussian Processes Chevron down icon Chevron up icon
Inference Engines Chevron down icon Chevron up icon
Where To Go Next? Chevron down icon Chevron up icon
Other Books You May Enjoy Chevron down icon Chevron up icon

Customer reviews

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Rating distribution
Full star icon Full star icon Full star icon Half star icon Empty star icon 3.2
(17 Ratings)
5 star 41.2%
4 star 5.9%
3 star 5.9%
2 star 23.5%
1 star 23.5%
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N/A Nov 19, 2023
Full star icon Full star icon Full star icon Full star icon Full star icon 5
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F. Salvi May 26, 2019
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Super Book . Really well written and with interesting examples on the associated github website.
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Daniel C. Thomas Feb 07, 2021
Full star icon Full star icon Full star icon Full star icon Full star icon 5
I purchased my copy through packt directly, not through amazon.Wanted to write a positive review because I saw a lot of the negative reviews here that didn't seem fair. This book is really great and the author - and the pymc folks in general - are really helpful and responsive. The errors in typesetting and graph colors, etc, mentioned by others a bit overblown (at least, in my version), and they are almost certainly due to the publishing platform. Packt is really good for fast, hot-topic publication, but they are not famous for typo-free books, etc. I think many of the ones listed by other reviews have already been corrected, and the typos I found were minor. They did not subtract from my ability to learn from this book - I learned a lot, really quickly. If the typos really bother you, you can always run the code on the jupyter notebook, the author is still keeping these up-to-date.Every example I followed (most in the book) worked, especially when I checked my code against the notebooks. I did have to use an earlier (3.7) version of python, which has more to do with where pymc3 is right now - pymc3 didn't play well with my python 3.8 installation.I really wanted to get out of R for my introduction to bayesian methods, and keep things in the scipy ecosystem. This was the only book that was up-to-date and that I found approachable for this. I hope the typos in the book are corrected, but it doesn't really matter, I still find this the best quick-and-dirty intro. And I hope the author writes more books.
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F. Salvi May 26, 2019
Full star icon Full star icon Full star icon Full star icon Full star icon 5
The book is really well written. The examples are accurate and interesting. The author has immediately answered a couple of my question. I definitely recommend it to learn bayesian analysis wit python.
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J. Lowry Dec 20, 2021
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Excellent
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