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Bayesian Analysis with Python: Introduction to Statistical Modeling

In recent years, Bayesian analysis has become an increasingly popular approach to statistical modeling, offering a powerful framework for making inferences and predictions from data. With the rise of Python as a leading programming language for data science, it’s now easier than ever to apply Bayesian methods to real-world problems. In this article, we’ll introduce the basics of Bayesian analysis and explore how to implement it using Python.

What is Bayesian Analysis?

Bayesian analysis is a statistical approach that uses Bayes’ theorem to update the probability of a hypothesis based on new data. The theorem, named after Thomas Bayes, describes how to update the probability of a hypothesis (H) given new evidence (E) using the following formula:

P(H|E) = P(E|H) * P(H) / P(E)

Where:

  • P(H|E) is the posterior probability of the hypothesis given the evidence
  • P(E|H) is the likelihood of the evidence given the hypothesis
  • P(H) is the prior probability of the hypothesis
  • P(E) is the probability of the evidence

Key Concepts in Bayesian Analysis

Before diving into the Python implementation, let’s cover some essential concepts in Bayesian analysis:

  1. Prior Distribution: The prior distribution represents our initial beliefs about the parameters of a model before observing any data.
  2. Likelihood Function: The likelihood function describes the probability of observing the data given the model parameters.
  3. Posterior Distribution: The posterior distribution represents the updated probability of the model parameters given the observed data.
  4. Markov Chain Monte Carlo (MCMC): MCMC is a computational method used to sample from the posterior distribution.

Python Libraries for Bayesian Analysis

Several Python libraries are available for Bayesian analysis, including:

  1. PyMC3: A popular library for Bayesian modeling and computation.
  2. Scipy: A scientific computing library that includes tools for Bayesian analysis.
  3. Statsmodels: A library for statistical modeling that includes Bayesian methods.

Example: Bayesian Linear Regression with PyMC3

Let’s consider a simple example of Bayesian linear regression using PyMC3. We’ll generate some sample data and fit a linear model using Bayesian methods.
python
import pymc3 as pm
import numpy as np

Generate sample data

np.random.seed(0)
X = np.random.normal(0, 1, 100)
y = 2 + 3 * X + np.random.normal(0, 1, 100)

Define the model

with pm.Model() as model:

Prior distributions

beta0 = pm.Normal('beta0', mu=0, sigma=10)
beta1 = pm.Normal('beta1', mu=0, sigma=10)
sigma = pm.HalfNormal('sigma', sigma=1)

# Likelihood function
mu = beta0 + beta1 * X
y_obs = pm.Normal('y_obs', mu=mu, sigma=sigma, observed=y)

# Sample from the posterior distribution
trace = pm.sample(1000, tune=500)

Print the posterior means

print(pm.summary(trace))

In this example, we define a linear regression model with two parameters (beta0 and beta1) and a noise term (sigma). We then sample from the posterior distribution using MCMC and print the posterior means.

Conclusion

Bayesian analysis offers a powerful framework for statistical modeling, and Python provides an ideal platform for implementation. With libraries like PyMC3, Scipy, and Statsmodels, you can easily apply Bayesian methods to a wide range of problems. In this article, we’ve introduced the basics of Bayesian analysis and provided a simple example using PyMC3. Whether you’re a seasoned statistician or just starting out, Bayesian analysis with Python is an exciting area to explore.

13 reviews for Bayesian Analysis with Python: Introduction to statistical m…

  1. J. Lowry –

    Excellent in conjunction with McElreath book
    Excellent

  2. Daniel C. Thomas –

    Really approachable hands-on intro to real-world python bayes, don’t worry about typos
    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.

  3. Stedeej –

    Great idea- killed by typos and poor publishing standards.
    This was a genuinely good idea for a book. Provide a good basic introduction that is actually applied. I’ve read most textbooks on Bayesian statistics that were designed for introductory level, as I teach statistics at the college level. Most of these books are good at providing theory, but students come away from them without feeling like they know how to run ANY actual analyses, even one as simple as a two group comparison or correlation, using Bayesian approaches. PYCM3 is a good program to use as well, and Python is emerging as the most important platform in data science. So great idea for a book! Unfortunately the application and publishing standards pretty close to kill the book here.Initially, I thought to myself “I’ll just make a note of all the typos and send them to the author to help out.” After dozens of typos in the first hundred pages, I just quit. Really typos in-text aren’t such a big deal, but when your book’s primary selling point is providing usable code for applied problems for emerging/learning practitioners you CANT have typos in the code!Also, I love the idea to use actual datasets in examples, as they do a few times in the text. However if you’re going to do so, perhaps you could actually provide the dataset so people can follow along. Without this, providing the code is kind of useless.Also, maybe when your book references the colors in figures you could actually provide color figures. Yes, perhaps this costs a bit more (though I’ve bought cheaper books that have color figures), but at least they provide it online, right? Good luck with that- I had to send terse emails to the publisher just to finally get the code without paying to subscribe to their website program. Maybe they are somewhere on the website- I stopped caring enough to continue to dig this stuff up.Further, the code that the publisher provides is from 2016! Remember Python in 2016!? Probably not. No pandas here, etc. and none of the code is updated since then (note that arviz doesn’t even use the plot_hpd anymore, you need to use plot_hdi, etc.). I can forgive missing the more recent program updates, but how hard is it to provide the code used in the actual most-recent version of the book? Of course, the code is laced with typos, formatting problems, etc. anyway. In a programming language in which spacing is paramount, how does a guidebook mess up tabs and formatting? Ever think to yourself “I wish I could just throw 2-3 hours in the trash trying to correct the code from a stats guidebook”-? Probably not.Great idea- like the author’s general approach here- but it’s mostly killed by poor publishing standards.

  4. Dario Romero –

    Content quality is excellent, but Book quality is bungling task.
    I respect the work of Osvaldo Marin on this book. I know he is an authority in the field, but the work of the editor (Packt Publishing) is definitively a bungling job. The images in the book are black & white and they refer you to a website where they have all images in color. Why they charged people with a book in pure black & white when other Publishing companies does the same and at the same price in full color. Really disappointed with Packt Publishing. To me this is dishonest marketing, therefore scam.

  5. Diz –

    Bayesian Methods explained practically with great examples
    I found this book to take statistical concepts that are typically shrouded with mathematical notional, and make them relatable to every people. For example Osvaldo explains out p( Pope | Argentinian) is 1 in ~44 million (The population of Argentina) but P(Argentinian | Pope) =1 because if you ask the pope what nationality he is, its obviously Argentinian!This example was much more memorable for me than trying memorizing P(a|b) is not always equal to P(b|a)The whole book contains similar examples, and hands on exercises. Definitely 5 out of 5 book.

  6. Thomas Wiecki –

    Great introduction with many practical examples
    This book is the ideal introduction to probabilistic programming if you are application focused. It’s not focused on theory but rather on solving applied problems. The many code examples and visualizations have you running you building your own models quickly.

  7. F. Salvi –

    A very good book.
    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.

  8. Gloria Fullerton –

    Very well written; very poorly formatted
    This is an excellent presentation. Concepts are well presented; examples are useful; and the code and libraries are good to get a jump-start. BUT THE KINDLE EDITION IS UNACCEPTABLE. Inline formulas are so tiny as to be illegible.Amazon tech support suggested I take a refund and purchase/download again to see if that fixed the issue. It. did. not.

  9. JimJohnny –

    This book is intended as an accessible introduction to Bayesian analysis using PyMC3. Although it emphasises practical aspects, it aims to provide sufficient treatment of the theoretical to serve as a useful starting point for Bayesian analysis in general. The author clearly makes an attempt to be entertaining and conversational in his writing style. So far, so good.That said, I was surprised to find the text littered with typographical errors. In Chapter 1, there is also the issue that plots are referred to by colour, when the images are in fact printed in grey scale.Perhaps more concerning are mathematical errors, for example in the formula given for Cohen’s d, in Chapter 2. One wonders what the purpose is of having a reviewer, if errors such as this one preside in the second edition of this book.With such errors present in the text, I wouldn’t recommend this book so much as an introduction to Bayesian analysis as such, but instead as an introduction to using PyMC3 for readers already familiar with the theoretical subject material. I very much hope that the errors will be resolved in a subsequent edition of the book.

  10. F. Salvi –

    Super Book . Really well written and with interesting examples on the associated github website.

  11. Lorenzo –

    The book seems good, but the formulas in the Kindle-Version are extremely small and very hardly readable.This must have been a problem in the generation of the eBook and needs to be fixed by the editor, pushing an update.

  12. Joe –

    If you’re buying this book then you probably already know that the writer is very capable and covers the topics very well.However, the mathematics and graphics in this book (second edition) are terrible in quality. Low DPI black and white makes it unpleasant to look at at the best of times and, in the case of complicated trace plots which rely on colour (eg page 233) or botched formulas (eg page 91), can make things unreadable. For the price, I would have expected better quality; if you’re expecting quality like an O’Reilly publication you will be disappointed.

  13. PJ –

    Bought this book to prepare some materials for teaching on using PyMC3. The book is useful and covers most of the practical aspects. However, the typsetting is a big let-down. For a book with serious mathematical nature, the quality is in fact surprisingly low. Wish the graphics were in colour to have improved visualisation, but understandably, will shoot the price of the book up.

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