Bayesian Statistical Modelling using Python

Python is a versatile and powerful programming language that has gained significant popularity in the field of data analysis and statistics. It offers a variety of libraries that can be used to perform Bayesian statistical modelling. This article will provide an overview of Python's capabilities in this area and introduce some of the most commonly used libraries for Bayesian analysis.

Introduction to Python

Python is an open-source, high-level programming language known for its simplicity and readability. It has a wide range of applications, from web development to machine learning, and is particularly popular in the field of data analysis due to its powerful libraries and tools.

Python's syntax is designed to be easy to understand and write, making it an excellent choice for beginners. However, it's also powerful enough to handle complex statistical analyses, making it a popular choice among professionals in the field.

Bayesian Libraries in Python

Python offers several libraries that are specifically designed for Bayesian analysis. Two of the most commonly used are PyMC3 and pystan.

PyMC3

PyMC3 is a Python library for probabilistic programming which allows you to write down models using an intuitive syntax to describe a data generating process.

Key features of PyMC3 include:

• Intuitive model specification syntax, for example, the "with model:" syntax.
• Powerful sampling algorithms, such as Hamiltonian Monte Carlo.
• Variety of built-in distributions, from simple ones like Uniform and Normal to more complex ones like the Exponential and Half-Cauchy distributions.

pystan

Stan is a state-of-the-art platform for statistical modeling and high-performance statistical computation. PyStan is the Python interface for Stan.

Key features of pystan include:

• Full Bayesian inference using the No-U-Turn Sampler (NUTS), a variant of Hamiltonian Monte Carlo.
• Variational inference: algorithms for approximate Bayesian inference.
• Optimization: penalized maximum likelihood estimation (MLE) and penalized maximum a posteriori estimation (MAP).

Hands-on Exercise

To get a feel for Bayesian analysis in Python, let's consider a simple example. Suppose we have a coin and we want to determine the probability that it lands heads when tossed. We can use PyMC3 to perform a Bayesian analysis of this problem.

import pymc3 as pm

# Number of coin flips and number of heads
n = 100
heads = 61

# Define the model
with pm.Model() as coin_flip_model:
    # Prior
    p = pm.Beta('p', alpha=2, beta=2)
    
    # Likelihood
    y = pm.Binomial('y', n=n, p=p, observed=heads)

# Perform MCMC
with coin_flip_model:
    trace = pm.sample(2000, tune=1000)

# Print the posterior mean
print("Posterior Mean: ", trace['p'].mean())

In this example, we define a prior distribution for the probability of heads as a Beta distribution. We then define the likelihood as a Binomial distribution with the number of trials equal to the number of coin flips and the number of successes equal to the number of heads. We then use PyMC3's sample function to perform Markov chain Monte Carlo (MCMC) and generate samples from the posterior distribution.

By the end of this unit, you should have a basic understanding of how to perform Bayesian statistical modelling in Python using libraries like PyMC3 and pystan.