MCMC Example

This is a simple example of Markov Chain Monte Carlo (MCMC) using Python and the PyMC3 library.

MCMC Overview

Markov Chain Monte Carlo (MCMC) is a family of algorithms used for sampling from complex probability distributions. It is particularly useful when direct sampling is impractical, and it provides a way to approximate the posterior distribution of model parameters given observed data. MCMC methods use Markov chains to explore the parameter space, and the samples generated converge to the desired distribution.

Key concepts of MCMC:

• Markov Chains: Sequences of random variables where each variable depends only on the previous one.
• Monte Carlo: Techniques for numerical integration using random sampling.
• Metropolis-Hastings Algorithm: A specific MCMC algorithm for sampling from a target distribution by iteratively proposing and accepting or rejecting new samples.
• Posterior Distribution: The distribution of model parameters given observed data.

MCMC is widely used in Bayesian statistics for inference and parameter estimation.

Python Source Code:

# Import necessary libraries
import pymc3 as pm
import arviz as az
import matplotlib.pyplot as plt
import numpy as np

# Generate synthetic data
np.random.seed(42)
observed_data = np.random.normal(loc=5, scale=2, size=100)

# Define Bayesian model using PyMC3
with pm.Model() as model:
    # Define priors
    mu = pm.Normal('mu', mu=0, sigma=10)
    sigma = pm.HalfNormal('sigma', sigma=10)
    
    # Define likelihood
    likelihood = pm.Normal('likelihood', mu=mu, sigma=sigma, observed=observed_data)
    
    # Use Metropolis-Hastings sampler for MCMC
    trace = pm.sample(2000, tune=1000, chains=2)

# Plot posterior distribution
az.plot_posterior(trace, var_names=['mu', 'sigma'])
plt.show()

Explanation:

• Import Libraries: Import necessary Python libraries, including PyMC3 for MCMC and ArviZ for visualization.
• Generate Synthetic Data: Generate synthetic data for demonstration purposes.
• Define Bayesian Model: Use PyMC3 to define a Bayesian model with priors for mean (mu) and standard deviation (sigma) and a normal likelihood distribution.
• Specify Priors and Likelihood: Define prior distributions for parameters and the likelihood of the observed data given the parameters.
• Run MCMC: Use the Metropolis-Hastings sampler to run MCMC and obtain samples from the posterior distribution.
• Plot Posterior Distribution: Visualize the posterior distribution of the model parameters using ArviZ.