How do you use MCMC sampling to perform Bayesian inference and hypothesis testing?

1 What is MCMC sampling?

MCMC sampling is a technique that allows you to approximate the posterior distribution of a parameter or a model by drawing random samples from it. The idea is to construct a Markov chain, a sequence of random variables that depend only on the previous one, that converges to the target distribution as the number of iterations increases. Each sample in the Markov chain is generated by applying a proposal function to the previous one and accepting or rejecting it based on a criterion that ensures the correct stationary distribution. There are different types of MCMC algorithms, such as Metropolis-Hastings, Gibbs sampling, and Hamiltonian Monte Carlo, that vary in their proposal functions and acceptance criteria.

2 Why use MCMC sampling?

MCMC sampling is useful when the posterior distribution is difficult or impossible to calculate analytically or numerically. For example, if the likelihood function is non-standard, the prior distribution is not conjugate, or the model has many parameters or hierarchical structure. MCMC sampling can handle these situations by generating samples that approximate the posterior distribution without requiring any explicit formula or integration. MCMC sampling can also be used to compare different models based on their marginal likelihoods or Bayes factors, which are also hard to compute directly.

3 How to use MCMC sampling?

To use MCMC sampling, you need to specify the model, the prior distribution, the likelihood function, and the data. Then, you need to choose an MCMC algorithm and its parameters, such as the proposal function, the acceptance criterion, the initial value, and the number of iterations. You also need to decide how to store and thin the samples, and how to check for convergence and mixing of the Markov chain. Finally, you need to analyze the samples and summarize the posterior distribution using descriptive statistics, plots, and credible intervals.

4 How to perform hypothesis testing?

Hypothesis testing is a way to evaluate the evidence for or against a claim about a parameter or a model. In Bayesian inference, hypothesis testing is done by comparing the posterior probabilities of different hypotheses given the data and the prior. For example, you can test whether a parameter is equal to, greater than, or less than a certain value, or whether a model is more likely than another one. To perform hypothesis testing using MCMC sampling, you need to calculate the posterior probabilities of the hypotheses based on the samples, and use a decision rule or a threshold to accept or reject them.

5 How to use statistical programming?

Statistical programming is the use of software tools and languages to perform statistical analysis and modeling. There are many options for statistical programming, but some of the most popular ones for Bayesian inference and MCMC sampling are R, Python, Stan, and JAGS. These tools provide functions and packages that allow you to define and fit Bayesian models, run MCMC algorithms, and visualize and interpret the results. For example, in R you can use the rstan package to interface with Stan, a probabilistic programming language that implements efficient MCMC algorithms. In Python, you can use the pymc3 package to build and sample from Bayesian models using a variety of MCMC methods.

6 How to learn more?

If you are interested in learning more about MCMC sampling and Bayesian inference, there are numerous resources available both online and offline. Books, courses, tutorials, blogs, podcasts, and videos can provide you with a comprehensive understanding of these topics. You can also join online communities and forums to ask questions, share your work, and receive feedback from other learners and experts. Examples of useful resources include Doing Bayesian Data Analysis by John Kruschke, Bayesian Data Analysis by Andrew Gelman et al., Statistical Rethinking by Richard McElreath, Bayesian Methods for Hackers by Cameron Davidson-Pilon, R for Data Science by Hadley Wickham and Garrett Grolemund, Python Data Science Handbook by Jake VanderPlas, Stan User's Guide, PyMC3 User's Guide, R-bloggers, PyData, The Bayesian Spectacles blog, The R Podcast, The Bayesian Conspiracy podcast, and The DataCamp platform.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?