By Michael Halls Moore

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We couple our prior beliefs with the data we have observed and update our beliefs accordingly. Luckily for us, if we use a beta distribution as our prior and a Bernoulli likelihood we also get a beta distribution as a posterior. These are known as conjugate priors. 5. Inference - Once we have a posterior belief we can estimate the coin’s fairness θ, predict the probability of heads on the next flip or even see how the results depend upon different choices of prior beliefs. The latter is known as model comparison.

Thomas Wiecki has also written a great blog post[102] explaining the rationale for MCMC. The PyMC3 project[10] also has some extremely useful documentation and some examples. 46 Chapter 5 Bayesian Linear Regression At this stage in our journey of Bayesian statistics we inferred a binomial proportion analytically with conjugate priors and have described the basics of Markov Chain Monte Carlo via the Metropolis algorithm. In this chapter we are going to introduce linear regression modelling in the Bayesian framework and carry out inference using the PyMC3 MCMC library.

Show() 22 Chapter 3 Bayesian Inference of a Binomial Proportion In the previous chapter we examined Bayes’ rule and considered how it allowed us to rationally update beliefs about uncertainty as new evidence came to light. We mentioned briefly that such techniques are becoming extremely important in the fields of data science and quantitative finance. In this chapter we are going to expand on the coin-flip example that we studied in the previous chapter by discussing the notion of Bernoulli trials, the beta distribution and conjugate priors.