By Michael Halls Moore

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4) To make a subsequent prediction yN +1 , given some new data xN +1 , we simply multiply the components of xN +1 by the associated β coefficients and obtain yN +1 . The important point here is that βˆ is a point estimate, meaning that it is a single value in real-valued p + 1-dimensional space–Rp+1 . In the Bayesian formulation we will see that the interpretation differs substantially. 2 Bayesian Linear Regression In a Bayesian framework linear regression is stated in a probabilistic manner. The above linear regression model is reformulated in probabilistic language.

In this chapter we will use PyMC3 to carry out a simple example of inferring a binomial proportion. This is sufficient to express the main ideas of MCMC without getting bogged down in implementation specifics. In later chapters we will explore more features of PyMC3 by carrying out inference on more sophisticated models. 5 Inferring a Binomial Proportion with Markov Chain Monte Carlo If you recall from the previous chapter on inferring a binomial proportion using conjugate priors our goal was to estimate the fairness of a coin, by carrying out a sequence of coin flips.

289, since this is the standard deviation of a uniform density (which itself implies no prior belief on any particular fairness of the coin). Let’s carry out an example now. 5, but I’m not particularly certain (hence I have a wider standard deviation). 1. What beta distribution is produced as a result? 2. 2: A beta distribution with α = 12 and β = 12. 5 but that there is significant uncertainty in this belief, represented by the width of the curve. 6 Using Bayes’ Rule to Calculate a Posterior We are finally in a position to be able to calculate our posterior beliefs using Bayes’ rule.