By Michael Halls Moore

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To do this we need to understand the range of values that θ can take and how likely we think each of those values are to occur. θ = 0 indicates a coin that always comes up tails, while θ = 1 implies a coin that always comes up heads. 5. Hence θ ∈ [0, 1]. This implies that our probability distribution must also exist on the interval [0, 1]. The task then becomes determining which probability distribution we utilise to quantify our beliefs about the coin. 1 Beta Distribution In this instance we are going to choose the beta distribution.

It uses a model specification syntax that is similar to how the R statistical language specifies models. To achieve this we make implicit use of the Patsy library. In the following snippet we are going to import PyMC3, utilise the with context manager, as described in the previous chapter on MCMC and then specify the model using the glm module. We are then going to find the maximum a posteriori (MAP) estimate for the MCMC sampler to begin sampling from. 1: Simulation of noisy linear data via Numpy, pandas and seaborn Calculates the Markov Chain Monte Carlo trace of a Generalised Linear Model Bayesian linear regression model on supplied data.

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. 18) This says that the posterior belief in the fairness θ, given z heads in N flips, is equal to the likelihood of seeing z heads in N flips, given a fairness θ, multiplied by our prior belief in θ, normalised by the evidence.