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2 Applying Bayes’ Rule for Bayesian Inference As we stated at the start of this chapter the basic idea of Bayesian inference is to continually update our prior beliefs about events as new evidence is presented. This is a very natural way to think about probabilistic events. As more and more evidence is accumulated our prior beliefs are steadily "washed out" by any new data. Consider a (rather nonsensical) prior belief that the Moon is going to collide with the Earth. For every night that passes, the application of Bayesian inference will tend to correct our prior belief to a posterior belief that the Moon is less and less likely to collide with the Earth, since it remains in orbit.

We can use the helpful traceplot method to plot both a kernel density estimate (KDE) of the histogram displayed above, as well as the trace itself. The trace plot is extremely useful for assessing convergence of an MCMC algorithm and whether we need to exclude a period of initial samples (known as the burn in). 2. In addition we can see that the MCMC sampling procedure has "converged to the distribution" since the sampling series looks stationary. 6 Bibliographic Note The algorithm described in this chapter is due to Metropolis[69].

What do we get out of this reformulation? There are two main reasons for doing so[99]: • Prior Distributions: If we have any prior knowledge about the parameters β then we can choose prior distributions that reflect this. If we do not then we can still choose 49 non-informative priors. • Posterior Distributions: I mentioned above that the frequentist MLE value for our ˆ was only a single point estimate. In the Bayesian formulation we regression coefficients, β, receive an entire probability distribution that characterises our uncertainty on the different β coefficients.

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