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By Michael Halls Moore

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Note however that a prior is only conjugate with respect to a particular likelihood function. 2 Why Is A Beta Prior Conjugate to the Bernoulli Likelihood? We can actually use a simple calculation to prove why the choice of the beta distribution for the prior, with a Bernoulli likelihood, gives a beta distribution for the posterior. 12) You can see that the form of the beta distribution is similar to the form of a Bernoulli likelihood. 13) Notice that the term on the right hand side of the proportionality sign has the same form as our prior (up to a normalising constant).

In the first sub-plot we have carried out no trials and hence our probability density function (in this case our prior density) is the uniform distribution. It states that we have equal belief in all values of θ representing the fairness of the coin. The next panel shows 2 trials carried out and they both come up heads. Our Bayesian procedure using the conjugate Beta distributions now allows us to update to a posterior density. Notice how the weight of the density is now shifted to the right hand side of the chart.

At this stage, it just allows us to easily create some visualisations below that emphasise the Bayesian procedure! In the following figure we can see 6 particular points at which we have carried out a number of Bernoulli trials (coin flips). In the first sub-plot we have carried out no trials and hence our probability density function (in this case our prior density) is the uniform distribution. It states that we have equal belief in all values of θ representing the fairness of the coin. The next panel shows 2 trials carried out and they both come up heads.

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