Thursday, 4 March 2010

Beta-Binomial Bayesian Post continued…

This is a belated continuation of the post on the Beta-Binomial distribution.

Firstly, if we set a=b=1, then

clip_image002

This result is a constant regardless of the value held byclip_image004.

In this instance, clip_image004[1] is equally likely prior to observing ANY data where the prior information is a flat Beta (1,1) distribution. The interest in this comes from the fact that this is what Bayes' verbal exposition of his argument was based on. In the early 1700's, Laplace proved this result mathematically.

Basically, if we have absolutely no idea of the value of the success rate, clip_image006, we cannot have any idea regarding the respective values of y.

Other non-informative priors also exist.

For instance, if we take a=b=0.5, then:

clip_image008

This instance is NOT constant. As n=1,2,3,... increases we see the distribution become a U shaped plot.

The distribution moves more and more towards a Beta(0.5, 0.5) density, but between 0 and n. This implies an expectation of more data close to 0 or n and little (if any) in the centre of the density plot (that is between 0 and n but not 0 or n). We expect this as the Beta(0.5, 0.5) density produces a higher probability close to clip_image010and clip_image012.

More on this later.

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