Wednesday, 17 February 2010

Bayesian Statistics

I am going to post a few details concerning a number of distributions for those wanting to learn more on this topic. I will start with a little on the Beta-Binomial probability density function and as a Bayesian, I shall start with a small discourse on predictive distributions. These are used to account for parameter uncertainty with respect to:
· Estimation
· Inference
· Comparing models
In general, the prior predictive distributions can be stated as:
clip_image002
Starting with the Y~Binomial clip_image004distribution, the distribution of the data as it would be modelled before we have made any observations can be expressed as:
clip_image006
Starting with the selection of a conjugate Beta(a,b) prior, then;
clip_image008
To prove the prior result, we need to use the following:
clip_image010
As (by definition), clip_image012
clip_image014
We can calculate the prior predictive density. This comes out as:
clip_image016
The above distribution is called the Beta-Binomial. I will detail some aspects of this distribution tomorrow.

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