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Understanding Computational Bayesian Statistics (Wiley Series in Computational Statistics)

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In theory, Bayesian statistics is very simple. The posterior is proportional to the prior times likelihood. This gives the shape of the posterior, but it is not a density so it cannot be used for inference. The exact scale factor needed to make this a density can be found only in a few special cases. For other cases, the scale factor requires a numerical integration, which may be difficult when there are multiple parameters. So in practice, Bayesian statistics is more difficult, and this has held back its use for applied statistical problems.

Computational Bayesian statistics has changed all this. It is based on the big idea that statistical inferences can be based on a random sample drawn from the posterior. The algorithms that are used allow us to draw samples from the exact posterior even when we only know its shape and we do not know the scale factor needed to make it an exact density. These algorithms include direct methods where a random sample drawn from an easily sampled distribution is reshaped by only accepting some candidate values into the final sample. More sophisticated algorithms are based on setting up a Markov chain that has the posterior as its long-run distribution. When the chain is allowed to run a sufficiently long time, a draw from the chain can be considered a random draw from the target (posterior) distribution. These algorithms are particularly well suited for complicated models with many parameters. This is revolutionizing applied statistics. Now applied statistics based on these computational Bayesian methods can be easily accomplished in practice.
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