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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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