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Concept# Nested sampling algorithm

Summary

The nested sampling algorithm is a computational approach to the Bayesian statistics problems of comparing models and generating samples from posterior distributions. It was developed in 2004 by physicist John Skilling.
Background
Bayes' theorem can be applied to a pair of competing models M_1 and M_2 for data D, one of which may be true (though which one is unknown) but which both cannot be true simultaneously. The posterior probability for M_1 may be calculated as:
:
\begin{align}
P(M_1\mid D) & = \frac{P(D\mid M_1) P(M_1)}{P(D)} \
& = \frac{P(D\mid M_1) P(M_1)}{P(D\mid M_1) P(M_1) + P(D\mid M_2) P(M_2)} \
& = \frac{1}{1 + \frac{P(D\mid M_2)}{P(D\mid M_1)} \frac{P(M_2)}{P(M_1)} }
\end{align}
The prior probabilities M_1 and M_2 are already known, as they are chosen by the researcher ahead of time. However, the remaining Bayes factor P(D\mid M_2)/P(D\mid M_1) i

Official source

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