Nested Sampling (NS) is a numerical algorithm for fitting models to data in the Bayesian setting, put forward by John Skilling in 2004. It has some advantages over Markov chain Monte Carlo algorithms; no starting point issues, no burn-in, no proposal distributions.
Nested Sampling calculates the Evidence Pr[data|I] directly; posterior samples are in some sense a by-product.
The "central problem" is the drawing of a likelihood-restricted prior sample at each compression step.
Consideration of new such sampling methods has led to some work on equidistribution testing.