  We introduce a novel technique within the Nested Sampling framework to
enhance efficiency of the computation of Bayesian evidence, a critical
component in scientific data analysis. In higher dimensions, Nested Sampling
relies on Markov Chain-based likelihood-constrained prior samplers, which
generate numerous 'phantom points' during parameter space exploration. These
points are too auto-correlated to be used in the standard Nested Sampling
scheme and so are conventionally discarded, leading to waste. Our approach
discovers a way to integrate these phantom points into the evidence
calculation, thereby improving the efficiency of Nested Sampling without
sacrificing accuracy. This is achieved by ensuring the points within the live
set remain asymptotically i.i.d. uniformly distributed, allowing these points
to contribute meaningfully to the final evidence estimation. We apply our
method on several models, demonstrating substantial enhancements in sampling
efficiency, that scales well in high-dimension. Our findings suggest that this
approach can reduce the number of required likelihood evaluations by at least a
factor of 5. This advancement holds considerable promise for improving the
robustness and speed of statistical analyses over a wide range of fields, from
astrophysics and cosmology to climate modelling.
