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In our latest research, we introduce a novel post-processing method designed to enhance the accuracy of Bayesian evidence calculations derived from Nested Sampling (NS) data, as detailed in our paper 2408.09889. Bayesian evidence is a cornerstone of model comparison, yet its precise computation remains a significant challenge, particularly in high-dimensional problems common in astrophysics and cosmology.

Nested Sampling, a powerful algorithm first proposed by John Skilling (10.1214/06-BA127), elegantly transforms the complex, multi-dimensional evidence integral into a one-dimensional problem. It achieves this by integrating the likelihood function, $\mathcal{L}(X)$, over a sequence of “nested” prior volumes, $X$. However, the accuracy of this method is fundamentally limited by the stochastic estimation of these prior volumes. At each step, the volume shrinks by a random factor, introducing a statistical error that propagates through the calculation and often dominates the final uncertainty in the log-evidence, $\ln \mathcal{Z}$.

A Field-Theoretic Approach to Evidence Refinement

Our work, led by Margret Westerkamp in collaboration with Jakob Roth, Philipp Frank, Will Handley, and Torsten Enßlin, presents an orthogonal approach to tackling this limitation. Instead of modifying the sampling process itself, we reframe the estimation of the prior volumes as a Bayesian inference problem. The core idea is to treat the entire likelihood-prior-volume relation, $L(X)$, not as a series of discrete points with statistical uncertainty, but as a single, continuous field to be reconstructed.

To perform this reconstruction, we employ the formalism of Information Field Theory (IFT), a comprehensive framework for Bayesian field inference ([10.1002/andp.201800127]). The method uses the likelihood values generated by a standard NS run as its input data. Crucially, it incorporates a powerful prior assumption: that the underlying $L(X)$ function is inherently smooth. This allows the algorithm to regularize the stochastic fluctuations from the NS run and infer a more robust relationship between likelihood and prior volume. This is a powerful synergy, as this technique can be applied to the output of modern samplers like PolyChord ([10.1093/mnras/stv1911]), which was co-developed by author Will Handley.

Methodology and Validation

The implementation models the derivative of the re-parameterized log-likelihood with respect to the log-prior-volume, d f(ln L)/d ln X, as a log-normal process. This innovative step ensures the reconstructed function is both smooth and monotonically decreasing, consistent with the theoretical properties of the likelihood-prior-volume curve.

We validated our method on two synthetic test cases where the true evidence is known analytically:

  • A 10-dimensional Gaussian likelihood.
  • A more complex 10-dimensional “spike-and-slab” likelihood, which features an abrupt change in the likelihood gradient.

The results are highly promising. For NS runs with a relatively low number of live points (e.g., fewer than 100), our post-processing step provides a significant improvement in the accuracy and a reduction in the uncertainty of the calculated log-evidence. This is particularly valuable for computationally expensive problems where increasing the number of live points is infeasible. While the current implementation can face numerical challenges with the extremely fine sampling from high-n_live runs, it establishes a powerful proof-of-concept.

By treating the prior volume estimation as a field inference problem, this work opens up a new avenue for enhancing the precision of Bayesian model comparison and pushes the boundaries of what can be achieved with Nested Sampling.

Will Handley

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