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Our latest paper, “Improving Gradient-guided Nested Sampling for Posterior Inference,” led by Pablo Lemos in collaboration with Nikolay Malkin, Will Handley, Yoshua Bengio, Yashar Hezaveh, and Laurence Perreault-Levasseur, introduces a powerful new algorithm designed to tackle challenging, high-dimensional Bayesian inference problems. This work stands at the intersection of fundamental statistical methods and modern machine learning, offering a significant leap forward in computational efficiency and scalability.

The Challenge of High-Dimensional Inference

Bayesian inference is a cornerstone of modern science, providing a principled framework for parameter estimation and model comparison. A key algorithm in this domain is nested sampling, introduced by John Skilling (10.1214/06-BA127), which excels at exploring complex, multimodal posterior distributions and calculating the Bayesian evidence (or marginal likelihood). This has made it an indispensable tool in fields from cosmology to particle physics.

However, the core challenge of nested sampling is to efficiently generate new sample points from the prior distribution subject to a hard likelihood constraint. Traditional methods often struggle as the number of dimensions increases. “Region samplers” like MultiNest (10.1111/j.1365-2966.2007.12353.x) scale exponentially, while “step samplers” typically exhibit quadratic scaling. This “curse of dimensionality” has historically limited the application of nested sampling to problems with hundreds, rather than thousands, of dimensions.

GGNS: A Gradient-Guided Approach

Our new algorithm, Gradient-guided Nested Sampling (GGNS), directly addresses this scaling problem by harnessing the power of differentiable programming and gradient information. GGNS combines several state-of-the-art techniques into a single, performant framework:

  • Hamiltonian Slice Sampling (HSS): At its core, GGNS uses HSS to generate new points. By simulating a particle’s trajectory that reflects off the hard likelihood boundary, we can efficiently explore the constrained parameter space. Crucially, we use the likelihood gradient—readily available through modern frameworks like PyTorch—to guide these reflections.
  • Adaptive Control & Parallelization: We introduce an adaptive time-step mechanism for the HSS integrator and a novel trajectory preservation scheme. This removes the need for tedious hyperparameter tuning that plagued previous HSS implementations. Combined with dynamic nested sampling principles that allow for massive parallelization on GPUs, the algorithm’s execution is dramatically accelerated.
  • Robustness and Mode Handling: To prevent the sampler from collapsing into a single mode in multimodal distributions, GGNS incorporates the sophisticated clustering algorithms developed for PolyChord. We also introduce a more robust termination criterion based on the trajectory of the posterior mass, enhancing the reliability of the evidence calculation.

The result of this unique combination is an algorithm that breaks the scaling bottleneck. Our experiments show that GGNS achieves near-linear scaling of likelihood evaluations with dimensionality, outperforming existing methods and making high-dimensional inference significantly more tractable.

Synergy with Generative Flow Networks

Beyond its standalone performance, we demonstrate the powerful potential of combining GGNS with Generative Flow Networks (GFlowNets), a novel class of generative models. We show that GGNS can be used to produce a high-quality dataset of posterior samples that effectively “teaches” a GFlowNet to navigate a complex, multimodal landscape. This synergy allows for faster mode discovery during training. In turn, the trained GFlowNet can generate vast quantities of posterior samples, effectively amortizing the inference cost for future use—a promising direction for combining traditional sampling with modern deep learning. GGNS can also be used for deep learning applications, such as in cases where the prior or likelihood are defined by neural networks, for example to accelerate Bayesian inference with normalizing flows (10.1093/mnras/staa1469).

By significantly improving scalability and integrating with cutting-edge machine learning techniques, GGNS provides a robust, general-purpose tool poised to expand the frontiers of scientific discovery.

Will Handley

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