GPU-Accelerated Nested Sampling

• GPU-accelerated nested sampling is a Bayesian inference technique restructured to exploit massive parallelism, transforming likelihood evaluations and posterior exploration.
• It uses batch replacement of live points, fixed-length MCMC chains, and vectorized proposal generation to overcome CPU bottlenecks in high-dimensional settings.
• Empirical benchmarks indicate up to 40× speedups with statistically validated results, making it essential for resource-intensive astrophysical and gravitational-wave analyses.

GPU-accelerated nested sampling refers to the application and adaptation of the nested sampling algorithm, a technique in Bayesian computation and statistical inference, onto graphics processing unit (GPU) architectures to achieve substantial speedups, scalability, and resource efficiency for high-dimensional and computationally demanding inference tasks in physics, astronomy, and related fields. The key methodological innovation is restructuring both the nested sampling workflow and proposal mechanisms to maximally exploit GPU parallelism while maintaining statistical fidelity of the posteriors and evidence estimates.

1. Foundations of Nested Sampling and Computational Bottlenecks

Nested sampling (NS), introduced for Bayesian evidence computation and posterior exploration, is fundamentally based on the progressive contraction of the prior volume subject to likelihood constraints, with the central equations: $$\sigma = \int_{\Omega} d\Phi\, \Vert\mathcal{M}(\Phi)\Vert^2$$

$$\sigma = \int_{[0,1]^D} d\theta\, \mathcal{L}(\theta)$$

The algorithm maintains a set of “live points” distributed according to the prior, iteratively replaces the lowest-likelihood point with a higher-likelihood proposal, and accumulates the evidence integral.

Traditional NS implementations are CPU-bound, suffering from bottlenecks in likelihood evaluation and constrained proposal generation, both of which scale unfavorably with dimension and sample size. The need to evaluate likelihoods for thousands or millions of candidate samples, and to generate new live points satisfying hard likelihood thresholds, is computationally expensive and often parallelizable only in a limited fashion. The scaling of cost with the Kullback–Leibler divergence between prior and posterior is a recognized limiting factor for standard NS approaches (Petrosyan et al., 2022).

2. Algorithmic Strategies for GPU Acceleration

Efficient GPU acceleration of nested sampling hinges on eliminating or refactoring steps in the traditional algorithm that induce thread divergence or serial dependencies. Notable strategies include:

• Batch Replacement of Live Points: Rather than sequentially updating one live point per NS iteration, a batch of points (typically $k \approx 0.5 \times n_\text{live}$) is replaced in parallel. Each point in the batch undergoes an independent Markov chain, resulting in a fixed-length chain per batch and eliminating thread divergence.
• Fixed MCMC Chain Lengths: Adaptivity in per-chain walk lengths would result in divergent threads; by fixing the MCMC length across the batch, all GPU threads execute coherent workloads (Prathaban et al., 4 Sep 2025).
• Vectorized Proposal Generation: Proposals, often following a Differential Evolution (DE) kernel, are generated by adding stochastic multiples of live point differences. The DE proposal is applied in parallel across the batch.
• Likelihood Evaluation Parallelization: When the likelihood is expensive (e.g., gravitational-wave waveform computations), the likelihood function is evaluated for all proposals simultaneously, employing GPU vectorization over both frequency bins and sample indices.

Summary pseudocode capturing GPU-accelerated NS is:

initialize n_live; set batch_size k = num_delete while not converged: identify k lowest-likelihood live points (L_min = batch max) # Parallel MCMC chains for each of k points on GPU: initialize MCMC at discarded point for fixed steps: propose new_point using DE kernel if likelihood(new_point) > L_min: accept new_point replace discarded points with accepted new proposals update batch acceptance rate; tune MCMC steps per batch

The GPU implementation requires an increased live point count to match prior-volume compression of sequential CPU NS: $$n_\text{GPU} \approx 2\ln(2)\,n_\text{CPU} \approx 1.4\,n_\text{CPU}$$ (Prathaban et al., 4 Sep 2025)

3. Performance and Statistical Validation

Empirical studies using the GPU-accelerated acceptance-walk method within the blackjax-ns framework, executing standard gravitational-wave analyses, report:

Test Case	CPU (16 Cores)	GPU (NVIDIA L4)	Speedup Factor	Cost Reduction
4-s BBH signal	38× slower	38× faster	38×	2.4×
8-s quadruple bins	20–40× slower	20–40× faster	20–40×	—
100-signal injection run	—	32× faster	32×	—

Statistical fidelity is demonstrated through:

• Statistical agreement of posterior distributions and log-evidence values with CPU bilby/dynesty reference implementations.
• PP-plot coverage analysis from injection studies, with coverage curves lying concordantly with the diagonal, indicating credible intervals are well-calibrated.
• Additional empirical validation via the Kolmogorov–Smirnov test (Prathaban et al., 4 Sep 2025).

4. Methodological Innovations in GPU-Friendly Nested Sampling

Achieving massive parallelism and reproducible results required algorithmic modifications:

• Acceptance–Walk Kernel: The standard NS proposal, a stochastic DE kernel with likelihood thresholding, is retained for statistical equivalence but adapted for parallel GPU computation. Each live point in the batch evolves independently, subject to the fixed likelihood constraint.
• Batch-Deletion and Live Point Cycling: GPU-based NS deletes a batch of live points, leading to a saw-tooth pattern in live point count. The derivation for the optimal number of live points to maintain volume contraction and convergence rates is provided in (Prathaban et al., 4 Sep 2025).
• Likelihood Vectorization: Core likelihood functions (e.g., IMRPhenomD waveforms in gravitational-wave inference) are vectorized both in sample batch and frequency bin dimensions, achieving maximal hardware utilization until resource saturation.

5. Impacts and Resource Scaling

The architectural shift to GPU delivers transformative runtime performance, resource cost reduction, and scalability to large datasets and complex models—while maintaining the statistical integrity required for precision astrophysical inference.

• Runtime and Cost: Benchmarking shows wall-time and cost reductions by factors of 20–40 for standard analyses, with performance gains scaling with problem complexity up to the limit of GPU memory and bandwidth.
• Resource Saturation: For very high-resolution signals or numerous frequency bins, GPU resource saturation may occur, prompting adoption of compressed-likelihood or heterodyned techniques.
• Enabling Algorithmic Benchmarks: A GPU-native NS implementation establishes a reference against which further algorithmic innovations—e.g., Hit-and-Run Slice Sampling—can be objectively compared, separating hardware speedup from methodological advances (Prathaban et al., 4 Sep 2025).

6. Future Directions and Synergies

The modular, vectorized GPU architecture is compatible with advanced proposal-generation strategies, including those utilizing normalizing flows, posterior repartitioning, and generative flow networks. NS methods leveraging gradient-guided proposals (Lemos et al., 2023) and posterior repartitioning (Petrosyan et al., 2022) can further augment GPU efficiency, accelerating inference convergence by reducing the effective KL divergence and amortizing sampling across parallel hardware resources.

A plausible implication is the integration of ML-augmented proposals or flow-model bootstrapping for further runtime reduction and mode discovery in multimodal and high-dimensional settings. As data volumes and model complexities grow in next-generation observatories, GPU-accelerated nested sampling is poised to underpin computationally feasible Bayesian inference pipelines.

7. Objective Summary

GPU-accelerated nested sampling, as implemented in the blackjax-ns framework using the acceptance-walk kernel, achieves reproducible and statistically validated posteriors and evidence estimates while delivering an order-of-magnitude improvement in both runtime and computational cost compared to legacy CPU implementations. By restructuring the sampling and proposal mechanisms for maximal GPU parallelism and optimizing resource utilization, this approach provides a scalable, community-standard foundation for large-scale Bayesian inference tasks, particularly in gravitational-wave data analysis. The resulting reference implementation enables rigorous benchmarking and comparative evaluation of further algorithmic advances in high-performance Bayesian computation (Prathaban et al., 4 Sep 2025).