Automatic Laplace Collapsed Sampling: Scalable Marginalisation of Latent Parameters via Automatic Differentiation
This provides a scalable method for Bayesian inference in complex models, though it appears incremental as it builds on existing techniques like nested sampling and Laplace approximations.
The authors tackled the problem of marginalizing latent parameters in Bayesian models by introducing Automatic Laplace Collapsed Sampling (ALCS), which reduces dimensionality from d_θ + d_z to d_θ using automatic differentiation and Laplace approximations, making evidence computation tractable for high-dimensional settings with GPU-accelerated performance.
We present Automatic Laplace Collapsed Sampling (ALCS), a general framework for marginalising latent parameters in Bayesian models using automatic differentiation, which we combine with nested sampling to explore the hyperparameter space in a robust and efficient manner. At each nested sampling likelihood evaluation, ALCS collapses the high-dimensional latent variables $z$ to a scalar contribution via maximum a posteriori (MAP) optimisation and a Laplace approximation, both computed using autodiff. This reduces the effective dimension from $d_θ+ d_z$ to just $d_θ$, making Bayesian evidence computation tractable for high-dimensional settings without hand-derived gradients or Hessians, and with minimal model-specific engineering. The MAP optimisation and Hessian evaluation are parallelised across live points on GPU-hardware, making the method practical at scale. We also show that automatic differentiation enables local approximations beyond Laplace to parametric families such as the Student-$t$, which improves evidence estimates for heavy-tailed latents. We validate ALCS on a suite of benchmarks spanning hierarchical, time-series, and discrete-likelihood models and establish where the Gaussian approximation holds. This enables a post-hoc ESS diagnostic that localises failures across hyperparameter space without expensive joint sampling.