QES — Quenched Ensemble Sampling

Evidence estimation by sequential Monte Carlo over a family of soft level ensembles

$$\rho_E(x) \propto \pi(x) (E - U(x))_+^{\nu}, \qquad U = -\log \mathcal{L},$$

with $E$ stepped downward. The level volume $G_\nu(E) = \mathrm{E}_\pi[(E-U)_+^\nu]$ is measured along the ladder by telescoped weight averages, and

$$Z = \frac{1}{\Gamma(\nu+1)} \int G_\nu(E) e^{-E} \mathrm{d}E .$$

$\nu$ interpolates the two incumbents: $\nu \to 0$ is nested sampling's hard constraint, whose score vanishes identically; $\nu \to \infty$ is tempering, one effective temperature instead of a band. The interior is usable by a gradient kernel and has no band of energies to skip.

The two paths on an asymmetric double well, both starting from the prior — QES quenching a level to depth beside tempered SMC annealing $\beta$, with each method's density and score:

Install and run

uv sync --extra dev

uv run python examples/spike_slab.py --D 10 --seeds 3

uv run pytest -q

Layout

qes/level.py     the family, the weights, the ESS dissection
qes/kernel.py    the inner MCMC kernels and the score metric
qes/adapt.py     step and metric adaptation
qes/qes.py       anchor, level loop, evidence estimator
qes/tempered.py  tempered SMC, matched in everything but the path
examples/        spike-slab, Gaussian, double well

Built on blackjax: the inner kernel chosen as MALA (mcmc.mala), and resampling, ESS, and the schedule bisection come from the SMC machinery. The kernel is preconditioned by a diagonal metric read from the previous level's particle scores, and a single scalar step size is held near the optimal acceptance rate by a constant-gain controller.

Citation

@misc{yallup2026quenchedensemblesampling,
      title={Quenched Ensemble Sampling}, 
      author={David Yallup},
      year={2026},
      eprint={2609.15894},
      archivePrefix={arXiv},
      primaryClass={stat.ML},
      url={https://arxiv.org/abs/2609.15894}, 
}

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© 2026 David Yallup. Released under the Apache 2.0 license.

This work was supported by a Google Research Grant