# Particle Metropolis-Hastings using gradient and Hessian information

arXiv: https://arxiv.org/abs/1311.0686 · HTML: https://mlsanity.com/paper/1311.0686/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2013.

## Built on by (most cited first, 4 of 4)

- [Efficient Learning of the Parameters of Non-Linear Models using Differentiable Resampling in Particle Filters](https://mlsanity.com/paper/2111.01409/) (2021)
- [Enhanced SMC2: Leveraging Gradient Information from Differentiable Particle Filters Within Langevin Proposals](https://mlsanity.com/paper/2407.17296/) (2024)
- [Hess-MC2: Sequential Monte Carlo Squared using Hessian Information and Second Order Proposals](https://mlsanity.com/paper/2507.07461/) (2025)
- [Dynestyx: A Probabilistic Programming Library for Dynamical Systems](https://mlsanity.com/paper/2606.16985/) (2026)

## Abstract (arXiv)

Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in nonlinear state space models by combining Markov chain Monte Carlo (MCMC) and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles. We propose a number of alternative versions of PMH that incorporate gradient and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood estimation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.