# Efficient inference for stochastic differential equation mixed-effects models using correlated particle pseudo-marginal algorithms

## TL;DR

This paper introduces an efficient Bayesian inference method for stochastic differential equation mixed-effects models using correlated particle pseudo-marginal algorithms, significantly improving computational speed for complex hierarchical models.

## Contribution

It develops a Gibbs sampling framework with correlated pseudo-marginal steps and blocking strategies, enhancing efficiency for SDEMEMs without requiring bespoke analytical derivations.

## Key findings

- Achieves approximately tenfold increase in computational efficiency.
- Demonstrates effectiveness on tumor growth and neuronal data.
- Applicable to a broad class of SDEMEMs.

## Abstract

Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that are able to account for random variability inherent in the underlying time-dynamics, as well as the variability between experimental units and, optionally, account for measurement error. Fully Bayesian inference for state-space SDEMEMs is performed, using data at discrete times that may be incomplete and subject to measurement error. However, the inference problem is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameter values of interest. The algorithm is made computationally efficient through careful use of blocking strategies and correlated pseudo-marginal Metropolis-Hastings steps within the Gibbs scheme. The resulting methodology is flexible and is able to deal with a large class of SDEMEMs. The methodology is demonstrated on three case studies, including tumor growth dynamics and neuronal data. The gains in terms of increased computational efficiency are model and data dependent, but unless bespoke sampling strategies requiring analytical derivations are possible for a given model, we generally observe an efficiency increase of one order of magnitude when using correlated particle methods together with our blocked-Gibbs strategy.