# Particle Methods for Stochastic Differential Equation Mixed Effects Models

## TL;DR

This paper develops advanced particle MCMC methods tailored for stochastic differential equation mixed effects models, addressing intractable likelihoods and improving inference efficiency in complex biological data.

## Contribution

It introduces three novel extensions to particle MCMC that exploit SDEMEM-specific features and correlated pseudo-marginal techniques for more efficient inference.

## Key findings

- Enhanced inference efficiency demonstrated on real and simulated data
- Extensions outperform naive approaches in computational speed
- Applicable to complex biological models with intractable likelihoods

## Abstract

Parameter inference for stochastic differential equation mixed effects models (SDEMEMs) is a challenging problem. Analytical solutions for these models are rarely available, which means that the likelihood is also intractable. In this case, exact inference is possible using the pseudo-marginal method, where the intractable likelihood is replaced by its nonnegative unbiased estimate. A useful application of this idea is particle MCMC, which uses a particle filter estimate of the likelihood. While the exact posterior is targeted by these methods, a naive implementation for SDEMEMs can be highly inefficient. We develop three extensions to the naive approach which exploits specific aspects of SDEMEMs and other advances such as correlated pseudo-marginal methods. We compare these methods on real and simulated data from a tumour xenography study on mice.