Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models

Abstract

Stochastic mathematical models are attractive for modelling biological phenomena because they naturally capture stochasticity and variability that is often evident in biological data. Stochastic models also often allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we aim to combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments. In particular we focus on models of cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between a fine-scale stochastic model and its associated coarse-grained continuum model. We form this statistical model based on a limited number of expensive model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using a statistical model of the discrepancy allows us to use the computationally inexpensive continuum model to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters and profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available on GitHub.

Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know

“…For example, BAE can be used to model the approximation errors due to the use of reduced order models or upscaled models. BAE has also been used to account for the errors due to the use of a mean-field model instead of an underlying high-fidelity stochastic model; see, e.g., [34]. Hence, the framework presented for OED under uncertainty in the present work can be extended to OED in inverse problems where the forward model is replaced with a low-fidelity approximate model instead of a computationally intensive high-fidelity model, as long the approximation error can be modeled adequately by a Gaussian.…”

“…See the review [1], for a survey of the recent literature in this area. There has also been an increased interest in parameter inversion and design of experiments in systems governed by uncertain forward models; see e.g., [6,14,24,26,32,34] for a small sample of the literature addressing inverse problems under uncertainty. Methods for OED in such inverse problems have been studied in [5,11,18,27].…”