Comprehensive benchmarking of Markov chain Monte Carlo methods for dynamical systems

Benjamin Ballnus,Fabian J Theis,Sabine Hug,Kathrin Hatz,Linus Görlitz,Jan Hasenauer

doi:10.1186/s12918-017-0433-1

Abstract

BackgroundIn quantitative biology, mathematical models are used to describe and analyze biological processes. The parameters of these models are usually unknown and need to be estimated from experimental data using statistical methods. In particular, Markov chain Monte Carlo (MCMC) methods have become increasingly popular as they allow for a rigorous analysis of parameter and prediction uncertainties without the need for assuming parameter identifiability or removing non-identifiable parameters. A broad spectrum of MCMC algorithms have been proposed, including single- and multi-chain approaches. However, selecting and tuning sampling algorithms suited for a given problem remains challenging and a comprehensive comparison of different methods is so far not available.ResultsWe present the results of a thorough benchmarking of state-of-the-art single- and multi-chain sampling methods, including Adaptive Metropolis, Delayed Rejection Adaptive Metropolis, Metropolis adjusted Langevin algorithm, Parallel Tempering and Parallel Hierarchical Sampling. Different initialization and adaptation schemes are considered. To ensure a comprehensive and fair comparison, we consider problems with a range of features such as bifurcations, periodical orbits, multistability of steady-state solutions and chaotic regimes. These problem properties give rise to various posterior distributions including uni- and multi-modal distributions and non-normally distributed mode tails. For an objective comparison, we developed a pipeline for the semi-automatic comparison of sampling results.ConclusionThe comparison of MCMC algorithms, initialization and adaptation schemes revealed that overall multi-chain algorithms perform better than single-chain algorithms. In some cases this performance can be further increased by using a preceding multi-start local optimization scheme. These results can inform the selection of sampling methods and the benchmark collection can serve for the evaluation of new algorithms. Furthermore, our results confirm the need to address exploration quality of MCMC chains before applying the commonly used quality measure of effective sample size to prevent false analysis conclusions.

Highlights

In quantitative biology, mathematical models are used to describe and analyze biological processes
Mechanistic modelling and parameter estimation We focus on ordinary differential equation (ODE) models for the mechanistic description of biological processes
Application of sampling methods to mRNA transcription model To illustrate the behavior and the properties of the different sampling methods, we consider the process of mRNA transcription ((M1), Fig. 6a)

Summary

Introduction

Mathematical models are used to describe and analyze biological processes. In the field of computational systems biology, mechanistic models are developed to explain experimental data, to gain a quantitative understanding of processes and to predict the process dynamics under new experimental conditions [1,2,3] The parameters of these mechanistic models are typically unknown and need to be estimated from available experimental data. Likelihoods and posterior distributions are frequently multi-modal and possess pronounced tails (see, e.g., [4, 5]), and many applications problems possess structural and practical non-identifiabilities (see, e.g., [13,14,15,16] and references therein) This is, among others, due to scares, noise-corrupted experimental data and a features of the underlying dynamical systems, such as bistability [17, 18], oscillation [19,20,21] and chaos [22,23,24]

Methods

Results

Discussion

Conclusion