An Introduction to Thermodynamic Integration and Application to Dynamic Causal Models

Abstract

In generative modeling of neuroimaging data, such as dynamic causal modeling (DCM), one typically considers several alternative models, either to determine the most plausible explanation for observed data (Bayesian model selection) or to account for model uncertainty (Bayesian model averaging). Both procedures rest on estimates of the model evidence, a principled trade-off between model accuracy and complexity. In the context of DCM, the log evidence is usually approximated using variational Bayes. Although this approach is highly efficient, it makes distributional assumptions and is vulnerable to local extrema. This paper introduces the use of thermodynamic integration (TI) for Bayesian model selection and averaging in the context of DCM. TI is based on Markov chain Monte Carlo sampling which is asymptotically exact but orders of magnitude slower than variational Bayes. In this paper, we explain the theoretical foundations of TI, covering key concepts such as the free energy and its origins in statistical physics. Our aim is to convey an in-depth understanding of the method starting from its historical origin in statistical physics. In addition, we demonstrate the practical application of TI via a series of examples which serve to guide the user in applying this method. Furthermore, these examples demonstrate that, given an efficient implementation and hardware capable of parallel processing, the challenge of high computational demand can be overcome successfully. The TI implementation presented in this paper is freely available as part of the open source software TAPAS. Author summaryWhen fitting computational models to data in the setting of Bayesian inference, a user has the choice between two broad classes of algorithms: variational inference and Monte Carlo simulation. While both methods have advantages and drawbacks, variational inference has become standard in the domain of modelling directed brain connectivity due to its computational efficiency, especially when the challenges to select between competing hypotheses that explain the observed data. By contrast, the high computational demand by Monte Carlo methods has so far prevented their widespread use for inference on brain connectivity, despite their capability to overcome some of the shortcomings of variational inference. In this paper, we introduce the user to thermodynamic integration (TI), a Monte Carlo method designed for model fitting and model selection. By covering its foundations and historical origins in statistical physics, we hope to convey an in-depth understanding of TI that goes beyond a purely technical treatment. In addition, we also provide examples for concrete applications, demonstrating that, given an efficient implementation and up-to-date hardware, the challenge of high computational demand can be overcome successfully.