In April 2015, the paper "Controversy in mechanistic modelling with Gaussian processes" by Benn Macdonald, Catherine Higham and Dirk Husmeier was accepted for presentation at the International Conference on Machine Learning (ICML) and subsequent publication in the conference proceedings. There were 1037 submissions to ICML this year. After a rigorous double-blind review process, the senior program committee accepted 270 papers.

What is the paper about?

Modern statistics and machine learning aims to extend inference to increasingly complex systems described by coupled differential equations (DEs). The differential equations typically don't have a closed form solution. This requires them to be integrated numerically every time the system parameters are to be adapted, leading to excessive computational costs.

To circumvent the high computational complexity of explicitly solving the DEs, various authors have proposed an approach based on gradient matching: in a first preliminary "smoothing" step, interpolate the time series data; in a second "inference" step, optimise the parameters so as to minimise some metric measuring the difference between the slopes of the tangents to the interpolants, and the parameter-dependent time derivatives from the DEs. In this way, the DEs never have to be solved explicitly, and the typically unknown initial conditions are effectively profiled over. A disadvantage of this two-step scheme is that the results of parameter inference may critically hinge on the quality of the initial interpolant. A better approach is to regularise the interpolants by the DEs themselves, and a method called GPODE, based on Gaussian processes, was proposed at ICML 2014.

In our paper we critically assess the viability of GPODE. Adopting an alternative mathematical representation, we work out its intrinsic disadvantages and limitations. We then present an alternative improved method. This method, which is also based on Gaussian processes, overcomes the shortcomings of GPODE, and we demonstrate its superiority on three independent benchmark data.

Why is the paper important?

ICML is the leading international machine learning conference. Our paper leads to deeper insight into statistical inference in mechanistic models, provides new practical inference tools, and is of direct relevance to our current EPSRC "Mathematics Centres for Health Care" proposal.

First published: 26 May 2015

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