Unbiased Inference for Complex Models

Large scale mathematical models play an essential role in many applications throughout science. A prime example of an application of a complex continuous model is the research on climate change. The climate is usually modelled as a non-linear mathematical-physical system based on the coupling of a model of the atmosphere and an ocean circulation model. Both of these models involve a big set of state variables, such as the temperature, humidity, the wind, flow speed and pressure, to name but a few. The state of the system is modelled as the collection of these parameters for every grid cell. The evolution of these parameters arises as a discretisation of the underlying continuous model. In order to make inference about the climate, the model needs to be run at a fixed discretisation. The accuracy of the inference depends on the number of times the model can be simulated under a finite computational budget which thus influences the level of the discretisation. This trade-off results in a widening gap between the models that we can simulate and the ones for which we can make sound statistical inference. Bayesian methods are ubiquitous in statistical modelling and machine learning for the analysis of data. The aim is to model the posterior distribution which is in most cases only available indirectly as an appropriate limit of a sequence of probability measures. The most prominent technique to access the posterior distribution is via MCMC algorithms. MCMC algorithms are fully flexible and generally applicable. However, they require simulations of the full complex model on each step which limits the number of steps that are possible for a fixed computational budget, thus requiring huge computational budgets if we want to keep a high level of accuracy. For this reason Monte Carlo methods are often neglected, even though they are the only methods targeting the correct posterior.

For a wide range of applications, for example in weather prediction, nuclear waste management and quantitative finance, the quantity to be inferred is often only given indirectly as an appropriate limit of distributions. The conventional approach to such an indirect representation requires a truncation. In most cases, the choice is to either run the MCMC only for a sufficiently long time, to fix a finite discretisation or to incorporate only a certain number of terms in the series expansion. However, all of these truncations lead to a systematic bias in the modelling of the target distribution and the exact accuracy of this bias is highly application-specific and very difficult to analyse.

The main motivation of this proposal is to establish, extend and improve schemes for direct unbiased inference and thus to remove systematic bias in Bayesian inference for complex models. In particular, it aims at improving established inference methods by an incorporation of a new class of unbiased estimators and to develop new inference algorithms that yield unbiased estimators for Bayesian inference. This approach naturally allows the distribution of computations across different discretisations which results in great computational gain which is beneficial for a wide range of applications. The BBC has, for example, recently reported on the possibility to use the mobile phone records in Western Africa in order to predict the spreading of Ebola. Mathematical models have also become a factor for the strategic planning of the Metropolitan police in London to predict crime and to identify the areas of high probabilities of certain crimes in order to increase the presence of the police accordingly. Many more of these very recent examples in the press can be found undermining the importance of the use of accurate models around us. In the medium term, a sound statistical method, allowing an unbiased evaluation of complex models would therefore optimise a wide range of production processes in industry, politics and medicine to name just a few.

The potential to extract understanding by analysing complex relationships becomes apparent on an almost daily basis. The BBC has recently reported on the possibility to use mobile phone records in Western Africa in order to predict the spread of Ebola and complex mathematical models have become a factor for the strategic planning of the Metropolitan Police to predict crime and to identify the areas of high probabilities for certain crimes to name just two examples. However, a breakthrough to a computationally efficient sound statistical method that is applicable to a wide range of complex models is still lacking and currently the growth in model complexity is outpacing the statistical tools to handle such complexity in reasonable computing times. This proposal offers a fundamentally novel approach to the challenges that we are facing by exploiting the impact of a recently established scheme for unbiased estimators, viable at low computational cost, onto other branches in statistical modelling and machine learning in academia. Not only the academic environment will profit from this proposal but also a wide range of UK industries. Computational statistics and machine learning are key analytical techniques in almost all applications such as the financial industry, pharmaceutical industry, the defence industry to name but a few. A sound statistical method to make inference on the basis of complex models would be highly beneficial for time critical responses to important decisions. Therefore, we expect the developments of this proposal to provide tools which help to maintain the UK's leading edge in its industries.

The impact of this research project will be obtained through new methods, novel ideas and a better understanding of inference for and optimisation of complex models. The reason for a high demand of these methods is the tremendous interest in the evaluation of uncertainty in complex models. The spread of the resulting methods and ideas will mostly happen indirectly, and will thus only become effective on a time range of 5 to 10 years. On a smaller time scale the impact of the proposed research will mainly focus on interdisciplinary research in academia ranging from statistical analysis of (social) networks to uncertainty quantification.

Apart from its impact on knowledge and applications in academia and the UK economy, the proposed research project will also benefit everyone who is working on this project. New tools will be developed, understood and applied and thus, research skills will be refined. In particular, it will improve the abilities of the PDRA to implement, develop and analyse complex models and related statistical inference algorithms. As demonstrated above, these skills are crucial for research in academia as well as in the industry.