Research

Bayes Linear Statistics

Bayes linear statistics is a means of comparing beliefs about uncertain systems without the explicit specification of distributions on prior parameters. It follows the ideas of de Finetti, viewing expectation (rather than a probability distribution) as a primitive when considering uncertain systems. All that is required to use the tools of Bayes linear statistics is the specification of the expectation, and of any second-order quantities (covariances and variances). The posterior prediction, correspondingly, is given in terms of the modified expectation and variance generated by the Bayes linear update equations: \[\text{E}_D[B] = \text{E}[B] + \text{Cov}[B, D]\text{Var}[D]^{-1}(D - \text{E}[D]),\] \[\text{Var}_D[B] = \text{Var}[B] - \text{Cov}[B, D]\text{Var}[D]^{-1}\text{Cov}[D, B].\] Here \(B\) are our prior beliefs, \(D\) is any data used to modify said beliefs, and \(\text{E}_D[B]\) (\(\text{Var}_D[B]\)) is the modified expectation (variance).

One can produce graphical interpretations of such belief adjustments, in order to determine the most informative data for producing the posterior specifications. Examples of (interactive) use of this approach can be found here:

Emulation

Many complex real-world phenomena are modelled using equally complex computer simulations. These can be extremely powerful for examining the underlying dynamics of the physical world, as well as investigating inputs that would give rise to observations, but their complexity often comes with a large computational overhead. Emulation is a statistical method by which complex models can be efficiently run across the entirety of their (usually large) parameter space without resorting to large numbers of model evaluations.

Let \(f(x)\) be a complex computer model which takes inputs \(x\in\mathbb{R}\) and returns a single value - the output of interest. Then we can create an emulator for this output as \[g(x) = \sum \beta_i h_i(x) + u(x).\] Here \(h_i(x)\) are basis functions in the variables \(x\), \(\beta_i\) the corresponding coefficients, and \(u(x)\) is a covariance structure. The coefficients \(\beta\) and covariance structure \(u(x)\) are not assumed to be known, but we can attach distributional statements to them. In light of data, we may update these prior beliefs to obtain posterior predictions.

This approach means that we can make predictions (with the appropriate uncertainty) across the whole parameter space on only a handful of runs (data) from the complex model itself, allowing a more thorough exploration of parameter space. We can simplify the task further by employing a Bayes linear approach, specifying only second-order quantities for \(\beta\) and \(u(x)\). Then the posterior prediction is obtained via the Bayes linear update formulae.

History Matching

Being able to evaluate a complex model more efficiently is one thing; however, our statistical approximation (emulator) is not a perfect representation of the model. Moreover, the model itself is not expected to be a perfect representation of reality, nor do we (necessarily) expect our observations of reality to which we want to match to be completely accurate. This necessitates a careful approach to what we consider a 'good' match to data.

Furthermore, if we were to use an approach like optimisation (say) to find acceptable combinations of parameters, there is no guarantee that the result will be representative of the true possible scope of values that could be taken to match to the data. This can cause severe problems if we want to predict into the future: for example, to forecast hospital occupancy numbers in light of a pandemic.

History matching is a method by which we can deal with the uncertainties and ensure that we obtain the complete set of parameter combinations that could give rise to observed reality. It works on the principle of complementarity; rather than looking for 'good' points, it instead seeks to rule out 'bad' points. When applied iteratively (in waves), this successively removes parts of parameter space that, even accounting for uncertainties, cannot give rise to observed reality.

In conjunction with emulation, we define an implausibility \[I(x)^2 = \frac{(\text{E}[g(x)]-z)^2}{\text{Var}[g(x)] + \text{Var}[\epsilon] + \text{Var}[e]},\] where \(z\) is the observation to match to, the \(g(x)\) quantities are the (adjusted) predictions from the emulator, \(\epsilon\) is the model discrepancy, and \(e\) the observational error. A high implausibility means that the point \(x\) is highly unlikely to result in an acceptable match to data when passed to the model.

For a simple example of emulation and history matching applied to a one-dimensional model, see here.

The process of history matching and emulation can be difficult to implement in the course of modelling a physical phenomenon; to that end, the hmer R package is designed to make the process as smooth as possible without requiring expert statistical knowledge. For examples of the package's use in both synthetic and real-world situations, see the following papers.

Past Research Interests

Instantons

An instanton is a soliton in Yang-Mills theory for electromagnetism: given the action for Yang-Mills \[S=\frac{1}{2}\int\text{Tr}(F_{\mu\nu}F^{\mu\nu})\mathrm{d}^{4}x\] we can find solutions that minimise the action \(S\), corresponding to solutions of the equations of motion. General solutions are hard to find, but a static solution (one which does not change over time) can be found using the ADHM construction: any such constructed object is an instanton solution. They can be thought of as 'lumps' of electromagnetic energy: the extent of each one is heavily localised.

While these are static, we can approximate motion using the solutions, provided the motion is slow enough. We can use this to consider evolution of instantons, scattering, and lots of other aspects of their behaviour. Instantons have a wide range of applicability: they appear in compactification of M-Theory and in D-brane configurations in supersymmetric string theory; they are higher-dimensional analogues of other soliton solutions, particularly vortices, and so can be related to superconductivity in metals.

For a detailed treatment of instantons and scattering simulations in noncommutative space, see my PhD thesis.