arXiv subjects

Daniel Williamson

Publications and source records attributed to Daniel Williamson.

14 recordsLinked to original sources

arXiv2026

Generalized and Scalable Deep Gaussian Process Emulation

Gaussian process (GP) emulators have become essential tools for approximating complex simulators, significantly reducing computational demands in optimization, sensitivity analysis, and model calibration. While traditional GP emulators effectively model continuous and Gaussian-distributed simulator outputs with homogeneous variability, they typically struggle with discrete, heteroskedastic Gaussian, or non-Gaussian data, limiting their applicability to increasingly common stochastic simulators. In this work, we introduce a scalable Generalized Deep Gaussian Process (GDGP) emulation framework designed to accommodate simulators with heteroskedastic Gaussian outputs and a wide range of non-Gaussian response distributions, including Poisson, negative binomial, and categorical distributions. The GDGP framework leverages the expressiveness of DGPs and extends them to latent GP structures, enabling it to capture the complex, non-stationary behavior inherent in many simulators while also modeling non-Gaussian simulator outputs. We make GDGP scalable by incorporating the Vecchia approximation for settings with a large number of input locations, while also developing efficient inference procedures for handling large numbers of replicates. In particular, we present methodological developments that further enhance the computation of the approach for heteroskedastic Gaussian responses. We demonstrate through a series of synthetic and empirical examples that these extensions deliver the practical application of GDGP emulators and a unified methodology capable of addressing diverse modeling challenges. The proposed GDGP framework is implemented in the open-source R package dgpsi.

stat.CO↗

arXiv2025

stCEG: An R Package for Modelling Events over Spatial Areas Using Chain Event Graphs

stCEG is an R package which allows a user to fully specify a Chain Event Graph (CEG) model from data and to produce interactive plots. It includes functions for the user to visualise spatial variables they wish to include in the model. There is also a web-based graphical user interface (GUI) provided, increasing ease of use for those without knowledge of R. We demonstrate stCEG using a dataset of homicides in London, which is included in the package. stCEG is the first software package for CEGs that allows for full model customisation.

stat.CO↗

arXiv2024

Deep Gaussian Process Emulation and Uncertainty Quantification for Large Computer Experiments

Computer models are used as a way to explore complex physical systems. Stationary Gaussian process emulators, with their accompanying uncertainty quantification, are popular surrogates for computer models. However, many computer models are not well represented by stationary Gaussian processes models. Deep Gaussian processes have been shown to be capable of capturing non-stationary behaviors and abrupt regime changes in the computer model response. In this paper, we explore the properties of two deep Gaussian process formulations within the context of computer model emulation. For one of these formulations, we introduce a new parameter that controls the amount of smoothness in the deep Gaussian process layers. We adapt a stochastic variational approach to inference for this model, allowing for prior specification and posterior exploration of the smoothness of the response surface. Our approach can be applied to a large class of computer models, and scales to arbitrarily large simulation designs. The proposed methodology was motivated by the need to emulate an astrophysical model of the formation of binary black hole mergers.

stat.ME↗

arXiv2024

Generalised Bayes Linear Inference

Motivated by big data and the vast parameter spaces in modern machine learning models, optimisation approaches to Bayesian inference have seen a surge in popularity in recent years. In this paper, we address the connection between the popular new methods termed generalised Bayesian inference and Bayes linear methods. We propose a further generalisation to Bayesian inference that unifies these and other recent approaches by considering the Bayesian inference problem as one of finding the closest point in a particular solution space to a data generating process, where these notions differ depending on user-specified geometries and foundational belief systems. Motivated by this framework, we propose a generalisation to Bayes linear approaches that enables fast and principled inferences that obey the coherence requirements implied by domain restrictions on random quantities. We demonstrate the efficacy of generalised Bayes linear inference on a number of examples, including monotonic regression and inference for spatial counts. This paper is accompanied by an R package available at github.com/astfalckl/bayeslinear.

stat.ME↗

arXiv2023

On the meaning of uncertainty for ethical AI: philosophy and practice

Whether and how data scientists, statisticians and modellers should be accountable for the AI systems they develop remains a controversial and highly debated topic, especially given the complexity of AI systems and the difficulties in comparing and synthesising competing claims arising from their deployment for data analysis. This paper proposes to address this issue by decreasing the opacity and heightening the accountability of decision making using AI systems, through the explicit acknowledgement of the statistical foundations that underpin their development and the ways in which these dictate how their results should be interpreted and acted upon by users. In turn, this enhances (1) the responsiveness of the models to feedback, (2) the quality and meaning of uncertainty on their outputs and (3) their transparency to evaluation. To exemplify this approach, we extend Posterior Belief Assessment to offer a route to belief ownership from complex and competing AI structures. We argue that this is a significant way to bring ethical considerations into mathematical reasoning, and to implement ethical AI in statistical practice. We demonstrate these ideas within the context of competing models used to advise the UK government on the spread of the Omicron variant of COVID-19 during December 2021.

cs.AI↗

arXiv2023

Linked Deep Gaussian Process Emulation for Model Networks

Modern scientific problems are often multi-disciplinary and require integration of computer models from different disciplines, each with distinct functional complexities, programming environments, and computation times. Linked Gaussian process (LGP) emulation tackles this challenge through a divide-and-conquer strategy that integrates Gaussian process emulators of the individual computer models in a network. However, the required stationarity of the component Gaussian process emulators within the LGP framework limits its applicability in many real-world applications. In this work, we conceptualize a network of computer models as a deep Gaussian process with partial exposure of its hidden layers. We develop a method for inference for these partially exposed deep networks that retains a key strength of the LGP framework, whereby each model can be emulated separately using a DGP and then linked together. We show in both synthetic and empirical examples that our linked deep Gaussian process emulators exhibit significantly better predictive performance than standard LGP emulators in terms of accuracy and uncertainty quantification. They also outperform single DGPs fitted to the network as a whole because they are able to integrate information from the partially exposed hidden layers. Our methods are implemented in an R package $\texttt{dgpsi}$ that is freely available on CRAN.

stat.ML↗

arXiv2021

Coexchangeable process modelling for uncertainty quantification in joint climate reconstruction

Any experiment with climate models relies on a potentially large set of spatio-temporal boundary conditions. These can represent both the initial state of the system and/or forcings driving the model output throughout the experiment. Whilst these boundary conditions are typically fixed using available reconstructions in climate modelling studies, they are highly uncertain, that uncertainty is unquantified, and the effect on the output of the experiment can be considerable. We develop efficient quantification of these uncertainties that combines relevant data from multiple models and observations. Starting from the coexchangeability model, we develop a coexchangable process model to capture multiple correlated spatio-temporal fields of variables. We demonstrate that further exchangeability judgements over the parameters within this representation lead to a Bayes linear analogy of a hierarchical model. We use the framework to provide a joint reconstruction of sea-surface temperature and sea-ice concentration boundary conditions at the last glacial maximum (19-23 ka) and use it to force an ensemble of ice-sheet simulations using the FAMOUS-Ice coupled atmosphere and ice-sheet model. We demonstrate that existing boundary conditions typically used in these experiments are implausible given our uncertainties and demonstrate the impact of using more plausible boundary conditions on ice-sheet simulation.

stat.AP↗

arXiv2021

Deep Gaussian Process Emulation using Stochastic Imputation

Deep Gaussian processes (DGPs) provide a rich class of models that can better represent functions with varying regimes or sharp changes, compared to conventional GPs. In this work, we propose a novel inference method for DGPs for computer model emulation. By stochastically imputing the latent layers, our approach transforms a DGP into a linked GP: a novel emulator developed for systems of linked computer models. This transformation permits an efficient DGP training procedure that only involves optimizations of conventional GPs. In addition, predictions from DGP emulators can be made in a fast and analytically tractable manner by naturally utilizing the closed form predictive means and variances of linked GP emulators. We demonstrate the method in a series of synthetic examples and empirical applications, and show that it is a competitive candidate for DGP surrogate inference, combining efficiency that is comparable to doubly stochastic variational inference and uncertainty quantification that is comparable to the fully-Bayesian approach. A $\texttt{Python}$ package $\texttt{dgpsi}$ implementing the method is also produced and available at https://github.com/mingdeyu/DGP.

stat.ML↗

arXiv2020

Cross-validation based adaptive sampling for Gaussian process models

In many real-world applications, we are interested in approximating black-box, costly functions as accurately as possible with the smallest number of function evaluations. A complex computer code is an example of such a function. In this work, a Gaussian process (GP) emulator is used to approximate the output of complex computer code. We consider the problem of extending an initial experiment (set of model runs) sequentially to improve the emulator. A sequential sampling approach based on leave-one-out (LOO) cross-validation is proposed that can be easily extended to a batch mode. This is a desirable property since it saves the user time when parallel computing is available. After fitting a GP to training data points, the expected squared LOO (ES-LOO) error is calculated at each design point. ES-LOO is used as a measure to identify important data points. More precisely, when this quantity is large at a point it means that the quality of prediction depends a great deal on that point and adding more samples nearby could improve the accuracy of the GP. As a result, it is reasonable to select the next sample where ES-LOO is maximised. However, ES-LOO is only known at the experimental design and needs to be estimated at unobserved points. To do this, a second GP is fitted to the ES-LOO errors and where the maximum of the modified expected improvement (EI) criterion occurs is chosen as the next sample. EI is a popular acquisition function in Bayesian optimisation and is used to trade-off between local/global search. However, it has a tendency towards exploitation, meaning that its maximum is close to the (current) "best" sample. To avoid clustering, a modified version of EI, called pseudo expected improvement, is employed which is more explorative than EI yet allows us to discover unexplored regions. Our results show that the proposed sampling method is promising.

stat.CO↗

arXiv2020

Classification of Computer Models with Labelled Outputs

Classification is a vital tool that is important for modelling many complex numerical models. A model or system may be such that, for certain areas of input space, the output either does not exist, or is not in a quantifiable form. Here, we present a new method for classification where the model outputs are given distinct classifying labels, which we model using a latent Gaussian process (GP). The latent variable is estimated using MCMC sampling, a unique likelihood and distinct prior specifications. Our classifier is then verified by calculating a misclassification rate across the input space. Comparisons are made with other existing classification methods including logistic regression, which models the probability of being classified into one of two regions. To make classification predictions we draw from an independent Bernoulli distribution, meaning that distance correlation is lost from the independent draws and so can result in many misclassifications. By modelling the labels using a latent GP, this problem does not occur in our method. We apply our novel method to a range of examples including a motivating example which models the hormones associated with the reproductive system in mammals, where the two labelled outputs are high and low rates of reproduction.

stat.ME↗

arXiv2019

Emulating computer models with step-discontinuous outputs using Gaussian processes

In many real-world applications we are interested in approximating costly functions that are analytically unknown, e.g. complex computer codes. An emulator provides a fast approximation of such functions relying on a limited number of evaluations. Gaussian processes (GPs) are commonplace emulators due to their statistical properties such as the ability to estimate their own uncertainty. GPs are essentially developed to fit smooth, continuous functions. However, the assumptions of continuity and smoothness is unwarranted in many situations. For example, in computer models where bifurcations or tipping points occur, the outputs can be discontinuous. This work examines the capacity of GPs for emulating step-discontinuous functions. Several approaches are proposed for this purpose. Two special covariance functions/kernels are adapted with the ability to model discontinuities. They are the neural network and Gibbs kernels whose properties are demonstrated using several examples. Another approach, which is called warping, is to transform the input space into a new space where a GP with a standard kernel, such as the Matern family, is able to predict the function well. The transformation is perform by a parametric map whose parameters are estimated by maximum likelihood. The results show that the proposed approaches have superior performance to GPs with standard kernels in capturing sharp jumps in the true function.

stat.ME↗

arXiv2019

Modelling Numerical Systems with Two Distinct Labelled Output Classes

We present a new method of modelling numerical systems where there are two distinct output solution classes, for example tipping points or bifurcations. Gaussian process emulation is a useful tool in understanding these complex systems and provides estimates of uncertainty, but we aim to include systems where there are discontinuities between the two output solutions. Due to continuity assumptions, we consider current methods of classification to split our input space into two output regions. Classification and logistic regression methods currently rely on drawing from an independent Bernoulli distribution, which neglects any information known in the neighbouring area. We build on this by including correlation between our input points. Gaussian processes are still a vital element, but used in latent space to model the two regions. Using the input values and an associated output class label, the latent variable is estimated using MCMC sampling and a unique likelihood. A threshold (usually at zero) defines the boundary. We apply our method to a motivating example provided by the hormones associated with the reproductive system in mammals, where the two solutions are associated with high and low rates of reproduction.

stat.ME↗

arXiv2015

Posterior Belief Assessment: Extracting Meaningful Subjective Judgements from Bayesian Analyses with Complex Statistical Models

In this paper, we are concerned with attributing meaning to the results of a Bayesian analysis for a problem which is sufficiently complex that we are unable to assert a precise correspondence between the expert probabilistic judgements of the analyst and the particular forms chosen for the prior specification and the likelihood for the analysis. In order to do this, we propose performing a finite collection of additional Bayesian analyses under alternative collections of prior and likelihood modelling judgements that we may also view as representative of our prior knowledge and the problem structure, and use these to compute posterior belief assessments for key quantities of interest. We show that these assessments are closer to our true underlying beliefs than the original Bayesian analysis and use the temporal sure preference principle to establish a probabilistic relationship between our true posterior judgements, our posterior belief assessment and our original Bayesian analysis to make this precise. We exploit second order exchangeability in order to generalise our approach to situations where there are infinitely many alternative Bayesian analyses we might consider as informative for our true judgements so that the method remains tractable even in these cases. We argue that posterior belief assessment is a tractable and powerful alternative to robust Bayesian analysis. We describe a methodology for computing posterior belief assessments in even the most complex of statistical models and illustrate with an example of calibrating an expensive ocean model in order to quantify uncertainty about global mean temperature in the real ocean.

math.ST↗

arXiv2013

Efficient uniform designs for multi-wave computer experiments

In this paper we tackle the problem of generating uniform designs in very small subregions of computer model input space that have been identified in previous experiments as worthy of further study. The method is capable of producing uniform designs in subregions of computer model input space defined by a membership function that consists of a continuous function passing a threshold test, and does so far more efficiently than current methods when these subregions are small. Our application is designing for regions of input space that are not ruled out by history matching, a statistical methodology applied in numerous diverse scientific applications whereby model runs are used to cut out regions of input space that are incompatible with real world observations. History matching defines a membership function for a region of input space that is not ruled out yet by observations in the form of a distance metric called implausibility. We use this distance metric to drive a new type of Evolutionary Monte Carlo algorithm with a uniform distribution on the not ruled out yet region as its target distribution. The algorithm can locate and generate uniform points within extremely small subspaces of the computer model input space with complex and even disconnected topologies. We illustrate the performance of the technique in comparison to current methods with a number of idealised examples. We then apply our algorithm to generating an optimal design for the not ruled out yet region of a galaxy simulation model called GALFORM following 4 previous waves of history matching where the target region is 0.001% the volume of the input space.

stat.ME↗