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Mathematics and Statistics

Photo of Prof Daniel Williamson

Prof Daniel Williamson

Associate Professor

 (Streatham) 5514

 01392 725514

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For codes to do history matching, Gaussian process emulation, emergent constraints and our UQ for Covid-19 project.

Twitter: @BayesExeter

The overarching theme in my research is the quantification of uncertainty to support decision making and I view these problems through the lens of a subjective Bayesian. I have worked on a diverse set of projects connected to this theme, from the development of proper underpinning foundations of subjective Bayesian inference for complex statistical models, through to Bayesian modelling for transport behaviour and decision support (to reduce congestion, improve health outcomes) and seasonal forecasting in meteorology. A large proportion of my work has been in Uncertainty Quantification (UQ) for complex computer models of physical systems, and in climate model tuning (through UQ). Climate models are a key part of local, national and international planning for adapting to climate change (e.g. in food security, planning of future renewable energy systems, water management etc.). Developing tools to properly capture the uncertainty in any forecast of a future system where future weather (obtained by climate model simulations) plays a forcing role, has been and continues to be a focus of my work.

Climate models are, arguably, the world's most complicated and expensive computer models, and existing statistical methods for calibrating models that would enable a solution to the tuning problem have been tried and do not scale up to climate models. One focus of my research has been and continues to be developing new Bayesian methods in UQ that scale up to climate models and to embed these new methods in climate modelling centres as tools that they regularly use in model development. This will change climate science, both in enabling tuning uncertainty to be routinely explored and quantified, leading to better climate models, and in enabling novel multi-model, perturbed parameter ensemble designs that can be used to go between and beyond scenarios for decision support.

The new methods and theory I am working on for UQ and decision making have wide applicability beyond climate models, with implications for environmental science, data science, data-centric engineering, policy support and Ethical AI. Theory for policy support and ethical AI are of particular current interest and I am seeking to link my foundational work on subjective Bayes with my UQ work to develop a new theme in Ethical AI around ownership of statements from complex models (or AI such as Neural Networks), in the subjectivist sense. I am also developing methods for decision support using linked models with application to decision making for landscapes. In particular linking policy models, climate models, and models for a range of ecosystem services. I lead a KTP with SRK Exploration in Cardiff and consult with researchers at Meteo France and Environment Canada on the development of their models.

Research Grants

  • InnovateUK KTP: SRK Explorations. £166,570, PI, Apr 2020 - Apr 2022.
  • UKRI "Constraining projections of ice sheet instabilities and future sea level rise" £1,500,000 Co-I (Exeter PI)  May 2019- Apr 2023 (with option for 3 year extension).

  • NERC/SPF JULES Emulator of Ecosystem Services, £50,670, Co-I, Sept 2019 - Feb 2020.

  • Horizon 2020: Tipping points in the Earth System €8,561,238. Awarded Sep 2019 - Sep 2022.

  • Alan Turing Institute project: Uncertainty quantification for black box computational models with application to machine decisions: £500,074. Co-I Oct 2019 - Oct 2022.

  • Alan Turing Institute project: Uncertainty quantification of multi-scale and multi-physics computer models: application to hazard and climate models. £850,000. Co-I (Exeter PI) Jan 2019 - Jan 2021.

  • Alan Turing Institute Fellowship: £7003. P.I. Oct 2018 - Oct 2020

  • EPSRC Past Earth Network: Searching for the deglaciation: spatio-temporal boundary condition uncertainty and its implications for understanding abrupt climate change. £24805. PI. May 2017 - Nov 2017

  • Agence Nationale de la Recherche (ANR) HIGH-TUNE project €405,000. (€24,600 to cover interaction with me and my research team.) Exeter PI. Dec 2016 - Sept 2020.

  • EPSRC centre for mathematical science in health care. £1,927,811, Co-I, Jan 2016 - Jan 2020.

  • NERC Robust spatial projections for real world climate change. £1,118,206, Co-I, Oct 2016 - Oct 2020.

  • InnovateUK: Engaged Smart Transport £292,229, Co-I, Nov 2015 - Jan 2018.

  • NERC Persistence of seasonal climate anomalies £1,111,600, Co-I, Dec 2014 - Dec 2018.

  • EPSRC fellowship: Uncertainty quantification for the linking of spatio-temporal computer model hierarchies and the real world. £219,400, Sept 2013 - Sept 2016.

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Copyright Notice: Any articles made available for download are for personal use only. Any other use requires prior permission of the author and the copyright holder.

| 2023 | 2022 | 2021 | 2020 | 2019 | 2018 | 2017 | 2015 | 2014 | 2013 | 2012 |










  • Williamson D, Blaker AT. (2014) Evolving Bayesian Emulators for Structured Chaotic Time Series, with application to large climate models, SIAM Journal on Uncertainty Quantification, volume 2, pages 1-28.



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Further information

Personal Homepage



Volodina, V., Williamson, D. B. (2020) Diagnostics-driven nonstationary emulators using kernel mixtures. SIAM Journal of Uncertainty Quantification 8(1) 1-26 DOI 10.1137/19M124438X 

Williamson, D. B., Sansom, P. G. (2020) How are emergent constraints quantifying uncertainty and what do they leave behind? BAMS, 100, 2571-2588, 

Dawkins, L. C. Williamson, D. B. Mengersen, K. L., Morawska, L.; Jayaratne, R., Shaddick, G. (2020) Where Is the Clean Air? A Bayesian Decision Framework for Personalised Cyclist Route Selection Using R-INLA. Bayesian Analysis, doi:10.1214/19-BA1193. 

Dawkins, L. C., Williamson, D. B., Barr, S. W., Lampkin, S. R. (2020). What drives commuter behaviour? A Bayesian clustering approach for understanding opposing behaviours in social surveys." J. Roy Stat. Soc. Ser. A, 183, 251-280,


Sansom, P.G., Williamson, D. B., Stephenson, D. B. (2019) State space models for intermittently coupled systems, J. Roy. Stat. Soc. Ser. C, 68, 1259-1280. doi:10.1111/rssc.12354

Salter, J.M., Williamson, D.B., Scinocca, J., Kharin, S. (2019) "Uncertainty quantification for spatio-temporal computer models with calibration-optimal bases" Journal of the American Statistical Association 114:528, 1800-1814, DOI: 10.1080/01621459.2018.1514306. 

Dawkins, L. C., Barr, S. W., Williamson, D. B., and Lampkin, S. R. (2018). "Influencing transport behaviour: A Bayesian modelling approach for segmentation of social surveys." Journal of Transport Geography, 70 91-103. 

Williamson, D. B., Blaker, A. T., Sinha, B. (2017) "Tuning without over-tuning: parametric uncertainty quantification for the NEMO ocean model.", Geoscientific Model Development, 10(4) 1789-1816.

Screen, J., Williamson, D. (2017) "Ice-free Arctic at $1.5^o$C?", Nature Climate Change 7(4) 230-231.

Hourdin, F., Mauritsen, T., Gettelman, A., Golaz, J-C., Balaji, V., Duan, Q., Folini, D,. Ji, D., Klocke, D., Qian, Y., Rauser, F., Rio, C., Tomassini, L., Watanabe, M., Williamson, D. (2017) "The art and science of climate model tuning" BAMS 589-602. 

Salter, J. M., Williamson, D. (2016) ``A comparison of statistical emulation methodologies for multi-wave calibration'', Environmetrics 27(8), 507-523.

Williamson, D., Goldstein M. (2015), "Posterior belief assessment: extracting meaningful subjective judgements from Bayesian analyses with complex statistical models", Bayesian Analysis, 10(4) 877-908. 

Williamson, D., Blaker, A. T., Hampton, C., Salter, J. (2015) ``Identifying and removing structural biases in climate models with history matching'',  Climate Dynamics, 45, 1299-1324.

Williamson, D. (2015), "Exploratory designs for computer experiments using k-extended Latin Hypercubes", Environmetrics 26(4) 268-283.

Williamson, D., Blaker, A.~T. (2014), ``Evolving Bayesian emulators for structured chaotic time series, with application to large climate models'', SIAM Journal of Uncertainty Quantification, 2(1).

Williamson, D., Vernon, I.R., (2013), ``Efficient uniform designs for multi-wave computer experiments'', arXiv:1309.3520

Williamson, D., Goldstein, M., Allison, L., Blaker, A., Challenor, P. Jackson, L., Yamazaki, K., (2013), ``History matching for exploring and reducing climate model parameter space using observations and a large perturbed physics ensemble'', Climate Dynamics, 41 1703--1729. 

Williamson, D., Goldstein, M. and Blaker, A. (2012), ``Fast Linked Analyses for Scenario based Hierarchies'', Journal of the Royal Statistical Society Series C. 61(5), 665-693.

Yamazaki, K., Rowlands, D.~J., Aina, T., Blaker, A., Bowery, A., Massey, N., Miller, J., Rye, C., Tett, S.~F.~B., Williamson, D., Yamazaki, Y.~H., Allen, M.~R. (2012), ``Obtaining diverse behaviours in a climate model without the use of flux adjustments'', Journal of Geophysical Research - Atmospheres, 118(7), 2781--2793.

Williamson, D. and Goldstein, M. (2012), ``Bayesian policy support for

  adaptive strategies using computer models for complex physical systems,'' Journal of the Operational Research Society, 63, 1021--1033.

Williamson, D. (2011) Comment on Gramacy, R. B. and Lee, H. K. H. Optimization under unknown constraints Bayesian Statistics 9, Oxford: Oxford University Press.

Williamson, D. (2010), ``Policy making using computer simulators for

  complex physical systems; Bayesian decision support for the development of

  adaptive strategies'', Ph.D. thesis, University of Durham,


Submitted and In Revision

Salter, J. M., Williamson, D.B., Gregoire, L.J., Edwards, T. L. (2020) "Quantifying spatio-temporal boundary condition uncertainty for the North American deglaciation". In Revision for SIAM Journal of Uncertainty Quantification.   Kimpton, L., Challenor, P., Williamson, D.B., (2020) Classification of computer models with labelled outputs. In Submission   Xu, W., Williamson, D.B., Challenor, P. (2020) Local Voronoi tessellations for robust multi-wave calibration of computer models. In Submission.   Couvreux, F., Hourdin, F., Williamson, D. B., Roehrig, R., Volodina, V., Villefranque, N., Rio, C., Audouin, O., Salter, J., Bazile, E., Brient, F., Favot, F., Honnert, R., Lefebvre, MP, Madeleine, JB, Rodier, Q. (2020) Process-based climate model development harnessing machine learning: I. a calibration tool for parameterization improvement.    Hourdin, F., Williamson, D. B., Rio, C., Couvreux, F., Roehrig, R., Villefranque, N., Musat, I., Fairhead, L., Diallo, F. B., Volodina, V. (2020) Process-based climate model development harnessing machine learning: II. model calibration from single column to global. In Submission to JAMES.  

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