Ian Vernon

Galaxy formation: a Bayesian uncertainty analysis

In many scientific disciplines complex computer models are used to understand the behaviour of large scale physical systems. An uncertainty anal- ysis of such a computer model known as Galform is presented. Galform models the creation and evolution of approximately one million galaxies from the begin- ning of the Universe until the current day, and is regarded as a state-of-the-art model within the cosmology community. It requires the specification of many in- put parameters in order to run the simulation, takes significant time to run, and provides various outputs that can be compared with real world data. A Bayes Linear approach is presented in order to identify the subset of the input space that could give rise to acceptable matches between model output and measured data. This approach takes account of the major sources of uncertainty in a consistent and unified manner, including input parameter uncertainty, function uncertainty, observational error, forcing function uncertainty and structural uncertainty. The approach is known as History Matching, and involves the use of an iterative suc- cession of emulators (stochastic belief specifications detailing beliefs about the Galform function), which are used to cut down the input parameter space. The analysis was successful in producing a large collection of model evaluations that exhibit good fits to the observed data.

The Parameter Space of Galaxy Formation

Semi-analytic models are a powerful tool for studying the formation of galaxies. However, these models inevitably involve a signicant number of poorly constrained parameters that must be adjusted to provide an acceptable match to the observed universe. In this paper, we set out to quantify the degree to which observational data-sets can constrain the model parameters. By revealing degeneracies in the parameter space we can hope to better understand the key physical processes probed by the data. We use novel mathematical techniques to explore the parameter space of the GALFORM semi-analytic model. We base our investigation on the Bower et al. 2006 version of GALFORM, adopting the same methodology of selecting model parameters based on an acceptable match to the local bJ and K luminosity functions. Since the GALFORM model is inherently approximate, we explicitly include a model discrepancy term when deciding if a match is acceptable or not. The model contains 16 parameters that are poorly constrained by our prior understanding of the galaxy formation processes and that can plausibly be adjusted between reasonable limits. We investigate this parameter space using the Model Emulator technique, constructing a Bayesian approximation to the GALFORM model that can be rapidly evaluated at any point in parameter space. The emulator returns both an expectation for the GALFORM model and an uncertainty which allows us to eliminate regions of parameter space in which it is implausible that a GALFORM run would match the luminosity function data. By combining successive waves of emulation, we show that only 0.26% of the initial volume is of interest for further exploration. However, within this region we show that the Bower et al. 2006 model is only one choice from an extended sub-space of model parameters that can provide equally acceptable ts to the luminosity function data. We explore the geometry of this region and begin to explore the physical connections between parameters that are exposed by this analysis. We also consider the impact of adding additional observational data to further constrain the parameter space. We see that the known tensions existing in the Bower et al. 2006 model lead to a further reduction in the successful parameter space.

Branes on the Horizon

Models with extra dimensions are often invoked to resolve cosmological problems. We investigate the possibility of apparent acausality as seen by a brane-based observer resulting from signal propagation through the extra dimensions. Null geodesics are first computed in static and cosmological single-brane models, following which we derive the equations of motion for the inter-brane distance in a two-brane scenario, which we use to examine possible acausality in this more complex setup. Despite observing significant effective acausality in some situations there is no a priori solution to the horizon problem using this mechanism. In the two-brane scenario there can be significant late time violation of gravitational Lorentz invariance, resulting in the gravitational horizon being larger than the particle horizon, leading to potential signals in gravitational wave detectors.Models with extra dimensions are often invoked to resolve cosmological problems. We investigate the possibility of apparent acausality as seen by a brane-based observer resulting from signal propagation through the extra dimensions. Null geodesics are first computed in static and cosmological single-brane models, following which we derive the equations of motion for the inter-brane distance in a two-brane scenario, which we use to examine possible acausality in this more complex setup. Despite observing significant effective acausality in some situations there is no a priori solution to the horizon problem using this mechanism. In the two-brane scenario there can be significant late time violation of gravitational Lorentz invariance, resulting in the gravitational horizon being larger than the particle horizon, leading to potential signals in gravitational wave detectors.

Bulk black holes radiating in non-Z2 brane-world spacetimes

In this paper we present a general asymmetric brane model involving arbitrary energy transport to and from an embedded 4D Friedmann–Robertson–Walker universe. We derive a locally defined mass function for the 5D spacetime and describe its time evolution on the brane. We then specialize our model to the two cases of graviton production in the early universe and radiating black holes in the bulk.

Bayesian History Matching of Complex Infectious Disease Models Using Emulation: A Tutorial and a Case Study on HIV in Uganda.

Advances in scientific computing have allowed the development of complex models that are being routinely applied to problems in disease epidemiology, public health and decision making. The utility of these models depends in part on how well they can reproduce empirical data. However, fitting such models to real world data is greatly hindered both by large numbers of input and output parameters, and by long run times, such that many modelling studies lack a formal calibration methodology. We present a novel method that has the potential to improve the calibration of complex infectious disease models (hereafter called simulators). We present this in the form of a tutorial and a case study where we history match a dynamic, event-driven, individual-based stochastic HIV simulator, using extensive demographic, behavioural and epidemiological data available from Uganda. The tutorial describes history matching and emulation. History matching is an iterative procedure that reduces the simulators input space by identifying and discarding areas that are unlikely to provide a good match to the empirical data. History matching relies on the computational efficiency of a Bayesian representation of the simulator, known as an emulator. Emulators mimic the simulators behaviour, but are often several orders of magnitude faster to evaluate. In the case study, we use a 22 input simulator, fitting its 18 outputs simultaneously. After 9 iterations of history matching, a non-implausible region of the simulator input space was identified that was times smaller than the original input space. Simulator evaluations made within this region were found to have a 65% probability of fitting all 18 outputs. History matching and emulation are useful additions to the toolbox of infectious disease modellers. Further research is required to explicitly address the stochastic nature of the simulator as well as to account for correlations between outputs.

Cosmological phase transitions in a brane world

In brane world scenarios the Friedmann equation is modified, resulting in an increased expansion at early times. This has important effects on cosmological phase transitions which we investigate, elucidating significant differences to the standard case. First order phase transitions require a higher nucleation rate to complete; baryogenesis and particle abundances could be suppressed. Topological defect evolution is also affected, though the current defect densities are largely unchanged. In particular, the increased expansion does not solve the usual monopole and domain wall problems.

Brane world cosmology without the Z2 symmetry

The Friedmann equation for a positive tension brane situated between two bulk spacetimes that posses the same 5D cosmological constant, but which does not posses a Z2 symmetry of the metric itself is derived, and the possible effects of dropping the Z2 symmetry on the expansion of our Universe are examined; cosmological constraints are discussed. We show the effect of this is an inflation-like period at very early times. The global solutions for the metric in the infinite extra dimension case are found and comparison with the symmetric case is made. We show that any brane world senario of this type must revert to a Z2 symmetric form at late times, and hence rule out certain proposed scenarios.

Galaxy Formation: Bayesian History Matching for the Observable Universe

Cosmologists at the Institute of Computational Cosmology, Durham University, have developed a state of the art model of galaxy formation known as Galform, intended to contribute to our understanding of the formation, growth and subsequent evolution of galaxies in the presence of dark matter. Galform requires the specification of many input parameters and takes a significant time to complete one simulation, making comparison between the model’s output and real observations of the Universe extremely challenging. This paper concerns the analysis of this problem using Bayesian emulation within an iterative history matching strategy, and represents the most detailed uncertainty analysis of a galaxy formation simulation yet performed.

Constraints on galaxy formation models from the galaxy stellar mass function and its evolution

We explore the parameter space of the semi-analytic galaxy formation model GALFORM, studying the constraints imposed by measurements of the galaxy stellar mass function (GSMF) and its evolution. We use the Bayesian emulator method to quickly eliminate vast implausible volumes of the parameter space and zoom in on the most interesting regions, allowing us to identify a set of models that match the observational data within model uncertainties. We find that the GSMF strongly constrains parameters related to quiescent star formation in discs, stellar and active galactic nucleus feedback and threshold for disc instabilities, but weakly restricts other parameters. Constraining the model using local data alone does not usually select models that match the evolution of the GSMF well. Nevertheless, we show that a small subset of models provides acceptable match to GSMF data out to redshift 1.5. We explore the physical significance of the parameters of these models, in particular exploring whether the model provides a better description if the mass loading of the galactic winds generated by starbursts (β0,burst) and quiescent discs (β0,disc) is different. Performing a principal component analysis of the plausible volume of the parameter space, we write a set of relations between parameters obeyed by plausible models with respect to GSMF evolution. We find that while β0,disc is strongly constrained by GSMF evolution data, constraints on β0,burst are weak. Although it is possible to find plausible models for which β0,burst = β0,disc, most plausible models have β0,burst > β0,disc, implying – for these – larger stellar feedback efficiency at higher redshifts.

Bayesian uncertainty analysis for complex systems biology models : emulation, global parameter searches and evaluation of gene functions.

BackgroundMany mathematical models have now been employed across every area of systems biology. These models increasingly involve large numbers of unknown parameters, have complex structure which can result in substantial evaluation time relative to the needs of the analysis, and need to be compared to observed data of various forms. The correct analysis of such models usually requires a global parameter search, over a high dimensional parameter space, that incorporates and respects the most important sources of uncertainty. This can be an extremely difficult task, but it is essential for any meaningful inference or prediction to be made about any biological system. It hence represents a fundamental challenge for the whole of systems biology.MethodsBayesian statistical methodology for the uncertainty analysis of complex models is introduced, which is designed to address the high dimensional global parameter search problem. Bayesian emulators that mimic the systems biology model but which are extremely fast to evaluate are embeded within an iterative history match: an efficient method to search high dimensional spaces within a more formal statistical setting, while incorporating major sources of uncertainty.ResultsThe approach is demonstrated via application to a model of hormonal crosstalk in Arabidopsis root development, which has 32 rate parameters, for which we identify the sets of rate parameter values that lead to acceptable matches between model output and observed trend data. The multiple insights into the model’s structure that this analysis provides are discussed. The methodology is applied to a second related model, and the biological consequences of the resulting comparison, including the evaluation of gene functions, are described.ConclusionsBayesian uncertainty analysis for complex models using both emulators and history matching is shown to be a powerful technique that can greatly aid the study of a large class of systems biology models. It both provides insight into model behaviour and identifies the sets of rate parameters of interest.