Alma Rahat

Alma Rahat, Tinkle Chugh, Jonathan Fieldsend et al.

Many methods for performing multi-objective optimisation of computationally expensive problems have been proposed recently. Typically, a probabilistic surrogate for each objective is constructed from an initial dataset. The surrogates can then be used to produce predictive densities in the objective space for any solution. Using the predictive densities, we can compute the expected hypervolume improvement (EHVI) due to a solution. Maximising the EHVI, we can locate the most promising solution that may be expensively evaluated next. There are closed-form expressions for computing the EHVI, integrating over the multivariate predictive densities. However, they require partitioning the objective space, which can be prohibitively expensive for more than three objectives. Furthermore, there are no closed-form expressions for a problem where the predictive densities are dependent, capturing the correlations between objectives. Monte Carlo approximation is used instead in such cases, which is not cheap. Hence, the need to develop new accurate but cheaper approximation methods remains. Here we investigate an alternative approach toward approximating the EHVI using Gauss-Hermite quadrature. We show that it can be an accurate alternative to Monte Carlo for both independent and correlated predictive densities with statistically significant rank correlations for a range of popular test problems.

Andy Gray, Alma Rahat, Tom Crick et al.

Assessment is a crucial part of education. Traditional marking is a source of inconsistencies and unconscious bias, placing a high cognitive load on the assessors. An approach to address these issues is comparative judgement (CJ). In CJ, the assessor is presented with a pair of items and is asked to select the better one. Following a series of comparisons, a rank is derived using a ranking model, for example, the BTM, based on the results. While CJ is considered a reliable method for marking, there are concerns around transparency, and the ideal number of pairwise comparisons to generate a reliable estimation of the rank order is not known. Additionally, there have been attempts to generate a method of selecting pairs that should be compared next in an informative manner, but some existing methods are known to have created their own bias within results inflating the reliability metric used. As a result, a random selection approach is usually deployed. We propose a novel Bayesian approach to CJ (BCJ) for determining the ranks of compared items alongside a new way to select the pairs to present to the marker(s) using active learning (AL), addressing the key shortcomings of traditional CJ. Furthermore, we demonstrate how the entire approach may provide transparency by providing the user insights into how it is making its decisions and, at the same time, being more efficient. Results from our experiments confirm that the proposed BCJ combined with entropy-driven AL pair-selection method is superior to other alternatives. We also find that the more comparisons done, the more accurate BCJ becomes, which solves the issue the current method has of the model deteriorating if too many comparisons are performed. As our approach can generate the complete predicted rank distribution for an item, we also show how this can be utilised in devising a predicted grade, guided by the assessor.

Xietao Wang Lin, Juan Ungredda, Max Butler et al.

Bayesian optimisation has proven to be a powerful tool for expensive global black-box optimisation problems. In this paper, we propose new Bayesian optimisation variants of the popular Knowledge Gradient acquisition functions for problems with \emph{decoupled} black-box constraints, in which subsets of the objective and constraint functions may be evaluated independently. In particular, our methods aim to take into account that often only a handful of the constraints may be binding at the optimum, and hence we should evaluate only relevant constraints when trying to optimise a function. We empirically benchmark these methods against existing methods and demonstrate their superiority over the state-of-the-art.

Andy Gray, Alma Rahat, Tom Crick et al.

Comparative Judgement (CJ) provides an alternative assessment approach by evaluating work holistically rather than breaking it into discrete criteria. This method leverages human ability to make nuanced comparisons, yielding more reliable and valid assessments. CJ aligns with real-world evaluations, where overall quality emerges from the interplay of various elements. However, rubrics remain widely used in education, offering structured criteria for grading and detailed feedback. This creates a gap between CJ's holistic ranking and the need for criterion-based performance breakdowns. This paper addresses this gap using a Bayesian approach. We build on Bayesian CJ (BCJ) by Gray et al., which directly models preferences instead of using likelihoods over total scores, allowing for expected ranks with uncertainty estimation. Their entropy-based active learning method selects the most informative pairwise comparisons for assessors. We extend BCJ to handle multiple independent learning outcome (LO) components, defined by a rubric, enabling both holistic and component-wise predictive rankings with uncertainty estimates. Additionally, we propose a method to aggregate entropies and identify the most informative comparison for assessors. Experiments on synthetic and real data demonstrate our method's effectiveness. Finally, we address a key limitation of BCJ, which is the inability to quantify assessor agreement. We show how to derive agreement levels, enhancing transparency in assessment.

Andy Gray, Alma Rahat, Stephen Lindsay et al.

Ensuring transparency in educational assessment is increasingly critical, particularly post-pandemic, as demand grows for fairer and more reliable evaluation methods. Comparative Judgement (CJ) offers a promising alternative to traditional assessments, yet concerns remain about its perceived opacity. This paper examines how Bayesian Comparative Judgement (BCJ) enhances transparency by integrating prior information into the judgement process, providing a structured, data-driven approach that improves interpretability and accountability. BCJ assigns probabilities to judgement outcomes, offering quantifiable measures of uncertainty and deeper insights into decision confidence. By systematically tracking how prior data and successive judgements inform final rankings, BCJ clarifies the assessment process and helps identify assessor disagreements. Multi-criteria BCJ extends this by evaluating multiple learning outcomes (LOs) independently, preserving the richness of CJ while producing transparent, granular rankings aligned with specific assessment goals. It also enables a holistic ranking derived from individual LOs, ensuring comprehensive evaluations without compromising detailed feedback. Using a real higher education dataset with professional markers in the UK, we demonstrate BCJ's quantitative rigour and ability to clarify ranking rationales. Through qualitative analysis and discussions with experienced CJ practitioners, we explore its effectiveness in contexts where transparency is crucial, such as high-stakes national assessments. We highlight the benefits and limitations of BCJ, offering insights into its real-world application across various educational settings.

Clym Stock-Williams, Tinkle Chugh, Alma Rahat et al.

Performing multi-objective Bayesian optimisation by scalarising the objectives avoids the computation of expensive multi-dimensional integral-based acquisition functions, instead of allowing one-dimensional standard acquisition functions\textemdash such as Expected Improvement\textemdash to be applied. Here, two infill criteria based on hypervolume improvement\textemdash one recently introduced and one novel\textemdash are compared with the multi-surrogate Expected Hypervolume Improvement. The reasons for the disparities in these methods' effectiveness in maximising the hypervolume of the acquired Pareto Front are investigated. In addition, the effect of the surrogate model mean function on exploration and exploitation is examined: careful choice of data normalisation is shown to be preferable to the exploration parameter commonly used with the Expected Improvement acquisition function. Finally, the effectiveness of all the methodological improvements defined here is demonstrated on a real-world problem: the optimisation of a wind turbine blade aerofoil for both aerodynamic performance and structural stiffness. With effective scalarisation, Bayesian optimisation finds a large number of new aerofoil shapes that strongly dominate standard designs.

Alma Rahat, Michael Wood

We are often interested in identifying the feasible subset of a decision space under multiple constraints to permit effective design exploration. If determining feasibility required computationally expensive simulations, the cost of exploration would be prohibitive. Bayesian search is data-efficient for such problems: starting from a small dataset, the central concept is to use Bayesian models of constraints with an acquisition function to locate promising solutions that may improve predictions of feasibility when the dataset is augmented. At the end of this sequential active learning approach with a limited number of expensive evaluations, the models can accurately predict the feasibility of any solution obviating the need for full simulations. In this paper, we propose a novel acquisition function that combines the probability that a solution lies at the boundary between feasible and infeasible spaces (representing exploitation) and the entropy in predictions (representing exploration). Experiments confirmed the efficacy of the proposed function.