Louis J. M. Aslett

Tianlin Yang, Hailiang Du, Louis Aslett

Probabilistic forecasting provides a principled framework for uncertainty quantification in dynamical systems by representing predictions as probability distributions rather than deterministic trajectories. However, existing forecasting approaches, whether physics-based or neural-network-based, remain fundamentally trajectory-oriented: predictive distributions are usually accessed through ensembles or sampling, rather than evolved directly as dynamical objects. A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions. The framework introduces a distributional encoding and decoding structure around a replaceable neural forecasting module, using kernel mean embeddings to represent input distributions and mixture density networks to parameterise output predictive distributions. This design enables recursive propagation of predictive uncertainty within a unified end-to-end neural architecture, with model training and evaluation carried out directly in terms of probabilistic forecast skill. The framework is demonstrated on the Lorenz63 chaotic dynamical system. Results show that the D2D model captures nontrivial distributional evolution under nonlinear dynamics, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with, and in some cases outperforms, a simplified perfect model benchmark. These findings point to a new paradigm for probabilistic forecasting, in which predictive distributions are learned and evolved directly rather than reconstructed indirectly through ensemble-based uncertainty propagation.

Louis Chislett, Louis JM Aslett, Alisha R Davies et al.

Clinical prediction models are statistical or machine learning models used to quantify the risk of a certain health outcome using patient data. These can then inform potential interventions on patients, causing an effect called performative prediction: predictions inform interventions which influence the outcome they were trying to predict, leading to a potential underestimation of risk in some patients if a model is updated on this data. One suggested resolution to this is the use of hold-out sets, in which a set of patients do not receive model derived risk scores, such that a model can be safely retrained. We present an overview of clinical and research ethics regarding potential implementation of hold-out sets for clinical prediction models in health settings. We focus on the ethical principles of beneficence, non-maleficence, autonomy and justice. We also discuss informed consent, clinical equipoise, and truth-telling. We present illustrative cases of potential hold-out set implementations and discuss statistical issues arising from different hold-out set sampling methods. We also discuss differences between hold-out sets and randomised control trials, in terms of ethics and statistical issues. Finally, we give practical recommendations for researchers interested in the use hold-out sets for clinical prediction models.

Sami Haidar-Wehbe, Samuel R Emerson, Louis J M Aslett et al.

Predictive risk scores for adverse outcomes are increasingly crucial in guiding health interventions. Such scores may need to be periodically updated due to change in the distributions they model. However, directly updating risk scores used to guide intervention can lead to biased risk estimates. To address this, we propose updating using a `holdout set' - a subset of the population that does not receive interventions guided by the risk score. Balancing the holdout set size is essential to ensure good performance of the updated risk score whilst minimising the number of held out samples. We prove that this approach reduces adverse outcome frequency to an asymptotically optimal level and argue that often there is no competitive alternative. We describe conditions under which an optimal holdout size (OHS) can be readily identified, and introduce parametric and semi-parametric algorithms for OHS estimation. We apply our methods to the ASPRE risk score for pre-eclampsia to recommend a plan for updating it in the presence of change in the underlying data distribution. We show that, in order to minimise the number of pre-eclampsia cases over time, this is best achieved using a holdout set of around 10,000 individuals.

James Liley, Samuel R Emerson, Bilal A Mateen et al.

Machine learning is increasingly being used to generate prediction models for use in a number of real-world settings, from credit risk assessment to clinical decision support. Recent discussions have highlighted potential problems in the updating of a predictive score for a binary outcome when an existing predictive score forms part of the standard workflow, driving interventions. In this setting, the existing score induces an additional causative pathway which leads to miscalibration when the original score is replaced. We propose a general causal framework to describe and address this problem, and demonstrate an equivalent formulation as a partially observed Markov decision process. We use this model to demonstrate the impact of such `naive updating' when performed repeatedly. Namely, we show that successive predictive scores may converge to a point where they predict their own effect, or may eventually tend toward a stable oscillation between two values, and we argue that neither outcome is desirable. Furthermore, we demonstrate that even if model-fitting procedures improve, actual performance may worsen. We complement these findings with a discussion of several potential routes to overcome these issues.

Pedro M. Esperança, Louis J. M. Aslett, Chris C. Holmes

Information that is stored in an encrypted format is, by definition, usually not amenable to statistical analysis or machine learning methods. In this paper we present detailed analysis of coordinate and accelerated gradient descent algorithms which are capable of fitting least squares and penalised ridge regression models, using data encrypted under a fully homomorphic encryption scheme. Gradient descent is shown to dominate in terms of encrypted computational speed, and theoretical results are proven to give parameter bounds which ensure correctness of decryption. The characteristics of encrypted computation are empirically shown to favour a non-standard acceleration technique. This demonstrates the possibility of approximating conventional statistical regression methods using encrypted data without compromising privacy.

Louis J. M. Aslett

The precise design of a system may be considered a trade secret which should be protected, whilst at the same time component manufacturers are sometimes reluctant to release full test data (perhaps only providing mean time to failure data). In this situation it seems impractical to both produce an accurate reliability assessment and satisfy all parties' privacy requirements. However, we present recent developments in cryptography which, when combined with the recently developed survival signature in reliability theory, allows almost total privacy to be maintained in a cryptographically strong manner in precisely this setting. Thus, the system designer does not have to reveal their trade secret design and the component manufacturer can retain component test data in-house.

Louis J. M. Aslett, Pedro M. Esperança, Chris C. Holmes

We present two new statistical machine learning methods designed to learn on fully homomorphic encrypted (FHE) data. The introduction of FHE schemes following Gentry (2009) opens up the prospect of privacy preserving statistical machine learning analysis and modelling of encrypted data without compromising security constraints. We propose tailored algorithms for applying extremely random forests, involving a new cryptographic stochastic fraction estimator, and naïve Bayes, involving a semi-parametric model for the class decision boundary, and show how they can be used to learn and predict from encrypted data. We demonstrate that these techniques perform competitively on a variety of classification data sets and provide detailed information about the computational practicalities of these and other FHE methods.

Louis J. M. Aslett, Pedro M. Esperança, Chris C. Holmes

Recent advances in cryptography promise to enable secure statistical computation on encrypted data, whereby a limited set of operations can be carried out without the need to first decrypt. We review these homomorphic encryption schemes in a manner accessible to statisticians and machine learners, focusing on pertinent limitations inherent in the current state of the art. These limitations restrict the kind of statistics and machine learning algorithms which can be implemented and we review those which have been successfully applied in the literature. Finally, we document a high performance R package implementing a recent homomorphic scheme in a general framework.