Chris Holmes

Robin Mitra, Sarah F. McGough, Tapabrata Chakraborti et al. · oxford

Missing data are an unavoidable complication in many machine learning tasks. When data are `missing at random' there exist a range of tools and techniques to deal with the issue. However, as machine learning studies become more ambitious, and seek to learn from ever-larger volumes of heterogeneous data, an increasingly encountered problem arises in which missing values exhibit an association or structure, either explicitly or implicitly. Such `structured missingness' raises a range of challenges that have not yet been systematically addressed, and presents a fundamental hindrance to machine learning at scale. Here, we outline the current literature and propose a set of grand challenges in learning from data with structured missingness.

Fabian Falck, Christopher Williams, Dominic Danks et al. · oxford

U-Net architectures are ubiquitous in state-of-the-art deep learning, however their regularisation properties and relationship to wavelets are understudied. In this paper, we formulate a multi-resolution framework which identifies U-Nets as finite-dimensional truncations of models on an infinite-dimensional function space. We provide theoretical results which prove that average pooling corresponds to projection within the space of square-integrable functions and show that U-Nets with average pooling implicitly learn a Haar wavelet basis representation of the data. We then leverage our framework to identify state-of-the-art hierarchical VAEs (HVAEs), which have a U-Net architecture, as a type of two-step forward Euler discretisation of multi-resolution diffusion processes which flow from a point mass, introducing sampling instabilities. We also demonstrate that HVAEs learn a representation of time which allows for improved parameter efficiency through weight-sharing. We use this observation to achieve state-of-the-art HVAE performance with half the number of parameters of existing models, exploiting the properties of our continuous-time formulation.

Jobie Budd, Kieran Baker, Emma Karoune et al.

The UK COVID-19 Vocal Audio Dataset is designed for the training and evaluation of machine learning models that classify SARS-CoV-2 infection status or associated respiratory symptoms using vocal audio. The UK Health Security Agency recruited voluntary participants through the national Test and Trace programme and the REACT-1 survey in England from March 2021 to March 2022, during dominant transmission of the Alpha and Delta SARS-CoV-2 variants and some Omicron variant sublineages. Audio recordings of volitional coughs, exhalations, and speech were collected in the 'Speak up to help beat coronavirus' digital survey alongside demographic, self-reported symptom and respiratory condition data, and linked to SARS-CoV-2 test results. The UK COVID-19 Vocal Audio Dataset represents the largest collection of SARS-CoV-2 PCR-referenced audio recordings to date. PCR results were linked to 70,794 of 72,999 participants and 24,155 of 25,776 positive cases. Respiratory symptoms were reported by 45.62% of participants. This dataset has additional potential uses for bioacoustics research, with 11.30% participants reporting asthma, and 27.20% with linked influenza PCR test results.

Oscar Clivio, Fabian Falck, Brieuc Lehmann et al. · oxford

Traditional methods for matching in causal inference are impractical for high-dimensional datasets. They suffer from the curse of dimensionality: exact matching and coarsened exact matching find exponentially fewer matches as the input dimension grows, and propensity score matching may match highly unrelated units together. To overcome this problem, we develop theoretical results which motivate the use of neural networks to obtain non-trivial, multivariate balancing scores of a chosen level of coarseness, in contrast to the classical, scalar propensity score. We leverage these balancing scores to perform matching for high-dimensional causal inference and call this procedure neural score matching. We show that our method is competitive against other matching approaches on semi-synthetic high-dimensional datasets, both in terms of treatment effect estimation and reducing imbalance.

Rob Cornish, Muhammad Faaiz Taufiq, Arnaud Doucet et al.

Digital twins are virtual systems designed to predict how a real-world process will evolve in response to interventions. This modelling paradigm holds substantial promise in many applications, but rigorous procedures for assessing their accuracy are essential for safety-critical settings. We consider how to assess the accuracy of a digital twin using real-world data. We formulate this as causal inference problem, which leads to a precise definition of what it means for a twin to be "correct" appropriate for many applications. Unfortunately, fundamental results from causal inference mean observational data cannot be used to certify that a twin is correct in this sense unless potentially tenuous assumptions are made, such as that the data are unconfounded. To avoid these assumptions, we propose instead to find situations in which the twin is not correct, and present a general-purpose statistical procedure for doing so. Our approach yields reliable and actionable information about the twin under only the assumption of an i.i.d. dataset of observational trajectories, and remains sound even if the data are confounded. We apply our methodology to a large-scale, real-world case study involving sepsis modelling within the Pulse Physiology Engine, which we assess using the MIMIC-III dataset of ICU patients.

Harry Coppock, George Nicholson, Ivan Kiskin et al.

Recent work has reported that AI classifiers trained on audio recordings can accurately predict severe acute respiratory syndrome coronavirus 2 (SARSCoV2) infection status. Here, we undertake a large scale study of audio-based deep learning classifiers, as part of the UK governments pandemic response. We collect and analyse a dataset of audio recordings from 67,842 individuals with linked metadata, including reverse transcription polymerase chain reaction (PCR) test outcomes, of whom 23,514 tested positive for SARS CoV 2. Subjects were recruited via the UK governments National Health Service Test-and-Trace programme and the REal-time Assessment of Community Transmission (REACT) randomised surveillance survey. In an unadjusted analysis of our dataset AI classifiers predict SARS-CoV-2 infection status with high accuracy (Receiver Operating Characteristic Area Under the Curve (ROCAUC) 0.846 [0.838, 0.854]) consistent with the findings of previous studies. However, after matching on measured confounders, such as age, gender, and self reported symptoms, our classifiers performance is much weaker (ROC-AUC 0.619 [0.594, 0.644]). Upon quantifying the utility of audio based classifiers in practical settings, we find them to be outperformed by simple predictive scores based on user reported symptoms.

Hannah Kerner, Snehal Chaudhari, Aninda Ghosh et al.

Crop field boundaries are foundational datasets for agricultural monitoring and assessments but are expensive to collect manually. Machine learning (ML) methods for automatically extracting field boundaries from remotely sensed images could help realize the demand for these datasets at a global scale. However, current ML methods for field instance segmentation lack sufficient geographic coverage, accuracy, and generalization capabilities. Further, research on improving ML methods is restricted by the lack of labeled datasets representing the diversity of global agricultural fields. We present Fields of The World (FTW) -- a novel ML benchmark dataset for agricultural field instance segmentation spanning 24 countries on four continents (Europe, Africa, Asia, and South America). FTW is an order of magnitude larger than previous datasets with 70,462 samples, each containing instance and semantic segmentation masks paired with multi-date, multi-spectral Sentinel-2 satellite images. We provide results from baseline models for the new FTW benchmark, show that models trained on FTW have better zero-shot and fine-tuning performance in held-out countries than models that aren't pre-trained with diverse datasets, and show positive qualitative zero-shot results of FTW models in a real-world scenario -- running on Sentinel-2 scenes over Ethiopia.

Jake Fawkes, Lucile Ter-Minassian, Desi Ivanova et al.

Merging datasets across institutions is a lengthy and costly procedure, especially when it involves private information. Data hosts may therefore want to prospectively gauge which datasets are most beneficial to merge with, without revealing sensitive information. For causal estimation this is particularly challenging as the value of a merge depends not only on reduction in epistemic uncertainty but also on improvement in overlap. To address this challenge, we introduce the first cryptographically secure information-theoretic approach for quantifying the value of a merge in the context of heterogeneous treatment effect estimation. We do this by evaluating the Expected Information Gain (EIG) using multi-party computation to ensure that no raw data is revealed. We further demonstrate that our approach can be combined with differential privacy (DP) to meet arbitrary privacy requirements whilst preserving more accurate computation compared to DP alone. To the best of our knowledge, this work presents the first privacy-preserving method for dataset acquisition tailored to causal estimation. We demonstrate the effectiveness and reliability of our method on a range of simulated and realistic benchmarks. Code is publicly available: https://github.com/LucileTerminassian/causal_prospective_merge.

Sahra Ghalebikesabi, Chris Holmes, Edwin Fong et al.

Bayesian methods are a popular choice for statistical inference in small-data regimes due to the regularization effect induced by the prior. In the context of density estimation, the standard nonparametric Bayesian approach is to target the posterior predictive of the Dirichlet process mixture model. In general, direct estimation of the posterior predictive is intractable and so methods typically resort to approximating the posterior distribution as an intermediate step. The recent development of quasi-Bayesian predictive copula updates, however, has made it possible to perform tractable predictive density estimation without the need for posterior approximation. Although these estimators are computationally appealing, they tend to struggle on non-smooth data distributions. This is due to the comparatively restrictive form of the likelihood models from which the proposed copula updates were derived. To address this shortcoming, we consider a Bayesian nonparametric model with an autoregressive likelihood decomposition and a Gaussian process prior. While the predictive update of such a model is typically intractable, we derive a quasi-Bayesian predictive update that achieves state-of-the-art results in small-data regimes.

Davide Pigoli, Kieran Baker, Jobie Budd et al.

Since early in the coronavirus disease 2019 (COVID-19) pandemic, there has been interest in using artificial intelligence methods to predict COVID-19 infection status based on vocal audio signals, for example cough recordings. However, existing studies have limitations in terms of data collection and of the assessment of the performances of the proposed predictive models. This paper rigorously assesses state-of-the-art machine learning techniques used to predict COVID-19 infection status based on vocal audio signals, using a dataset collected by the UK Health Security Agency. This dataset includes acoustic recordings and extensive study participant meta-data. We provide guidelines on testing the performance of methods to classify COVID-19 infection status based on acoustic features and we discuss how these can be extended more generally to the development and assessment of predictive methods based on public health datasets.

Lucile Ter-Minassian, Oscar Clivio, Karla Diaz-Ordaz et al.

Predictive black-box models can exhibit high accuracy but their opaque nature hinders their uptake in safety-critical deployment environments. Explanation methods (XAI) can provide confidence for decision-making through increased transparency. However, existing XAI methods are not tailored towards models in sensitive domains where one predictor is of special interest, such as a treatment effect in a clinical model, or ethnicity in policy models. We introduce Path-Wise Shapley effects (PWSHAP), a framework for assessing the targeted effect of a binary (e.g.~treatment) variable from a complex outcome model. Our approach augments the predictive model with a user-defined directed acyclic graph (DAG). The method then uses the graph alongside on-manifold Shapley values to identify effects along causal pathways whilst maintaining robustness to adversarial attacks. We establish error bounds for the identified path-wise Shapley effects and for Shapley values. We show PWSHAP can perform local bias and mediation analyses with faithfulness to the model. Further, if the targeted variable is randomised we can quantify local effect modification. We demonstrate the resolution, interpretability, and true locality of our approach on examples and a real-world experiment.

Jack Jewson, Sahra Ghalebikesabi, Chris Holmes

Differential privacy guarantees allow the results of a statistical analysis involving sensitive data to be released without compromising the privacy of any individual taking part. Achieving such guarantees generally requires the injection of noise, either directly into parameter estimates or into the estimation process. Instead of artificially introducing perturbations, sampling from Bayesian posterior distributions has been shown to be a special case of the exponential mechanism, producing consistent, and efficient private estimates without altering the data generative process. The application of current approaches has, however, been limited by their strong bounding assumptions which do not hold for basic models, such as simple linear regressors. To ameliorate this, we propose $β$D-Bayes, a posterior sampling scheme from a generalised posterior targeting the minimisation of the $β$-divergence between the model and the data generating process. This provides private estimation that is generally applicable without requiring changes to the underlying model and consistently learns the data generating parameter. We show that $β$D-Bayes produces more precise inference estimation for the same privacy guarantees, and further facilitates differentially private estimation via posterior sampling for complex classifiers and continuous regression models such as neural networks for the first time.

Oscar Clivio, Avi Feller, Chris Holmes

Reweighting a distribution to minimize a distance to a target distribution is a powerful and flexible strategy for estimating a wide range of causal effects, but can be challenging in practice because optimal weights typically depend on knowledge of the underlying data generating process. In this paper, we focus on design-based weights, which do not incorporate outcome information; prominent examples include prospective cohort studies, survey weighting, and the weighting portion of augmented weighting estimators. In such applications, we explore the central role of representation learning in finding desirable weights in practice. Unlike the common approach of assuming a well-specified representation, we highlight the error due to the choice of a representation and outline a general framework for finding suitable representations that minimize this error. Building on recent work that combines balancing weights and neural networks, we propose an end-to-end estimation procedure that learns a flexible representation, while retaining promising theoretical properties. We show that this approach is competitive in a range of common causal inference tasks.

Sheheryar Zaidi, Arber Zela, Thomas Elsken et al.

Ensembles of neural networks achieve superior performance compared to stand-alone networks in terms of accuracy, uncertainty calibration and robustness to dataset shift. \emph{Deep ensembles}, a state-of-the-art method for uncertainty estimation, only ensemble random initializations of a \emph{fixed} architecture. Instead, we propose two methods for automatically constructing ensembles with \emph{varying} architectures, which implicitly trade-off individual architectures' strengths against the ensemble's diversity and exploit architectural variation as a source of diversity. On a variety of classification tasks and modern architecture search spaces, we show that the resulting ensembles outperform deep ensembles not only in terms of accuracy but also uncertainty calibration and robustness to dataset shift. Our further analysis and ablation studies provide evidence of higher ensemble diversity due to architectural variation, resulting in ensembles that can outperform deep ensembles, even when having weaker average base learners. To foster reproducibility, our code is available: \url{https://github.com/automl/nes}

Vik Shirvaikar, Andrea Storås, Xi Lin et al.

The identification of heterogeneous treatment effects (HTE) across subgroups is of significant interest in clinical trial analysis. Several state-of-the-art HTE estimation methods, including causal forests, apply recursive partitioning for non-parametric identification of relevant covariates and interactions. However, the partitioning criterion is typically based on differences in absolute risk. This can dilute statistical power by masking variation in the relative risk, which is often a more appropriate quantity of clinical interest. In this work, we propose and implement a methodology for modifying causal forests to target relative risk, using a novel node-splitting procedure based on exhaustive generalized linear model comparison. We present results from simulated data that suggest relative risk causal forests can capture otherwise undetected sources of heterogeneity. We implement our method on real-world trial data to explore HTEs for liraglutide in patients with type 2 diabetes.

Yudi Pawitan, Chris Holmes

There is a growing literature on reasoning by large language models (LLMs), but the discussion on the uncertainty in their responses is still lacking. Our aim is to assess the extent of confidence that LLMs have in their answers and how it correlates with accuracy. Confidence is measured (i) qualitatively in terms of persistence in keeping their answer when prompted to reconsider, and (ii) quantitatively in terms of self-reported confidence score. We investigate the performance of three LLMs -- GPT4o, GPT4-turbo and Mistral -- on two benchmark sets of questions on causal judgement and formal fallacies and a set of probability and statistical puzzles and paradoxes. Although the LLMs show significantly better performance than random guessing, there is a wide variability in their tendency to change their initial answers. There is a positive correlation between qualitative confidence and accuracy, but the overall accuracy for the second answer is often worse than for the first answer. There is a strong tendency to overstate the self-reported confidence score. Confidence is only partially explained by the underlying token-level probability. The material effects of prompting on qualitative confidence and the strong tendency for overconfidence indicate that current LLMs do not have any internally coherent sense of confidence.

Lucile Ter-Minassian, Sahra Ghalebikesabi, Karla Diaz-Ordaz et al.

With the adoption of machine learning into routine clinical practice comes the need for Explainable AI methods tailored to medical applications. Shapley values have sparked wide interest for locally explaining models. Here, we demonstrate their interpretation strongly depends on both the summary statistic and the estimator for it, which in turn define what we identify as an 'anchor point'. We show that the convention of using a mean anchor point may generate misleading interpretations for survival analysis and introduce median-SHAP, a method for explaining black-box models predicting individual survival times.

Ziyu Wang, Chris Holmes

Bayesian modelling allows for the quantification of predictive uncertainty which is crucial in safety-critical applications. Yet for many machine learning (ML) algorithms, it is difficult to construct or implement their Bayesian counterpart. In this work we present a promising approach to address this challenge, based on the hypothesis that commonly used ML algorithms are efficient across a wide variety of tasks and may thus be near Bayes-optimal w.r.t. an unknown task distribution. We prove that it is possible to recover the Bayesian posterior defined by the task distribution, which is unknown but optimal in this setting, by building a martingale posterior using the algorithm. We further propose a practical uncertainty quantification method that apply to general ML algorithms. Experiments based on a variety of non-NN and NN algorithms demonstrate the efficacy of our method.

Lucile Ter-Minassian, Liran Szlak, Ehud Karavani et al.

Interpretability and transparency are essential for incorporating causal effect models from observational data into policy decision-making. They can provide trust for the model in the absence of ground truth labels to evaluate the accuracy of such models. To date, attempts at transparent causal effect estimation consist of applying post hoc explanation methods to black-box models, which are not interpretable. Here, we present BICauseTree: an interpretable balancing method that identifies clusters where natural experiments occur locally. Our approach builds on decision trees with a customized objective function to improve balancing and reduce treatment allocation bias. Consequently, it can additionally detect subgroups presenting positivity violations, exclude them, and provide a covariate-based definition of the target population we can infer from and generalize to. We evaluate the method's performance using synthetic and realistic datasets, explore its bias-interpretability tradeoff, and show that it is comparable with existing approaches.

Sam Dauncey, Chris Holmes, Christopher Williams et al.

Likelihood-based deep generative models such as score-based diffusion models and variational autoencoders are state-of-the-art machine learning models approximating high-dimensional distributions of data such as images, text, or audio. One of many downstream tasks they can be naturally applied to is out-of-distribution (OOD) detection. However, seminal work by Nalisnick et al. which we reproduce showed that deep generative models consistently infer higher log-likelihoods for OOD data than data they were trained on, marking an open problem. In this work, we analyse using the gradient of a data point with respect to the parameters of the deep generative model for OOD detection, based on the simple intuition that OOD data should have larger gradient norms than training data. We formalise measuring the size of the gradient as approximating the Fisher information metric. We show that the Fisher information matrix (FIM) has large absolute diagonal values, motivating the use of chi-square distributed, layer-wise gradient norms as features. We combine these features to make a simple, model-agnostic and hyperparameter-free method for OOD detection which estimates the joint density of the layer-wise gradient norms for a given data point. We find that these layer-wise gradient norms are weakly correlated, rendering their combined usage informative, and prove that the layer-wise gradient norms satisfy the principle of (data representation) invariance. Our empirical results indicate that this method outperforms the Typicality test for most deep generative models and image dataset pairings.

Oscar Clivio, Alexander D'Amour, Alexander Franks et al.

Overlap, also known as positivity, is a key condition for causal treatment effect estimation. Many popular estimators suffer from high variance and become brittle when features differ strongly across treatment groups. This is especially challenging in high dimensions: the curse of dimensionality can make overlap implausible. To address this, we propose a class of feature representations called deconfounding scores, which preserve both identification and the target of estimation; the classical propensity and prognostic scores are two special cases. We characterize the problem of finding a representation with better overlap as minimizing an overlap divergence under a deconfounding score constraint. We then derive closed-form expressions for a class of deconfounding scores under a broad family of generalized linear models with Gaussian features and show that prognostic scores are overlap-optimal within this class. We conduct extensive experiments to assess this behavior empirically.

Ziyu Wang, Chris Holmes

Applications of large language models often involve the generation of free-form responses, in which case uncertainty quantification becomes challenging. This is due to the need to identify task-specific uncertainties (e.g., about the semantics) which appears difficult to define in general cases. This work addresses these challenges from a perspective of Bayesian decision theory, starting from the assumption that our utility is characterized by a similarity measure that compares a generated response with a hypothetical true response. We discuss how this assumption enables principled quantification of the model's subjective uncertainty and its calibration. We further derive a measure for epistemic uncertainty, based on a missing data perspective and its characterization as an excess risk. The proposed methods can be applied to black-box language models. We illustrate the methods on question answering and machine translation tasks. Our experiments provide a principled evaluation of task-specific calibration, and demonstrate that epistemic uncertainty offers a promising deferral strategy for efficient data acquisition in in-context learning.

Fabian Falck, Ziyu Wang, Chris Holmes

In-context learning (ICL) has emerged as a particularly remarkable characteristic of Large Language Models (LLM): given a pretrained LLM and an observed dataset, LLMs can make predictions for new data points from the same distribution without fine-tuning. Numerous works have postulated ICL as approximately Bayesian inference, rendering this a natural hypothesis. In this work, we analyse this hypothesis from a new angle through the martingale property, a fundamental requirement of a Bayesian learning system for exchangeable data. We show that the martingale property is a necessary condition for unambiguous predictions in such scenarios, and enables a principled, decomposed notion of uncertainty vital in trustworthy, safety-critical systems. We derive actionable checks with corresponding theory and test statistics which must hold if the martingale property is satisfied. We also examine if uncertainty in LLMs decreases as expected in Bayesian learning when more data is observed. In three experiments, we provide evidence for violations of the martingale property, and deviations from a Bayesian scaling behaviour of uncertainty, falsifying the hypothesis that ICL is Bayesian.

Christopher Williams, Fabian Falck, George Deligiannidis et al.

U-Nets are a go-to, state-of-the-art neural architecture across numerous tasks for continuous signals on a square such as images and Partial Differential Equations (PDE), however their design and architecture is understudied. In this paper, we provide a framework for designing and analysing general U-Net architectures. We present theoretical results which characterise the role of the encoder and decoder in a U-Net, their high-resolution scaling limits and their conjugacy to ResNets via preconditioning. We propose Multi-ResNets, U-Nets with a simplified, wavelet-based encoder without learnable parameters. Further, we show how to design novel U-Net architectures which encode function constraints, natural bases, or the geometry of the data. In diffusion models, our framework enables us to identify that high-frequency information is dominated by noise exponentially faster, and show how U-Nets with average pooling exploit this. In our experiments, we demonstrate how Multi-ResNets achieve competitive and often superior performance compared to classical U-Nets in image segmentation, PDE surrogate modelling, and generative modelling with diffusion models. Our U-Net framework paves the way to study the theoretical properties of U-Nets and design natural, scalable neural architectures for a multitude of problems beyond the square.

Natalia Garcia Martin, Stefano Malacrino, Marta Wojciechowska et al.

Multiplexed immunofluorescence provides an unprecedented opportunity for studying specific cell-to-cell and cell microenvironment interactions. We employ graph neural networks to combine features obtained from tissue morphology with measurements of protein expression to profile the tumour microenvironment associated with different tumour stages. Our framework presents a new approach to analysing and processing these complex multi-dimensional datasets that overcomes some of the key challenges in analysing these data and opens up the opportunity to abstract biologically meaningful interactions.

Sahra Ghalebikesabi, Harrison Wilde, Jack Jewson et al.

Increasing interest in privacy-preserving machine learning has led to new and evolved approaches for generating private synthetic data from undisclosed real data. However, mechanisms of privacy preservation can significantly reduce the utility of synthetic data, which in turn impacts downstream tasks such as learning predictive models or inference. We propose several re-weighting strategies using privatised likelihood ratios that not only mitigate statistical bias of downstream estimators but also have general applicability to differentially private generative models. Through large-scale empirical evaluation, we show that private importance weighting provides simple and effective privacy-compliant augmentation for general applications of synthetic data.

Sahra Ghalebikesabi, Lucile Ter-Minassian, Karla Diaz-Ordaz et al.

Shapley values provide model agnostic feature attributions for model outcome at a particular instance by simulating feature absence under a global population distribution. The use of a global population can lead to potentially misleading results when local model behaviour is of interest. Hence we consider the formulation of neighbourhood reference distributions that improve the local interpretability of Shapley values. By doing so, we find that the Nadaraya-Watson estimator, a well-studied kernel regressor, can be expressed as a self-normalised importance sampling estimator. Empirically, we observe that Neighbourhood Shapley values identify meaningful sparse feature relevance attributions that provide insight into local model behaviour, complimenting conventional Shapley analysis. They also increase on-manifold explainability and robustness to the construction of adversarial classifiers.

Fabian Falck, Haoting Zhang, Matthew Willetts et al.

Work in deep clustering focuses on finding a single partition of data. However, high-dimensional data, such as images, typically feature multiple interesting characteristics one could cluster over. For example, images of objects against a background could be clustered over the shape of the object and separately by the colour of the background. In this paper, we introduce Multi-Facet Clustering Variational Autoencoders (MFCVAE), a novel class of variational autoencoders with a hierarchy of latent variables, each with a Mixture-of-Gaussians prior, that learns multiple clusterings simultaneously, and is trained fully unsupervised and end-to-end. MFCVAE uses a progressively-trained ladder architecture which leads to highly stable performance. We provide novel theoretical results for optimising the ELBO analytically with respect to the categorical variational posterior distribution, correcting earlier influential theoretical work. On image benchmarks, we demonstrate that our approach separates out and clusters over different aspects of the data in a disentangled manner. We also show other advantages of our model: the compositionality of its latent space and that it provides controlled generation of samples.

Sahra Ghalebikesabi, Rob Cornish, Luke J. Kelly et al.

We propose a variational autoencoder architecture to model both ignorable and nonignorable missing data using pattern-set mixtures as proposed by Little (1993). Our model explicitly learns to cluster the missing data into missingness pattern sets based on the observed data and missingness masks. Underpinning our approach is the assumption that the data distribution under missingness is probabilistically semi-supervised by samples from the observed data distribution. Our setup trades off the characteristics of ignorable and nonignorable missingness and can thus be applied to data of both types. We evaluate our method on a wide range of data sets with different types of missingness and achieve state-of-the-art imputation performance. Our model outperforms many common imputation algorithms, especially when the amount of missing data is high and the missingness mechanism is nonignorable.

Alexander Camuto, Xiaoyu Wang, Lingjiong Zhu et al.

Gaussian noise injections (GNIs) are a family of simple and widely-used regularisation methods for training neural networks, where one injects additive or multiplicative Gaussian noise to the network activations at every iteration of the optimisation algorithm, which is typically chosen as stochastic gradient descent (SGD). In this paper we focus on the so-called `implicit effect' of GNIs, which is the effect of the injected noise on the dynamics of SGD. We show that this effect induces an asymmetric heavy-tailed noise on SGD gradient updates. In order to model this modified dynamics, we first develop a Langevin-like stochastic differential equation that is driven by a general family of asymmetric heavy-tailed noise. Using this model we then formally prove that GNIs induce an `implicit bias', which varies depending on the heaviness of the tails and the level of asymmetry. Our empirical results confirm that different types of neural networks trained with GNIs are well-modelled by the proposed dynamics and that the implicit effect of these injections induces a bias that degrades the performance of networks.

Harrison Wilde, Jack Jewson, Sebastian Vollmer et al.

There is significant growth and interest in the use of synthetic data as an enabler for machine learning in environments where the release of real data is restricted due to privacy or availability constraints. Despite a large number of methods for synthetic data generation, there are comparatively few results on the statistical properties of models learnt on synthetic data, and fewer still for situations where a researcher wishes to augment real data with another party's synthesised data. We use a Bayesian paradigm to characterise the updating of model parameters when learning in these settings, demonstrating that caution should be taken when applying conventional learning algorithms without appropriate consideration of the synthetic data generating process and learning task. Recent results from general Bayesian updating support a novel and robust approach to Bayesian synthetic-learning founded on decision theory that outperforms standard approaches across repeated experiments on supervised learning and inference problems.

Alexander Camuto, Matthew Willetts, Umut Şimşekli et al.

We study the regularisation induced in neural networks by Gaussian noise injections (GNIs). Though such injections have been extensively studied when applied to data, there have been few studies on understanding the regularising effect they induce when applied to network activations. Here we derive the explicit regulariser of GNIs, obtained by marginalising out the injected noise, and show that it penalises functions with high-frequency components in the Fourier domain; particularly in layers closer to a neural network's output. We show analytically and empirically that such regularisation produces calibrated classifiers with large classification margins.

Alexander Camuto, Matthew Willetts, Stephen Roberts et al.

We make inroads into understanding the robustness of Variational Autoencoders (VAEs) to adversarial attacks and other input perturbations. While previous work has developed algorithmic approaches to attacking and defending VAEs, there remains a lack of formalization for what it means for a VAE to be robust. To address this, we develop a novel criterion for robustness in probabilistic models: $r$-robustness. We then use this to construct the first theoretical results for the robustness of VAEs, deriving margins in the input space for which we can provide guarantees about the resulting reconstruction. Informally, we are able to define a region within which any perturbation will produce a reconstruction that is similar to the original reconstruction. To support our analysis, we show that VAEs trained using disentangling methods not only score well under our robustness metrics, but that the reasons for this can be interpreted through our theoretical results.

Matthew Willetts, Xenia Miscouridou, Stephen Roberts et al.

Successfully training Variational Autoencoders (VAEs) with a hierarchy of discrete latent variables remains an area of active research. Vector-Quantised VAEs are a powerful approach to discrete VAEs, but naive hierarchical extensions can be unstable when training. Leveraging insights from classical methods of inference we introduce \textit{Relaxed-Responsibility Vector-Quantisation}, a novel way to parameterise discrete latent variables, a refinement of relaxed Vector-Quantisation that gives better performance and more stable training. This enables a novel approach to hierarchical discrete variational autoencoders with numerous layers of latent variables (here up to 32) that we train end-to-end. Within hierarchical probabilistic deep generative models with discrete latent variables trained end-to-end, we achieve state-of-the-art bits-per-dim results for various standard datasets. % Unlike discrete VAEs with a single layer of latent variables, we can produce samples by ancestral sampling: it is not essential to train a second autoregressive generative model over the learnt latent representations to then sample from and then decode. % Moreover, that latter approach in these deep hierarchical models would require thousands of forward passes to generate a single sample. Further, we observe different layers of our model become associated with different aspects of the data.

Tom Lovett, Mark Briers, Marcos Charalambides et al.

The Covid-19 pandemic has resulted in a variety of approaches for managing infection outbreaks in international populations. One example is mobile phone applications, which attempt to alert infected individuals and their contacts by automatically inferring two key components of infection risk: the proximity to an individual who may be infected, and the duration of proximity. The former component, proximity, relies on Bluetooth Low Energy (BLE) Received Signal Strength Indicator(RSSI) as a distance sensor, and this has been shown to be problematic; not least because of unpredictable variations caused by different device types, device location on-body, device orientation, the local environment and the general noise associated with radio frequency propagation. In this paper, we present an approach that infers posterior probabilities over distance given sequences of RSSI values. Using a single-dimensional Unscented Kalman Smoother (UKS) for non-linear state space modelling, we outline several Gaussian process observation transforms, including: a generative model that directly captures sources of variation; and a discriminative model that learns a suitable observation function from training data using both distance and infection risk as optimisation objective functions. Our results show that good risk prediction can be achieved in $\mathcal{O}(n)$ time on real-world data sets, with the UKS outperforming more traditional classification methods learned from the same training data.

Alexander Camuto, Matthew Willetts, Brooks Paige et al.

Separating high-dimensional data like images into independent latent factors, i.e independent component analysis (ICA), remains an open research problem. As we show, existing probabilistic deep generative models (DGMs), which are tailor-made for image data, underperform on non-linear ICA tasks. To address this, we propose a DGM which combines bijective feature maps with a linear ICA model to learn interpretable latent structures for high-dimensional data. Given the complexities of jointly training such a hybrid model, we introduce novel theory that constrains linear ICA to lie close to the manifold of orthogonal rectangular matrices, the Stiefel manifold. By doing so we create models that converge quickly, are easy to train, and achieve better unsupervised latent factor discovery than flow-based models, linear ICA, and Variational Autoencoders on images.

Matthew Willetts, Alexander Camuto, Stephen Roberts et al.

We develop a new method for regularising neural networks. We learn a probability distribution over the activations of all layers of the model and then insert imputed values into the network during training. We obtain a posterior for an arbitrary subset of activations conditioned on the remainder. This is a generalisation of data augmentation to the hidden layers of a network, and a form of data-aware dropout. We demonstrate that our training method leads to higher test accuracy and lower test-set cross-entropy for neural networks trained on CIFAR-10 and SVHN compared to standard regularisation baselines: our approach leads to networks with better calibrated uncertainty over the class posteriors all the while delivering greater test-set accuracy.

Matthew Willetts, Stephen Roberts, Chris Holmes

In clustering we normally output one cluster variable for each datapoint. However it is not necessarily the case that there is only one way to partition a given dataset into cluster components. For example, one could cluster objects by their colour, or by their type. Different attributes form a hierarchy, and we could wish to cluster in any of them. By disentangling the learnt latent representations of some dataset into different layers for different attributes we can then cluster in those latent spaces. We call this "disentangled clustering". Extending Variational Ladder Autoencoders (Zhao et al., 2017), we propose a clustering algorithm, VLAC, that outperforms a Gaussian Mixture DGM in cluster accuracy over digit identity on the test set of SVHN. We also demonstrate learning clusters jointly over numerous layers of the hierarchy of latent variables for the data, and show component-wise generation from this hierarchical model.

Matthew Willetts, Alexander Camuto, Tom Rainforth et al.

Variational autoencoders (VAEs) have recently been shown to be vulnerable to adversarial attacks, wherein they are fooled into reconstructing a chosen target image. However, how to defend against such attacks remains an open problem. We make significant advances in addressing this issue by introducing methods for producing adversarially robust VAEs. Namely, we first demonstrate that methods proposed to obtain disentangled latent representations produce VAEs that are more robust to these attacks. However, this robustness comes at the cost of reducing the quality of the reconstructions. We ameliorate this by applying disentangling methods to hierarchical VAEs. The resulting models produce high-fidelity autoencoders that are also adversarially robust. We confirm their capabilities on several different datasets and with current state-of-the-art VAE adversarial attacks, and also show that they increase the robustness of downstream tasks to attack.

Edwin Fong, Chris Holmes

In Bayesian statistics, the marginal likelihood, also known as the evidence, is used to evaluate model fit as it quantifies the joint probability of the data under the prior. In contrast, non-Bayesian models are typically compared using cross-validation on held-out data, either through $k$-fold partitioning or leave-$p$-out subsampling. We show that the marginal likelihood is formally equivalent to exhaustive leave-$p$-out cross-validation averaged over all values of $p$ and all held-out test sets when using the log posterior predictive probability as the scoring rule. Moreover, the log posterior predictive is the only coherent scoring rule under data exchangeability. This offers new insight into the marginal likelihood and cross-validation and highlights the potential sensitivity of the marginal likelihood to the choice of the prior. We suggest an alternative approach using cumulative cross-validation following a preparatory training phase. Our work has connections to prequential analysis and intrinsic Bayes factors but is motivated through a different course.

Edwin Fong, Simon Lyddon, Chris Holmes

Increasingly complex datasets pose a number of challenges for Bayesian inference. Conventional posterior sampling based on Markov chain Monte Carlo can be too computationally intensive, is serial in nature and mixes poorly between posterior modes. Further, all models are misspecified, which brings into question the validity of the conventional Bayesian update. We present a scalable Bayesian nonparametric learning routine that enables posterior sampling through the optimization of suitably randomized objective functions. A Dirichlet process prior on the unknown data distribution accounts for model misspecification, and admits an embarrassingly parallel posterior bootstrap algorithm that generates independent and exact samples from the nonparametric posterior distribution. Our method is particularly adept at sampling from multimodal posterior distributions via a random restart mechanism. We demonstrate our method on Gaussian mixture model and sparse logistic regression examples.

Sebastian Vollmer, Bilal A. Mateen, Gergo Bohner et al.

Machine learning (ML), artificial intelligence (AI) and other modern statistical methods are providing new opportunities to operationalize previously untapped and rapidly growing sources of data for patient benefit. Whilst there is a lot of promising research currently being undertaken, the literature as a whole lacks: transparency; clear reporting to facilitate replicability; exploration for potential ethical concerns; and, clear demonstrations of effectiveness. There are many reasons for why these issues exist, but one of the most important that we provide a preliminary solution for here is the current lack of ML/AI- specific best practice guidance. Although there is no consensus on what best practice looks in this field, we believe that interdisciplinary groups pursuing research and impact projects in the ML/AI for health domain would benefit from answering a series of questions based on the important issues that exist when undertaking work of this nature. Here we present 20 questions that span the entire project life cycle, from inception, data analysis, and model evaluation, to implementation, as a means to facilitate project planning and post-hoc (structured) independent evaluation. By beginning to answer these questions in different settings, we can start to understand what constitutes a good answer, and we expect that the resulting discussion will be central to developing an international consensus framework for transparent, replicable, ethical and effective research in artificial intelligence (AI-TREE) for health.

Matthew Willetts, Aiden Doherty, Stephen Roberts et al.

We introduce 'semi-unsupervised learning', a problem regime related to transfer learning and zero-shot learning where, in the training data, some classes are sparsely labelled and others entirely unlabelled. Models able to learn from training data of this type are potentially of great use as many real-world datasets are like this. Here we demonstrate a new deep generative model for classification in this regime. Our model, a Gaussian mixture deep generative model, demonstrates superior semi-unsupervised classification performance on MNIST to model M2 from Kingma and Welling (2014). We apply the model to human accelerometer data, performing activity classification and structure discovery on windows of time series data.

Simon Lyddon, Chris Holmes, Stephen Walker

In this paper we revisit the weighted likelihood bootstrap, a method that generates samples from an approximate Bayesian posterior of a parametric model. We show that the same method can be derived, without approximation, under a Bayesian nonparametric model with the parameter of interest defined as minimising an expected negative log-likelihood under an unknown sampling distribution. This interpretation enables us to extend the weighted likelihood bootstrap to posterior sampling for parameters minimizing an expected loss. We call this method the loss-likelihood bootstrap. We make a connection between this and general Bayesian updating, which is a way of updating prior belief distributions without needing to construct a global probability model, yet requires the calibration of two forms of loss function. The loss-likelihood bootstrap is used to calibrate the general Bayesian posterior by matching asymptotic Fisher information. We demonstrate the methodology on a number of examples.

Rémi Bardenet, Arnaud Doucet, Chris Holmes

Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis-Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algorithms. The aims of this article are as follows. First, we present a comprehensive review of the existing literature, commenting on the underlying assumptions and theoretical guarantees of each method. Second, by leveraging our understanding of these limitations, we propose an original subsampling-based approach which samples from a distribution provably close to the posterior distribution of interest, yet can require less than $O(n)$ data point likelihood evaluations at each iteration for certain statistical models in favourable scenarios. Finally, we have only been able so far to propose subsampling-based methods which display good performance in scenarios where the Bernstein-von Mises approximation of the target posterior distribution is excellent. It remains an open challenge to develop such methods in scenarios where the Bernstein-von Mises approximation is poor.