On Uncertainty Quantification for Near-Bayes Optimal Algorithms

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.

Approximations to the Fisher Information Metric of Deep Generative Models for Out-Of-Distribution Detection

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.

Hierarchical Bias-Driven Stratification for Interpretable Causal Effect Estimation

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.

Explainable AI for survival analysis: a median-SHAP approach

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.

Targeting Relative Risk Heterogeneity with Causal Forests

Treatment effect heterogeneity (TEH), or variability in treatment effect for different subgroups within a population, is of significant interest in clinical trial analysis. Causal forests (Wager and Athey, 2018) is a highly popular method for this problem, but like many other methods for detecting TEH, its criterion for separating subgroups focuses on differences in absolute risk. This can dilute statistical power by masking nuance 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 generalized linear model (GLM) comparison. We present results on simulated and real-world data that suggest relative risk causal forests can capture otherwise unobserved sources of heterogeneity.

Differentially Private Statistical Inference through $β$-Divergence One Posterior Sampling

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 $\beta$D-Bayes, a posterior sampling scheme from a generalised posterior targeting the minimisation of the $\beta$-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 $\beta$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.

PWSHAP: A Path-Wise Explanation Model for Targeted Variables

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.

A Unified Framework for U-Net Design and Analysis

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.

Learning from data with structured missingness

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.

A Multi-Resolution Framework for U-Nets with Applications to Hierarchical VAEs

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.