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1. On Uncertainty, Tempering, and Data Augmentation in Bayesian Classification

   Kapoor, Sanyam; Maddox, Wesley; Izmailov, Pavel; Wilson, Andrew Gordon (October 2022, NeurIPS)

   Aleatoric uncertainty captures the inherent randomness of the data, such as measurement noise. In Bayesian regression, we often use a Gaussian observation model, where we control the level of aleatoric uncertainty with a noise variance parameter. By contrast, for Bayesian classification we use a categorical distribution with no mechanism to represent our beliefs about aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric uncertainty significantly improves the performance of Bayesian neural networks. We note that many standard benchmarks, such as CIFAR-10, have essentially no aleatoric uncertainty. Moreover, we show that data augmentation in approximate inference softens the likelihood, leading to underconfidence and misrepresenting our beliefs about aleatoric uncertainty. Accordingly, we find that a cold posterior, tempered by a power greater than one, often more honestly reflects our beliefs about aleatoric uncertainty than no tempering --- providing an explicit link between data augmentation and cold posteriors. We further show that we can match or exceed the performance of posterior tempering by using a Dirichlet observation model, where we explicitly control the level of aleatoric uncertainty, without any need for tempering. 

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2. A tutorial on the Bayesian statistical approach to inverse problems

   https://doi.org/10.1063/5.0154773  

   Waqar, Faaiq_G; Patel, Swati; Simon, Cory_M (November 2023, APL Machine Learning)

   Inverse problems are ubiquitous in science and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input–output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a “solution” that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our Tutorial aims to reach a wide range of scientists and engineers. 

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3. Kernel Interpolation for Scalable Online Gaussian Processes

   Stanton, S.; Maddox, W.J.; Delbridge, I.; Wilson, A.G. (January 2021, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics)

   null (Ed.)

   Gaussian processes (GPs) provide a gold standard for performance in online settings, such as sample-efficient control and black box optimization, where we need to update a posterior distribution as we acquire data in a sequential fashion. However, updating a GP posterior to accommodate even a single new observation after having observed n points incurs at least O(n) computations in the exact setting. We show how to use structured kernel interpolation to efficiently recycle computations for constant-time O(1) online updates with respect to the number of points n, while retaining exact inference. We demonstrate the promise of our approach in a range of online regression and classification settings, Bayesian optimization, and active sampling to reduce error in malaria incidence forecasting. 

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4. Estimating Epistemic and Aleatoric Uncertainty with a Single Model

   Chan, Matthew A; Molina, Maria J; Metzler, Christopher A (December 2024, Advances in neural information processing systems)

   Estimating and disentangling epistemic uncertainty, uncertainty that is reducible with more training data, and aleatoric uncertainty, uncertainty that is inherent to the task at hand, is critically important when applying machine learning to highstakes applications such as medical imaging and weather forecasting. Conditional diffusion models’ breakthrough ability to accurately and efficiently sample from the posterior distribution of a dataset now makes uncertainty estimation conceptually straightforward: One need only train and sample from a large ensemble of diffusion models. Unfortunately, training such an ensemble becomes computationally intractable as the complexity of the model architecture grows. In this work we introduce a new approach to ensembling, hyper-diffusion models (HyperDM), which allows one to accurately estimate both epistemic and aleatoric uncertainty with a single model. Unlike existing single-model uncertainty methods like Monte-Carlo dropout and Bayesian neural networks, HyperDM offers prediction accuracy on par with, and in some cases superior to, multi-model ensembles. Furthermore, our proposed approach scales to modern network architectures such as Attention U-Net and yields more accurate uncertainty estimates compared to existing methods. We validate our method on two distinct real-world tasks: x-ray computed tomography reconstruction and weather temperature forecasting. Source code is publicly available at https://github.com/matthewachan/hyperdm. 

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5. Fairness through Aleatoric Uncertainty

   https://doi.org/10.1145/3583780.3614875  

   Tahir, Anique; Cheng, Lu; Liu, Huan (October 2023, ACM)

   We propose a simple yet effective solution to tackle the often-competing goals of fairness and utility in classification tasks. While fairness ensures that the model's predictions are unbiased and do not discriminate against any particular group or individual, utility focuses on maximizing the model's predictive performance. This work introduces the idea of leveraging aleatoric uncertainty (e.g., data ambiguity) to improve the fairness-utility trade-off. Our central hypothesis is that aleatoric uncertainty is a key factor for algorithmic fairness and samples with low aleatoric uncertainty are modeled more accurately and fairly than those with high aleatoric uncertainty. We then propose a principled model to improve fairness when aleatoric uncertainty is high and improve utility elsewhere. Our approach first intervenes in the data distribution to better decouple aleatoric uncertainty and epistemic uncertainty. It then introduces a fairness-utility bi-objective loss defined based on the estimated aleatoric uncertainty. Our approach is theoretically guaranteed to improve the fairness-utility trade-off. Experimental results on both tabular and image datasets show that the proposed approach outperforms state-of-the-art methods w.r.t. the fairness-utility trade-off and w.r.t. both group and individual fairness metrics. This work presents a fresh perspective on the trade-off between utility and algorithmic fairness and opens a key avenue for the potential of using prediction uncertainty in fair machine learning. 

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