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1. Conformal Frequency Estimation with Sketched Data

   Sesia, Matteo; Favaro, Stefano (July 2022, Advances in neural information processing systems)

   A flexible conformal inference method is developed to construct confidence intervals for the frequencies of queried objects in very large data sets, based on a much smaller sketch of those data. The approach is data-adaptive and requires no knowledge of the data distribution or of the details of the sketching algorithm; instead, it constructs provably valid frequentist confidence intervals under the sole assumption of data exchangeability. Although our solution is broadly applicable, this paper focuses on applications involving the count-min sketch algorithm and a non-linear variation thereof. The performance is compared to that of frequentist and Bayesian alternatives through simulations and experiments with data sets of SARS-CoV-2 DNA sequences and classic English literature. 

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2. Exact Inference with Approximate Computation for Differentially Private Data via Perturbations

   https://doi.org/10.29012/jpc.797  

   Gong, Ruobin (November 2022, Journal of Privacy and Confidentiality)

   This paper discusses how two classes of approximate computation algorithms can be adapted, in a modular fashion, to achieve exact statistical inference from differentially private data products. Considered are approximate Bayesian computation for Bayesian inference, and Monte Carlo Expectation-Maximization for likelihood inference. Up to Monte Carlo error, inference from these algorithms is exact with respect to the joint specification of both the analyst's original data model, and the curator's differential privacy mechanism. Highlighted is a duality between approximate computation on exact data, and exact computation on approximate data, which can be leveraged by a well-designed computational procedure for statistical inference. 

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3. Efficient derivative-free Bayesian inference for large-scale inverse problems

   https://doi.org/10.1088/1361-6420/ac99fa  

   Huang, Daniel Zhengyu; Huang, Jiaoyang; Reich, Sebastian; Stuart, Andrew M (October 2022, Inverse Problems)

   Abstract We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model. This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O ( 1 0 4 ) model runs, or more. Moreover, the forward model is often given as a black box or is impractical to differentiate. Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems. The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system, into which the inverse problem is embedded as an observation operator. Theoretical properties are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it. This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior. Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach. The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model. Moreover, the stochastic ensemble Kalman filter and various ensemble square-root Kalman filters are all employed and are compared numerically. The results demonstrate that the proposed method, based on exponential convergence to the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems. 

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4. Scalable Gaussian Process Inference with Finite-data Mean and Variance Guarantees

   Huggins, Jonathan H.; Campbell, Trevor; Kasprzak, Mikolaj; Broderick, Tamara (April 2019, Proceedings of Machine Learning Research)

   Gaussian processes (GPs) offer a flexible class of priors for nonparametric Bayesian regression, but popular GP posterior inference methods are typically prohibitively slow or lack desirable finite-data guarantees on quality. We develop a scalable approach to approximate GP regression, with finite-data guarantees on the accuracy of our pointwise posterior mean and variance estimates. Our main contribution is a novel objective for approximate inference in the nonparametric setting: the preconditioned Fisher (pF) divergence. We show that unlike the Kullback–Leibler divergence (used in variational inference), the pF divergence bounds bounds the 2-Wasserstein distance, which in turn provides tight bounds on the pointwise error of mean and variance estimates. We demonstrate that, for sparse GP likelihood approximations, we can minimize the pF divergence bounds efficiently. Our experiments show that optimizing the pF divergence bounds has the same computational requirements as variational sparse GPs while providing comparable empirical performance—in addition to our novel finite-data quality guarantees. 

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5. A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows

   https://doi.org/10.1016/j.cma.2023.116682  

   Dasgupta, Agnimitra; Patel, Dhruv V; Ray, Deep; Johnson, Erik A; Oberai, Assad A (February 2024, Computer Methods in Applied Mechanics and Engineering)

   We propose a novel modular inference approach combining two different generative models — generative adversarial networks (GAN) and normalizing flows — to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. GAN-Flow does not involve Markov chain Monte Carlo simulation, making it particularly suitable for solving large-scale inverse problems. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows. Notably, one of the applications we consider involves a 65,536-dimensional inverse problem of phase retrieval wherein an object is reconstructed from sparse noisy measurements of the magnitude of its Fourier transform. 

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