We establish a general Bernstein--von Mises theorem for approximately linear semiparametric functionals of fractional posterior distributions based on nonparametric priors. This is illustrated in a number of nonparametric settings and for different classes of prior distributions, including Gaussian process priors. We show that fractional posterior credible sets can provide reliable semiparametric uncertainty quantification, but have inflated size. To remedy this, we further propose a \textit{shifted-and-rescaled} fractional posterior set that is an efficient confidence set having optimal size under regularity conditions. As part of our proofs, we also refine existing contraction rate results for fractional posteriors by sharpening the dependence of the rate on the fractional exponent.

Stochastic evolution equations describing the dynamics of systems under the influence of both deterministic and stochastic forces are prevalent in all fields of science. Yet, identifying these systems from sparse-in-time observations remains still a challenging endeavour. Existing approaches focus either on the temporal structure of the observations by relying on conditional expectations, discarding thereby information ingrained in the geometry of the system's invariant density; or employ geometric approximations of the invariant density, which are nevertheless restricted to systems with conservative forces. Here we propose a method that reconciles these two paradigms. We introduce a new data-driven path augmentation scheme that takes the local observation geometry into account. By employing non-parametric inference on the augmented paths, we can efficiently identify the deterministic driving forces of the underlying system for systems observed at low sampling rates.

Machine learning currently plays an increasingly important role in people's lives in areas such as credit scoring, auto-driving, disease diagnosing, and insurance quoting. However, in many of these areas, machine learning models have performed unfair behaviors against some sub-populations, such as some particular groups of race, sex, and age. These unfair behaviors can be on account of the pre-existing bias in the training dataset due to historical and social factors. In this paper, we focus on a real-world application of credit scoring and construct a fair prediction model by introducing latent variables to remove the correlation between protected attributes, such as sex and age, with the observable feature inputs, including house and job. For detailed implementation, we apply Bayesian approaches, including the Markov Chain Monte Carlo simulation, to estimate our proposed fair model.

Since the meaning representations are detailed and accurate annotations which express fine-grained sequence-level semtantics, it is usually hard to train discriminative semantic parsers via Maximum Likelihood Estimation (MLE) in an autoregressive fashion. In this paper, we propose a semantic-aware contrastive learning algorithm, which can learn to distinguish fine-grained meaning representations and take the overall sequence-level semantic into consideration. Specifically, a multi-level online sampling algorithm is proposed to sample confusing and diverse instances. Three semantic-aware similarity functions are designed to accurately measure the distance between meaning representations as a whole. And a ranked contrastive loss is proposed to pull the representations of the semantic-identical instances together and push negative instances away. Experiments on two standard datasets show that our approach achieves significant improvements over MLE baselines and gets state-of-the-art performances by simply applying semantic-aware contrastive learning on a vanilla Seq2Seq model.

Bayesian inference for high-dimensional inverse problems is computationally costly and requires selecting a suitable prior distribution. Amortized variational inference addresses these challenges via a neural network that approximates the posterior distribution not only for one instance of data, but a distribution of data pertaining to a specific inverse problem. During inference, the neural network -- in our case a conditional normalizing flow -- provides posterior samples at virtually no cost. However, the accuracy of amortized variational inference relies on the availability of high-fidelity training data, which seldom exists in geophysical inverse problems due to the Earth's heterogeneity. In addition, the network is prone to errors if evaluated over out-of-distribution data. As such, we propose to increase the resilience of amortized variational inference in the presence of moderate data distribution shifts. We achieve this via a correction to the latent distribution that improves the posterior distribution approximation for the data at hand. The correction involves relaxing the standard Gaussian assumption on the latent distribution and parameterizing it via a Gaussian distribution with an unknown mean and (diagonal) covariance. These unknowns are then estimated by minimizing the Kullback-Leibler divergence between the corrected and the (physics-based) true posterior distributions. While generic and applicable to other inverse problems, by means of a linearized seismic imaging example, we show that our correction step improves the robustness of amortized variational inference with respect to changes in the number of seismic sources, noise variance, and shifts in the prior distribution. This approach provides a seismic image with limited artifacts and an assessment of its uncertainty at approximately the same cost as five reverse-time migrations.

Partial differential equations (PDEs) are widely used for description of physical and engineering phenomena. Some key parameters involved in PDEs, which represents certain physical properties with important scientific interpretations, are difficult or even impossible to be measured directly. Estimation of these parameters from noisy and sparse experimental data of related physical quantities is an important task. Many methods for PDE parameter inference involve a large number of evaluations of numerical solution of PDE through algorithms such as finite element method, which can be time-consuming especially for nonlinear PDEs. In this paper, we propose a novel method for estimating unknown parameters in PDEs, called PDE-Informed Gaussian Process Inference (PIGPI). Through modeling the PDE solution as a Gaussian process (GP), we derive the manifold constraints induced by the (linear) PDE structure such that under the constraints, the GP satisfies the PDE. For nonlinear PDEs, we propose an augmentation method that transfers the nonlinear PDE into an equivalent PDE system linear in all derivatives that our PIGPI can handle. PIGPI can be applied to multi-dimensional PDE systems and PDE systems with unobserved components. The method completely bypasses the numerical solver for PDE, thus achieving drastic savings in computation time, especially for nonlinear PDEs. Moreover, the PIGPI method can give the uncertainty quantification for both the unknown parameters and the PDE solution. The proposed method is demonstrated by several application examples from different areas.

Gaussian process regression in its most simplified form assumes normal homoscedastic noise and utilizes analytically tractable mean and covariance functions of predictive posterior distribution using Gaussian conditioning. Its hyperparameters are estimated by maximizing the evidence, commonly known as type II maximum likelihood estimation. Unfortunately, Bayesian inference based on Gaussian likelihood is not robust to outliers, which are often present in the observational training data sets. To overcome this problem, we propose a robust process model in the Gaussian process framework with the likelihood of observed data expressed as the Huber probability distribution. The proposed model employs weights based on projection statistics to scale residuals and bound the influence of vertical outliers and bad leverage points on the latent functions estimates while exhibiting a high statistical efficiency at the Gaussian and thick tailed noise distributions. The proposed method is demonstrated by two real world problems and two numerical examples using datasets with additive errors following thick tailed distributions such as Students t, Laplace, and Cauchy distribution.

Many types of data from fields including natural language processing, computer vision, and bioinformatics, are well represented by discrete, compositional structures such as trees, sequences, or matchings. Latent structure models are a powerful tool for learning to extract such representations, offering a way to incorporate structural bias, discover insight about the data, and interpret decisions. However, effective training is challenging, as neural networks are typically designed for continuous computation. This text explores three broad strategies for learning with discrete latent structure: continuous relaxation, surrogate gradients, and probabilistic estimation. Our presentation relies on consistent notations for a wide range of models. As such, we reveal many new connections between latent structure learning strategies, showing how most consist of the same small set of fundamental building blocks, but use them differently, leading to substantially different applicability and properties.

Neural Networks are infamously sensitive to small perturbations in their inputs, making them vulnerable to adversarial attacks. This project evaluates the performance of Denoising Diffusion Probabilistic Models (DDPM) as a purification technique to defend against adversarial attacks. This works by adding noise to an adversarial example before removing it through the reverse process of the diffusion model. We evaluate the approach on the PatchCamelyon data set for histopathologic scans of lymph node sections and find an improvement of the robust accuracy by up to 88\% of the original model's accuracy, constituting a considerable improvement over the vanilla model and our baselines. The project code is located at https://github.com/ankile/Adversarial-Diffusion.

Representing probability distributions by the gradient of their density functions has proven effective in modeling a wide range of continuous data modalities. However, this representation is not applicable in discrete domains where the gradient is undefined. To this end, we propose an analogous score function called the "Concrete score", a generalization of the (Stein) score for discrete settings. Given a predefined neighborhood structure, the Concrete score of any input is defined by the rate of change of the probabilities with respect to local directional changes of the input. This formulation allows us to recover the (Stein) score in continuous domains when measuring such changes by the Euclidean distance, while using the Manhattan distance leads to our novel score function in discrete domains. Finally, we introduce a new framework to learn such scores from samples called Concrete Score Matching (CSM), and propose an efficient training objective to scale our approach to high dimensions. Empirically, we demonstrate the efficacy of CSM on density estimation tasks on a mixture of synthetic, tabular, and high-dimensional image datasets, and demonstrate that it performs favorably relative to existing baselines for modeling discrete data.