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Missing data arise in most applied settings and are ubiquitous in electronic health records (EHR). When data are missing not at random (MNAR) with respect to measured covariates, sensitivity analyses are often considered. These post-hoc solutions, however, are often unsatisfying in that they are not guaranteed to yield concrete conclusions. Motivated by an EHR-based study of long-term outcomes following bariatric surgery, we consider the use of double sampling as a means to mitigate MNAR outcome data when the statistical goals are estimation and inference regarding causal effects. We describe assumptions that are sufficient for the identification of the joint distribution of confounders, treatment, and outcome under this design. Additionally, we derive efficient and robust estimators of the average causal treatment effect under a nonparametric model and under a model assuming outcomes were, in fact, initially missing at random (MAR). We compare these in simulations to an approach that adaptively estimates based on evidence of violation of the MAR assumption. Finally, we also show that the proposed double sampling design can be extended to handle arbitrary coarsening mechanisms, and derive nonparametric efficient estimators of any smooth full data functional.

In this paper, we provide a novel framework for the analysis of generalization error of first-order optimization algorithms for statistical learning when the gradient can only be accessed through partial observations given by an oracle. Our analysis relies on the regularity of the gradient w.r.t. the data samples, and allows to derive near matching upper and lower bounds for the generalization error of multiple learning problems, including supervised learning, transfer learning, robust learning, distributed learning and communication efficient learning using gradient quantization. These results hold for smooth and strongly-convex optimization problems, as well as smooth non-convex optimization problems verifying a Polyak-Lojasiewicz assumption. In particular, our upper and lower bounds depend on a novel quantity that extends the notion of conditional standard deviation, and is a measure of the extent to which the gradient can be approximated by having access to the oracle. As a consequence, our analysis provides a precise meaning to the intuition that optimization of the statistical learning objective is as hard as the estimation of its gradient. Finally, we show that, in the case of standard supervised learning, mini-batch gradient descent with increasing batch sizes and a warm start can reach a generalization error that is optimal up to a multiplicative factor, thus motivating the use of this optimization scheme in practical applications.

With the rise of Extended Reality (XR) technology, there is a growing need for real-time light field generation from sparse view inputs. Existing methods can be classified into offline techniques, which can generate high-quality novel views but at the cost of long inference/training time, and online methods, which either lack generalizability or produce unsatisfactory results. However, we have observed that the intrinsic sparse manifold of Multi-plane Images (MPI) enables a significant acceleration of light field generation while maintaining rendering quality. Based on this insight, we introduce EffLiFe, a novel light field optimization method, which leverages the proposed Hierarchical Sparse Gradient Descent (HSGD) to produce high-quality light fields from sparse view images in real time. Technically, the coarse MPI of a scene is first generated using a 3D CNN, and it is further sparsely optimized by focusing only on important MPI gradients in a few iterations. Nevertheless, relying solely on optimization can lead to artifacts at occlusion boundaries. Therefore, we propose an occlusion-aware iterative refinement module that removes visual artifacts in occluded regions by iteratively filtering the input. Extensive experiments demonstrate that our method achieves comparable visual quality while being 100x faster on average than state-of-the-art offline methods and delivering better performance (about 2 dB higher in PSNR) compared to other online approaches.

We introduce a sufficient graphical model by applying the recently developed nonlinear sufficient dimension reduction techniques to the evaluation of conditional independence. The graphical model is nonparametric in nature, as it does not make distributional assumptions such as the Gaussian or copula Gaussian assumptions. However, unlike a fully nonparametric graphical model, which relies on the high-dimensional kernel to characterize conditional independence, our graphical model is based on conditional independence given a set of sufficient predictors with a substantially reduced dimension. In this way we avoid the curse of dimensionality that comes with a high-dimensional kernel. We develop the population-level properties, convergence rate, and variable selection consistency of our estimate. By simulation comparisons and an analysis of the DREAM 4 Challenge data set, we demonstrate that our method outperforms the existing methods when the Gaussian or copula Gaussian assumptions are violated, and its performance remains excellent in the high-dimensional setting.

Inverse Uncertainty Quantification (IUQ) method has been widely used to quantify the uncertainty of Physical Model Parameters (PMPs) in nuclear Thermal Hydraulics (TH) systems. This paper introduces a novel hierarchical Bayesian model which aims to mitigate two existing challenges in IUQ: the high variability of PMPs under varying experimental conditions, and unknown model discrepancies or outliers causing over-fitting issues. The proposed hierarchical model is compared with the conventional single-level Bayesian model using TRACE code and the measured void fraction data in the BFBT benchmark. A Hamiltonian Monte Carlo Method - No U-Turn Sampler (NUTS) is used for posterior sampling. The results demonstrate the effectiveness of the proposed hierarchical model in providing better estimates of the posterior distributions of PMPs and being less prone to over-fitting. The proposed method also demonstrates a promising approach for generalizing IUQ to larger databases with broad ranges of experimental conditions.

We consider the problem of uncertainty quantification in change point regressions, where the signal can be piecewise polynomial of arbitrary but fixed degree. That is we seek disjoint intervals which, uniformly at a given confidence level, must each contain a change point location. We propose a procedure based on performing local tests at a number of scales and locations on a sparse grid, which adapts to the choice of grid in the sense that by choosing a sparser grid one explicitly pays a lower price for multiple testing. The procedure is fast as its computational complexity is always of the order $\mathcal{O} (n \log (n))$ where $n$ is the length of the data, and optimal in the sense that under certain mild conditions every change point is detected with high probability and the widths of the intervals returned match the mini-max localisation rates for the associated change point problem up to log factors. A detailed simulation study shows our procedure is competitive against state of the art algorithms for similar problems. Our procedure is implemented in the R package ChangePointInference which is available via //github.com/gaviosha/ChangePointInference.

We study the effect of approximation errors in assessing the extreme behaviour of univariate functionals of random objects. We build our framework into a general setting where estimation of the extreme value index and extreme quantiles of the functional is based on some approximated value instead of the true one. As an example, we consider the effect of discretisation errors in computation of the norms of paths of stochastic processes. In particular, we quantify connections between the sample size $n$ (the number of observed paths), the number of the discretisation points $m$, and the modulus of continuity function $\phi$ describing the path continuity of the underlying stochastic process. As an interesting example fitting into our framework, we consider processes of form $Y(t) = \mathcal{R}Z(t)$, where $\mathcal{R}$ is a heavy-tailed random variable and the increments of the process $Z$ have lighter tails compared to $\mathcal{R}$.

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.

It has been extensively studied in the literature that solving Maxwell equations is very sensitive to the mesh structure, space conformity and solution regularity. Roughly speaking, for almost all the methods in the literature, optimal convergence for low-regularity solutions heavily relies on conforming spaces and highly-regular simplicial meshes. This can be a significant limitation for many popular methods based on polytopal meshes in the case of inhomogeneous media, as the discontinuity of electromagnetic parameters can lead to quite low regularity of solutions near media interfaces, and potentially worsened by geometric singularities, making many popular methods based on broken spaces, non-conforming or polytopal meshes particularly challenging to apply. In this article, we present a virtual element method for solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media with quite arbitrary polytopal meshes, and the media interface is allowed to have geometric singularity to cause low regularity. There are two key novelties: (i) the proposed method is theoretically guaranteed to achieve robust optimal convergence for solutions with merely $\mathbf{H}^{\theta}$ regularity, $\theta\in(1/2,1]$; (ii) the polytopal element shape can be highly anisotropic and shrinking, and an explicit formula is established to describe the relationship between the shape regularity and solution regularity. Extensive numerical experiments will be given to demonstrate the effectiveness of the proposed method.

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.