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Causal investigations in observational studies pose a great challenge in scientific research where randomized trials or intervention-based studies are not feasible. Leveraging Shannon's seminal work on information theory, we consider a framework of asymmetry where any causal link between putative cause and effect must be explained through a mechanism governing the cause as well as a generative process yielding an effect of the cause. Under weak assumptions, this framework enables the assessment of whether X is a stronger predictor of Y or vice-versa. Under stronger identifiability assumptions our framework is able to distinguish between cause and effect using observational data. We establish key statistical properties of this framework. Our proposed methodology relies on scalable non-parametric density estimation using fast Fourier transformation. The resulting estimation method is manyfold faster than the classical bandwidth-based density estimation while maintaining comparable mean integrated squared error rates. We investigate key asymptotic properties of our methodology and introduce a data-splitting technique to facilitate inference. The key attraction of our framework is its inference toolkit, which allows researchers to quantify uncertainty in causal discovery findings. We illustrate the performance of our methodology through simulation studies as well as multiple real data examples.

Causal investigations in observational studies pose a great challenge in research where randomized trials or intervention-based studies are not feasible. We develop an information geometric causal discovery and inference framework of "predictive asymmetry". For $(X, Y)$, predictive asymmetry enables assessment of whether $X$ is more likely to cause $Y$ or vice-versa. The asymmetry between cause and effect becomes particularly simple if $X$ and $Y$ are deterministically related. We propose a new metric called the Directed Mutual Information ($DMI$) and establish its key statistical properties. $DMI$ is not only able to detect complex non-linear association patterns in bivariate data, but also is able to detect and infer causal relations. Our proposed methodology relies on scalable non-parametric density estimation using Fourier transform. The resulting estimation method is manyfold faster than the classical bandwidth-based density estimation. We investigate key asymptotic properties of the $DMI$ methodology and a data-splitting technique is utilized to facilitate causal inference using the $DMI$. Through simulation studies and an application, we illustrate the performance of $DMI$.

We combine Kronecker products, and quantitative information flow, to give a novel formal analysis for the fine-grained verification of utility in complex privacy pipelines. The combination explains a surprising anomaly in the behaviour of utility of privacy-preserving pipelines -- that sometimes a reduction in privacy results also in a decrease in utility. We use the standard measure of utility for Bayesian analysis, introduced by Ghosh at al., to produce tractable and rigorous proofs of the fine-grained statistical behaviour leading to the anomaly. More generally, we offer the prospect of formal-analysis tools for utility that complement extant formal analyses of privacy. We demonstrate our results on a number of common privacy-preserving designs.

This work presents a novel global digital image correlation (DIC) method, based on a newly developed convolution finite element (C-FE) approximation. The convolution approximation can rely on the mesh of linear finite elements and enables arbitrarily high order approximations without adding more degrees of freedom. Therefore, the C-FE based DIC can be more accurate than {the} usual FE based DIC by providing highly smooth and accurate displacement and strain results with the same element size. The detailed formulation and implementation of the method have been discussed in this work. The controlling parameters in the method include the polynomial order, patch size, and dilation. A general choice of the parameters and their potential adaptivity have been discussed. The proposed DIC method has been tested by several representative examples, including the DIC challenge 2.0 benchmark problems, with comparison to the usual FE based DIC. C-FE outperformed FE in all the DIC results for the tested examples. This work demonstrates the potential of C-FE and opens a new avenue to enable highly smooth, accurate, and robust DIC analysis for full-field displacement and strain measurements.

This paper introduces an assumption-lean method that constructs valid and efficient lower predictive bounds (LPBs) for survival times with censored data. We build on recent work by Cand\`es et al. (2021), whose approach first subsets the data to discard any data points with early censoring times, and then uses a reweighting technique (namely, weighted conformal inference (Tibshirani et al., 2019)) to correct for the distribution shift introduced by this subsetting procedure. For our new method, instead of constraining to a fixed threshold for the censoring time when subsetting the data, we allow for a covariate-dependent and data-adaptive subsetting step, which is better able to capture the heterogeneity of the censoring mechanism. As a result, our method can lead to LPBs that are less conservative and give more accurate information. We show that in the Type I right-censoring setting, if either of the censoring mechanism or the conditional quantile of survival time is well estimated, our proposed procedure achieves nearly exact marginal coverage, where in the latter case we additionally have approximate conditional coverage. We evaluate the validity and efficiency of our proposed algorithm in numerical experiments, illustrating its advantage when compared with other competing methods. Finally, our method is applied to a real dataset to generate LPBs for users' active times on a mobile app.

We study optimal data pooling for shared learning in two common maintenance operations: condition-based maintenance and spare parts management. We consider a set of systems subject to Poisson input -- the degradation or demand process -- that are coupled through an a-priori unknown rate. Decision problems involving these systems are high-dimensional Markov decision processes (MDPs) and hence notoriously difficult to solve. We present a decomposition result that reduces such an MDP to two-dimensional MDPs, enabling structural analyses and computations. Leveraging this decomposition, we (i) demonstrate that pooling data can lead to significant cost reductions compared to not pooling, and (ii) show that the optimal policy for the condition-based maintenance problem is a control limit policy, while for the spare parts management problem, it is an order-up-to level policy, both dependent on the pooled data.

Safe reinforcement learning (RL) with hard constraint guarantees is a promising optimal control direction for multi-energy management systems. It only requires the environment-specific constraint functions itself a priori and not a complete model. The project-specific upfront and ongoing engineering efforts are therefore still reduced, better representations of the underlying system dynamics can still be learnt, and modelling bias is kept to a minimum. However, even the constraint functions alone are not always trivial to accurately provide in advance, leading to potentially unsafe behaviour. In this paper, we present two novel advancements: (I) combining the OptLayer and SafeFallback method, named OptLayerPolicy, to increase the initial utility while keeping a high sample efficiency and the possibility to formulate equality constraints. (II) introducing self-improving hard constraints, to increase the accuracy of the constraint functions as more and new data becomes available so that better policies can be learnt. Both advancements keep the constraint formulation decoupled from the RL formulation, so new (presumably better) RL algorithms can act as drop-in replacements. We have shown that, in a simulated multi-energy system case study, the initial utility is increased to 92.4% (OptLayerPolicy) compared to 86.1% (OptLayer) and that the policy after training is increased to 104.9% (GreyOptLayerPolicy) compared to 103.4% (OptLayer) - all relative to a vanilla RL benchmark. Although introducing surrogate functions into the optimisation problem requires special attention, we conclude that the newly presented GreyOptLayerPolicy method is the most advantageous.

This paper introduces an extended tensor decomposition (XTD) method for model reduction. The proposed method is based on a sparse non-separated enrichment to the conventional tensor decomposition, which is expected to improve the approximation accuracy and the reducibility (compressibility) in highly nonlinear and singular cases. The proposed XTD method can be a powerful tool for solving nonlinear space-time parametric problems. The method has been successfully applied to parametric elastic-plastic problems and real time additive manufacturing residual stress predictions with uncertainty quantification. Furthermore, a combined XTD-SCA (self-consistent clustering analysis) strategy has been presented for multi-scale material modeling, which enables real time multi-scale multi-parametric simulations. The efficiency of the method is demonstrated with comparison to finite element analysis. The proposed method enables a novel framework for fast manufacturing and material design with uncertainties.

We study two classic variants of block-structured integer programming. Two-stage stochastic programs are integer programs of the form $\{A_i \mathbf{x} + D_i \mathbf{y}_i = \mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where $A_i$ and $D_i$ are bounded-size matrices. On the other hand, $n$-fold programs are integer programs of the form $\{{\sum_{i=1}^n C_i\mathbf{y}_i=\mathbf{a}} \textrm{ and } D_i\mathbf{y}_i=\mathbf{b}_i\textrm{ for all }i=1,\ldots,n\}$, where again $C_i$ and $D_i$ are bounded-size matrices. It is known that solving these kind of programs is fixed-parameter tractable when parameterized by the maximum dimension among the relevant matrices $A_i,C_i,D_i$ and the maximum absolute value of any entry appearing in the constraint matrix. We show that the parameterized tractability results for two-stage stochastic and $n$-fold programs persist even when one allows large entries in the global part of the program. More precisely, we prove that: - The feasibility problem for two-stage stochastic programs is fixed-parameter tractable when parameterized by the dimensions of matrices $A_i,D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we allow matrices $A_i$ to have arbitrarily large entries. - The linear optimization problem for $n$-fold integer programs that are uniform -- all matrices $C_i$ are equal -- is fixed-parameter tractable when parameterized by the dimensions of matrices $C_i$ and $D_i$ and by the maximum absolute value of the entries of matrices $D_i$. That is, we require that $C_i=C$ for all $i=1,\ldots,n$, but we allow $C$ to have arbitrarily large entries. In the second result, the uniformity assumption is necessary; otherwise the problem is $\mathsf{NP}$-hard already when the parameters take constant values. Both our algorithms are weakly polynomial: the running time is measured in the total bitsize of the input.

We study stability properties of the expected utility function in Bayesian optimal experimental design. We provide a framework for this problem in a non-parametric setting and prove a convergence rate of the expected utility with respect to a likelihood perturbation. This rate is uniform over the design space and its sharpness in the general setting is demonstrated by proving a lower bound in a special case. To make the problem more concrete we proceed by considering non-linear Bayesian inverse problems with Gaussian likelihood and prove that the assumptions set out for the general case are satisfied and regain the stability of the expected utility with respect to perturbations to the observation map. Theoretical convergence rates are demonstrated numerically in three different examples.