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Deep Learning (DL) holds great promise in reshaping the healthcare systems given its precision, efficiency, and objectivity. However, the brittleness of DL models to noisy and out-of-distribution inputs is ailing their deployment in the clinic. Most systems produce point estimates without further information about model uncertainty or confidence. This paper introduces a new Bayesian deep learning framework for uncertainty quantification in segmentation neural networks, specifically encoder-decoder architectures. The proposed framework uses the first-order Taylor series approximation to propagate and learn the first two moments (mean and covariance) of the distribution of the model parameters given the training data by maximizing the evidence lower bound. The output consists of two maps: the segmented image and the uncertainty map of the segmentation. The uncertainty in the segmentation decisions is captured by the covariance matrix of the predictive distribution. We evaluate the proposed framework on medical image segmentation data from Magnetic Resonances Imaging and Computed Tomography scans. Our experiments on multiple benchmark datasets demonstrate that the proposed framework is more robust to noise and adversarial attacks as compared to state-of-the-art segmentation models. Moreover, the uncertainty map of the proposed framework associates low confidence (or equivalently high uncertainty) to patches in the test input images that are corrupted with noise, artifacts or adversarial attacks. Thus, the model can self-assess its segmentation decisions when it makes an erroneous prediction or misses part of the segmentation structures, e.g., tumor, by presenting higher values in the uncertainty map.

Defeaturing consists in simplifying geometrical models by removing the geometrical features that are considered not relevant for a given simulation. Feature removal and simplification of computer-aided design models enables faster simulations for engineering analysis problems, and simplifies the meshing problem that is otherwise often unfeasible. The effects of defeaturing on the analysis are then neglected and, as of today, there are basically very few strategies to quantitatively evaluate such an impact. Understanding well the effects of this process is an important step for automatic integration of design and analysis. We formalize the process of defeaturing by understanding its effect on the solution of Poisson equation defined on the geometrical model of interest containing a single feature, with Neumann boundary conditions on the feature itself. We derive an a posteriori estimator of the energy error between the solutions of the exact and the defeatured geometries in $\mathbb{R}^n$, $n\in\{2,3\}$, that is simple, reliable and efficient up to oscillations. The dependence of the estimator upon the size of the features is explicit.

Data from both a randomized trial and an observational study are sometimes simultaneously available for evaluating the effect of an intervention. The randomized data typically allows for reliable estimation of average treatment effects but may be limited in sample size and patient heterogeneity for estimating conditional average treatment effects for a broad range of patients. Estimates from the observational study can potentially compensate for these limitations, but there may be concerns about whether confounding and treatment effect heterogeneity have been adequately addressed. We propose an approach for combining conditional treatment effect estimators from each source such that it aggressively weights toward the randomized estimator when bias in the observational estimator is detected. This allows the combination to be consistent for a conditional causal effect, regardless of whether assumptions required for consistent estimation in the observational study are satisfied. When the bias is negligible, the estimators from each source are combined for optimal efficiency. We show the problem can be formulated as a penalized least squares problem and consider its asymptotic properties. Simulations demonstrate the robustness and efficiency of the method in finite samples, in scenarios with bias or no bias in the observational estimator. We illustrate the method by estimating the effects of hormone replacement therapy on the risk of developing coronary heart disease in data from the Women's Health Initiative.

Instrumental variables (IVs) can be used to provide evidence as to whether a treatment X has a causal effect on Y. Z is a valid instrument if it satisfies the three core IV assumptions of relevance, independence and the exclusion restriction. Even if the instrument satisfies these assumptions, further assumptions are required to estimate the average causal effect (ACE) of X on Y. Sufficient assumptions for this include: homogeneity in the causal effect of X on Y; homogeneity in the association of Z with X; and No Effect Modification (NEM). Here, we describe the NO Simultaneous Heterogeneity (NOSH) assumption, which requires the heterogeneity in the X-Y causal effect to be independent of both Z and heterogeneity in the Z-X association. We describe the necessary conditions for NOSH to hold, in which case conventional IV methods are consistent for the ACE even if both homogeneity assumptions and NEM are violated. We illustrate these ideas using simulations and by re-examining selected published studies.

Integrity constraints such as functional dependencies (FD) and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold {\em exactly}. However, many applications today consider constraints that hold only {\em approximately}. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the {\em relaxation problem}: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Then, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Finally, we show how some of the results in the paper can be derived using the {\em I-measure} theory, which relates between information theoretic measures and set theory. Our results recover, and sometimes extend, previously known results about the implication problem: the implication of MVDs and FDs can be checked by considering only 2-tuple relations.

We study downlink channel estimation in a multi-cell Massive multiple-input multiple-output (MIMO) system operating in time-division duplex. The users must know their effective channel gains to decode their received downlink data. Previous works have used the mean value as the estimate, motivated by channel hardening. However, this is associated with a performance loss in non-isotropic scattering environments. We propose two novel estimation methods that can be applied without downlink pilots. The first method is model-based and asymptotic arguments are utilized to identify a connection between the effective channel gain and the average received power during a coherence interval. The second method is data-driven and trains a neural network to identify a mapping between the available information and the effective channel gain. Both methods can be utilized for any channel distribution and precoding. For the model-aided method, we derive all expressions in closed form for the case when maximum ratio or zero-forcing precoding is used. We compare the proposed methods with the state-of-the-art using the normalized mean-squared error and spectral efficiency (SE). The results suggest that the two proposed methods provide better SE than the state-of-the-art when there is a low level of channel hardening, while the performance difference is relatively small with the uncorrelated channel model.

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.

We study the problem of density estimation for a random vector ${\boldsymbol X}$ in $\mathbb R^d$ with probability density $f(\boldsymbol x)$. For a spanning tree $T$ defined on the vertex set $\{1,\dots ,d\}$, the tree density $f_{T}$ is a product of bivariate conditional densities. The optimal spanning tree $T^*$ is the spanning tree $T$, for which the Kullback-Leibler divergence of $f$ and $f_{T}$ is the smallest. From i.i.d. data we identify the optimal tree $T^*$ and computationally efficiently construct a tree density estimate $f_n$ such that, without any regularity conditions on the density $f$, one has that $\lim_{n\to \infty} \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x=0$ a.s. For Lipschitz continuous $f$ with bounded support, $\mathbb E\{ \int |f_n(\boldsymbol x)-f_{T^*}(\boldsymbol x)|d\boldsymbol x\}=O(n^{-1/4})$.

The estimation of information measures of continuous distributions based on samples is a fundamental problem in statistics and machine learning. In this paper, we analyze estimates of differential entropy in $K$-dimensional Euclidean space, computed from a finite number of samples, when the probability density function belongs to a predetermined convex family $\mathcal{P}$. First, estimating differential entropy to any accuracy is shown to be infeasible if the differential entropy of densities in $\mathcal{P}$ is unbounded, clearly showing the necessity of additional assumptions. Subsequently, we investigate sufficient conditions that enable confidence bounds for the estimation of differential entropy. In particular, we provide confidence bounds for simple histogram based estimation of differential entropy from a fixed number of samples, assuming that the probability density function is Lipschitz continuous with known Lipschitz constant and known, bounded support. Our focus is on differential entropy, but we provide examples that show that similar results hold for mutual information and relative entropy as well.

This paper considers the integrated problem of quay crane assignment, quay crane scheduling, yard location assignment, and vehicle dispatching operations at a container terminal. The main objective is to minimize vessel turnover times and maximize the terminal throughput, which are key economic drivers in terminal operations. Due to their computational complexities, these problems are not optimized jointly in existing work. This paper revisits this limitation and proposes Mixed Integer Programming (MIP) and Constraint Programming (CP) models for the integrated problem, under some realistic assumptions. Experimental results show that the MIP formulation can only solve small instances, while the CP model finds optimal solutions in reasonable times for realistic instances derived from actual container terminal operations.