Many scientifically well-motivated statistical models in natural, engineering, and environmental sciences are specified through a generative process. However, in some cases, it may not be possible to write down the likelihood for these models analytically. Approximate Bayesian computation (ABC) methods allow Bayesian inference in such situations. The procedures are nonetheless typically computationally intensive. Recently, computationally attractive empirical likelihood-based ABC methods have been suggested in the literature. All of these methods rely on the availability of several suitable analytically tractable estimating equations, and this is sometimes problematic. We propose an easy-to-use empirical likelihood ABC method in this article. First, by using a variational approximation argument as a motivation, we show that the target log-posterior can be approximated as a sum of an expected joint log-likelihood and the differential entropy of the data generating density. The expected log-likelihood is then estimated by an empirical likelihood where the only inputs required are a choice of summary statistic, it's observed value, and the ability to simulate the chosen summary statistics for any parameter value under the model. The differential entropy is estimated from the simulated summaries using traditional methods. Posterior consistency is established for the method, and we discuss the bounds for the required number of simulated summaries in detail. The performance of the proposed method is explored in various examples.

Dempster-Shafer Theory (DST) generalizes Bayesian probability theory, offering useful additional information, but suffers from a much higher computational burden. A lot of work has been done to reduce the time complexity of information fusion with Dempster's rule, which is a pointwise multiplication of two zeta transforms, and optimal general algorithms have been found to get the complete definition of these transforms. Yet, it is shown in this paper that the zeta transform and its inverse, the M\"obius transform, can be exactly simplified, fitting the quantity of information contained in belief functions. Beyond that, this simplification actually works for any function on any partially ordered set. It relies on a new notion that we call focal point and that constitutes the smallest domain on which both the zeta and M\"obius transforms can be defined. We demonstrate the interest of these general results for DST, not only for the reduction in complexity of most transformations between belief representations and their fusion, but also for theoretical purposes. Indeed, we provide a new generalization of the conjunctive decomposition of evidence and formulas uncovering how each decomposition weight is tied to the corresponding mass function.

Many of us who follow the NBA Draft closely have a keen desire to know the precise ordering of the Draft, eg where each player will be taken ("where" having a dual meaning here, both the team and draft slot). To this end, it is common behavior to ingest the information we get from the dozen or more "prominent" mock drafts available leading up to the Draft. There are different ways analysts and media try to aggregate this information into what I like to call "Meta Mocks". Typically this just involves very simple operations like averaging ranks or looking at the minimum and maximum draft position. Chris Feller just the other day shared a very cool approach using survival analysis.

Recently it has also become increasingly important in marketing, medicine and political sciences. Process systems engineering community adopts mathematical models successfully in various endeavors such as product and plant design, control system design, operations optimization, etc. (Pantelides and Renfro, 2013; Fung et al., 2016; Safdarnejad et al., 2016). A mathematical model is usually an abstract representation of a true system via sets of equations (algebraic, ordinary differential or partial differential), inequalities (e.g., a range of model validity), and logical conditions. Model development is usually divided into three major steps a) identification of the model structure, b) design and realization of the experiments, and c) estimation of the unknown parameters. In the latter phase, one often realizes maximum-likelihood estimation via least-squares methodology as he/she assumes--knowingly or not--that the measurement error present in the measured data is statistically distributed as a white Gaussian noise. Once the parameter estimates are known, the experimenter commonly determines the quality of the obtained model. This can be done either by using some validation data--if available--or via assessing the joint-confidence regions of the estimated parameters (Beale, 1960; Bates and Watts, 1988; Rooney and Biegler, 2001; Seber and Wild, 2003).

In this comprehensive review, we describe a new mathematical problem in artificial intelligence (AI) from a mathematical modeling perspective, following the philosophy stated by Rudolf E. Kalman that "Once you get the physics right, the rest is mathematics". The new problem is called "Generalized Constraints (GCs)", and we adopt GCs as a general term to describe any type of prior information in modelings. To understand better about GCs to be a general problem, we compare them with the conventional constraints (CCs) and list their extra challenges over CCs. In the construction of AI machines, we basically encounter more often GCs for modeling, rather than CCs with well-defined forms. Furthermore, we discuss the ultimate goals of AI and redefine transparent, interpretable, and explainable AI in terms of comprehension levels about machines. We review the studies in relation to the GC problems although most of them do not take the notion of GCs. We demonstrate that if AI machines are simplified by a coupling with both knowledge-driven submodel and data-driven submodel, GCs will play a critical role in a knowledge-driven submodel as well as in the coupling form between the two submodels. Examples are given to show that the studies in view of a generalized constraint problem will help us perceive and explore novel subjects in AI, or even in mathematics, such as generalized constraint learning (GCL).

Probability trees are one of the simplest models of causal generative processes. They possess clean semantics and -- unlike causal Bayesian networks -- they can represent context-specific causal dependencies, which are necessary for e.g. causal induction. Yet, they have received little attention from the AI and ML community. Here we present concrete algorithms for causal reasoning in discrete probability trees that cover the entire causal hierarchy (association, intervention, and counterfactuals), and operate on arbitrary propositional and causal events. Our work expands the domain of causal reasoning to a very general class of discrete stochastic processes.

High-dimensional probability density estimation for inference suffers from the "curse of dimensionality". For many physical inference problems, the full posterior distribution is unwieldy and seldom used in practice. Instead, we propose direct estimation of lower-dimensional marginal distributions, bypassing high-dimensional density estimation or high-dimensional Markov chain Monte Carlo (MCMC) sampling. By evaluating the two-dimensional marginal posteriors we can unveil the full-dimensional parameter covariance structure. We additionally propose constructing a simple hierarchy of fast neural regression models, called Moment Networks, that compute increasing moments of any desired lower-dimensional marginal posterior density; these reproduce exact results from analytic posteriors and those obtained from Masked Autoregressive Flows. We demonstrate marginal posterior density estimation using high-dimensional LIGO-like gravitational wave time series and describe applications for problems of fundamental cosmology.

Recent advances in incorporating neural networks into particle filters provide the desired flexibility to apply particle filters in large-scale real-world applications. The dynamic and measurement models in this framework are learnable through the differentiable implementation of particle filters. Past efforts in optimising such models often require the knowledge of true states which can be expensive to obtain or even unavailable in practice. In this paper, in order to reduce the demand for annotated data, we present an end-to-end learning objective based upon the maximisation of a pseudo-likelihood function which can improve the estimation of states when large portion of true states are unknown. We assess performance of the proposed method in state estimation tasks in robotics with simulated and real-world datasets.

Deep neural networks (DNNs) have made a revolution in numerous fields during the last decade. However, in tasks with high safety requirements, such as medical or autonomous driving applications, providing an assessment of the models reliability can be vital. Uncertainty estimation for DNNs has been addressed using Bayesian methods, providing mathematically founded models for reliability assessment. These model are computationally expensive and generally impractical for many real-time use cases. Recently, non-Bayesian methods were proposed to tackle uncertainty estimation more efficiently. We propose an efficient method for uncertainty estimation in DNNs achieving high accuracy. We simulate the notion of multi-task learning on single-task problems by producing parallel predictions from similar models differing by their loss. This multi-loss approach allows one-phase training for single-task learning with uncertainty estimation. We keep our inference time relatively low by leveraging the advantage proposed by the Deep-Sub-Ensembles method. The novelty of this work resides in the proposed accurate variational inference with a simple and convenient training procedure, while remaining competitive in terms of computational time. We conduct experiments on SVHN, CIFAR10, CIFAR100 as well as Image-Net using different architectures. Our results show improved accuracy on the classification task and competitive results on several uncertainty measures.

Ride-hailing platforms generally provide various service options to customers, such as solo ride services, shared ride services, etc. It is generally expected that demands for different service modes are correlated, and the prediction of demand for one service mode can benefit from historical observations of demands for other service modes. Moreover, an accurate joint prediction of demands for multiple service modes can help the platforms better allocate and dispatch vehicle resources. Although there is a large stream of literature on ride-hailing demand predictions for one specific service mode, little efforts have been paid towards joint predictions of ride-hailing demands for multiple service modes. To address this issue, we propose a deep multi-task multi-graph learning approach, which combines two components: (1) multiple multi-graph convolutional (MGC) networks for predicting demands for different service modes, and (2) multi-task learning modules that enable knowledge sharing across multiple MGC networks. More specifically, two multi-task learning structures are established. The first one is the regularized cross-task learning, which builds cross-task connections among the inputs and outputs of multiple MGC networks. The second one is the multi-linear relationship learning, which imposes a prior tensor normal distribution on the weights of various MGC networks. Although there are no concrete bridges between different MGC networks, the weights of these networks are constrained by each other and subject to a common prior distribution. Evaluated with the for-hire-vehicle datasets in Manhattan, we show that our propose approach outperforms the benchmark algorithms in prediction accuracy for different ride-hailing modes.