To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the resolution of the parameter inverse problem from the observed spatiotemporal data a challenging endeavor. Starting from the observed data obtained from such systems, we propose a novel framework that facilitates the investigation of parameter identification for multi-state systems governed by spatiotemporal varying parametric partial differential equations. Our framework consists of two integral components: a constrained self-adaptive physics-informed neural network, encompassing a sub-network, as our methodology for parameter identification, and a finite mixture model approach to detect regions of probable parameter variations. Through our scheme, we can precisely ascertain the unknown varying parameters of the complex multi-state system, thereby accomplishing the inversion of the varying parameters. Furthermore, we have showcased the efficacy of our framework on two numerical cases: the 1D Burgers' equation with time-varying parameters and the 2D wave equation with a space-varying parameter.

Statistical relational AI and probabilistic logic programming have so far mostly focused on discrete probabilistic models. The reasons for this is that one needs to provide constructs to succinctly model the independencies in such models, and also provide efficient inference. Three types of independencies are important to represent and exploit for scalable inference in hybrid models: conditional independencies elegantly modeled in Bayesian networks, context-specific independencies naturally represented by logical rules, and independencies amongst attributes of related objects in relational models succinctly expressed by combining rules. This paper introduces a hybrid probabilistic logic programming language, DC#, which integrates distributional clauses' syntax and semantics principles of Bayesian logic programs. It represents the three types of independencies qualitatively. More importantly, we also introduce the scalable inference algorithm FO-CS-LW for DC#. FO-CS-LW is a first-order extension of the context-specific likelihood weighting algorithm (CS-LW), a novel sampling method that exploits conditional independencies and context-specific independencies in ground models. The FO-CS-LW algorithm upgrades CS-LW with unification and combining rules to the first-order case.

The common cause principle for two random variables $A$ and $B$ is examined in the case of causal insufficiency, when their common cause $C$ is known to exist, but only the joint probability of $A$ and $B$ is observed. As a result, $C$ cannot be uniquely identified (the latent confounder problem). We show that the generalized maximum likelihood method can be applied to this situation and allows identification of $C$ that is consistent with the common cause principle. It closely relates to the maximum entropy principle. Investigation of the two binary symmetric variables reveals a non-analytic behavior of conditional probabilities reminiscent of a second-order phase transition. This occurs during the transition from correlation to anti-correlation in the observed probability distribution. The relation between the generalized likelihood approach and alternative methods, such as predictive likelihood and the minimum common cause entropy, is discussed. The consideration of the common cause for three observed variables (and one hidden cause) uncovers causal structures that defy representation through directed acyclic graphs with the Markov condition.

Generative modeling has been used frequently in synthetic data generation. Fairness and privacy are two big concerns for synthetic data. Although Recent GAN [Goodfellow et al. (2014)] based methods show good results in preserving privacy, the generated data may be more biased. At the same time, these methods require high computation resources. We show the effectiveness of our method theoretically and empirically. We show that models trained on data generated by the proposed method can perform well (in inference stage) on real application scenarios. Synthetic data [Rubin (1993)] is data that is artificially created rather than being generated by actual events.

With the explosion of applications of Data Science, the field is has come loose from its foundations. This article argues for a new program of applied research in areas familiar to researchers in Bayesian methods in AI that are needed to ground the practice of Data Science by borrowing from AI techniques for model formulation that we term ``Decision Modelling.'' This article briefly reviews the formulation process as building a causal graphical model, then discusses the process in terms of six principles that comprise \emph{Decision Quality}, a framework from the popular business literature. We claim that any successful applied ML modelling effort must include these six principles. We explain how Decision Modelling combines a conventional machine learning model with an explicit value model. To give a specific example we show how this is done by integrating a model's ROC curve with a utility model.

The features in many prediction models naturally take the form of a hierarchy. The lower levels represent individuals or events. These units group naturally into locations and intervals or other aggregates, often at multiple levels. Levels of groupings may intersect and join, much as relational database tables do. Besides representing the structure of the data, predictive features in hierarchical models can be assigned to their proper levels. Such models lend themselves to hierarchical Bayes solution methods that ``share'' results of inference between groups by generalizing over the case of individual models for each group versus one model that aggregates all groups into one. In this paper we show our work-in-progress applying a hierarchical Bayesian model to forecast purchases throughout the day at store franchises, with groupings over locations and days of the week. We demonstrate using the \textsf{stan} package on individual sales transaction data collected over the course of a year. We show how this solves the dilemma of having limited data and hence modest accuracy for each day and location, while being able to scale to a large number of locations with improved accuracy.

We study sampling problems associated with potentials that lack smoothness. The potentials can be either convex or non-convex. Departing from the standard smooth setting, the potentials are only assumed to be weakly smooth or non-smooth, or the summation of multiple such functions. We develop a sampling algorithm that resembles proximal algorithms in optimization for this challenging sampling task. Our algorithm is based on a special case of Gibbs sampling known as the alternating sampling framework (ASF). The key contribution of this work is a practical realization of the ASF based on rejection sampling for both non-convex and convex potentials that are not necessarily smooth. In almost all the cases of sampling considered in this work, our proximal sampling algorithm achieves better complexity than all existing methods.

Statistical machine learning methods often face the challenge of limited data available from the population of interest. One remedy is to leverage data from auxiliary source populations, which share some conditional distributions or are linked in other ways with the target domain. Techniques leveraging such \emph{dataset shift} conditions are known as \emph{domain adaptation} or \emph{transfer learning}. Despite extensive literature on dataset shift, limited works address how to efficiently use the auxiliary populations to improve the accuracy of risk evaluation for a given machine learning task in the target population. In this paper, we study the general problem of efficiently estimating target population risk under various dataset shift conditions, leveraging semiparametric efficiency theory. We consider a general class of dataset shift conditions, which includes three popular conditions -- covariate, label and concept shift -- as special cases. We allow for partially non-overlapping support between the source and target populations. We develop efficient and multiply robust estimators along with a straightforward specification test of these dataset shift conditions. We also derive efficiency bounds for two other dataset shift conditions, posterior drift and location-scale shift. Simulation studies support the efficiency gains due to leveraging plausible dataset shift conditions.

We focus on the problem of uncertainty informed allocation of medical resources (vaccines) to heterogeneous populations for managing epidemic spread. We tackle two related questions: (1) For a compartmental ordinary differential equation (ODE) model of epidemic spread, how can we estimate and integrate parameter uncertainty into resource allocation decisions? (2) How can we computationally handle both nonlinear ODE constraints and parameter uncertainties for a generic stochastic optimization problem for resource allocation? To the best of our knowledge current literature does not fully resolve these questions. Here, we develop a data-driven approach to represent parameter uncertainty accurately and tractably in a novel stochastic optimization problem formulation. We first generate a tractable scenario set by estimating the distribution on ODE model parameters using Bayesian inference with Gaussian processes. Next, we develop a parallelized solution algorithm that accounts for scenario-dependent nonlinear ODE constraints. Our scenario-set generation procedure and solution approach are flexible in that they can handle any compartmental epidemiological ODE model. Our computational experiments on two different non-linear ODE models (SEIR and SEPIHR) indicate that accounting for uncertainty in key epidemiological parameters can improve the efficacy of time-critical allocation decisions by 4-8%. This improvement can be attributed to data-driven and optimal (strategic) nature of vaccine allocations, especially in the early stages of the epidemic when the allocation strategy can crucially impact the long-term trajectory of the disease.

Recently, a new class of non-convex optimization problems motivated by the statistical problem of learning an acyclic directed graphical model from data has attracted significant interest. While existing work uses standard first-order optimization schemes to solve this problem, proving the global optimality of such approaches has proven elusive. The difficulty lies in the fact that unlike other non-convex problems in the literature, this problem is not "benign", and possesses multiple spurious solutions that standard approaches can easily get trapped in. In this paper, we prove that a simple path-following optimization scheme globally converges to the global minimum of the population loss in the bivariate setting.