Probability distributions are central to Bayesian accounts of cognition, but behavioral assessments do not directly measure them. Posterior distributions are typically computed from collections of individual participant actions, yet are used to draw conclusions about the internal structure of participant beliefs. Also not explicitly measured are the prior distributions that distinguish Bayesian models from others by representing initial states of belief. Instead, priors are usually derived from experimenters' intuitions or model assumptions and applied equally to all participants. Here we present three experiments using "Plinko", a behavioral task in which participants estimate distributions of ball drops over all available outcomes and where distributions are explicitly measured before any observations. In Experiment 1, we show that participant priors cluster around prototypical probability distributions (Gaussian, bimodal, etc.), and that prior cluster membership may indicate learning ability. In Experiment 2, we highlight participants' ability to update to unannounced changes of presented distributions and how this ability is affected by environmental manipulation. Finally, in Experiment 3, we verify that individual participant priors are reliable representations and that learning is not impeded when faced with a physically implausible ball drop distribution that is dynamically defined according to individual participant input. This task will prove useful in more closely examining mechanisms of statistical learning and mental model updating without requiring many of the assumptions made by more traditional computational modeling methodologies.

Machine learning models always make a prediction, even when it is likely to be inaccurate. This behavior should be avoided in many decision support applications, where mistakes can have severe consequences. Albeit already studied in 1970, machine learning with a reject option recently gained interest. This machine learning subfield enables machine learning models to abstain from making a prediction when likely to make a mistake. This survey aims to provide an overview on machine learning with a reject option. We introduce the conditions leading to two types of rejection, ambiguity and novelty rejection. Moreover, we define the existing architectures for models with a reject option, describe the standard learning strategies to train such models and relate traditional machine learning techniques to rejection. Additionally, we review strategies to evaluate a model's predictive and rejective quality. Finally, we provide examples of relevant application domains and show how machine learning with rejection relates to other machine learning research areas.

Over the past three decades, the problem of learning directed graphical models from data has received enormous amount of attention since they provide a compact and flexible way to represent the joint distribution of the data, especially when the associated graph is a directed acyclic graph (DAG). A directed graph is called a DAG if it does not contain directed cycles. DAG models are popular in practice with applications in biology, genetics, machine learning and causal inference (Sachs et al., 2005; Zhang et al., 2013; Koller and Friedman, 2009; Spirtes et al., 2000). There exists an extensive literature on learning the graph structure from data under the assumption that the graph is a DAG. For a summary, see the survey of Drton and Maathuis (2017); Heinze-Deml et al. (2018). Existing approaches generally fall into two categories, constrain-based methods (Spirtes et al., 2000; Pearl, 2009) and score-based methods (Chickering, 2002). Constraint-based methods utilize conditional independence test to determine whether there exists an edge between two nodes and then orient the edges in the graph, such that the resulting graph is compatible with the conditional independencies seen in the data. Score-based methods formulate the structure learning task as optimizing a score function based on the unknown graph and the data. A polytree is a DAG which does not contain any cycles even if the directions of all edges are ignored.

Rough Set based Spanning Sets were recently proposed to deal with uncertainties arising in the problem in domain of natural language processing problems. This paper presents a novel span measure using upper approximations. The key contribution of this paper is to propose another uncertainty measure of span and spanning sets. Firstly, this paper proposes a new definition of computing span which use upper approximation instead of boundary regions. This is useful in situations where computing upper approximations are much more convenient that computing boundary region. Secondly, properties of novel span and relation with earlier span measure are discussed. Thirdly, the paper presents application areas where the proposed span measure can be utilized.

In this work, we propose a domain generalization (DG) approach to learn on several labeled source domains and transfer knowledge to a target domain that is inaccessible in training. Considering the inherent conditional and label shifts, we would expect the alignment of $p(x|y)$ and $p(y)$. However, the widely used domain invariant feature learning (IFL) methods relies on aligning the marginal concept shift w.r.t. $p(x)$, which rests on an unrealistic assumption that $p(y)$ is invariant across domains. We thereby propose a novel variational Bayesian inference framework to enforce the conditional distribution alignment w.r.t. $p(x|y)$ via the prior distribution matching in a latent space, which also takes the marginal label shift w.r.t. $p(y)$ into consideration with the posterior alignment. Extensive experiments on various benchmarks demonstrate that our framework is robust to the label shift and the cross-domain accuracy is significantly improved, thereby achieving superior performance over the conventional IFL counterparts.

Gaussian Bayesian networks (a.k.a. linear Gaussian structural equation models) are widely used to model causal interactions among continuous variables. In this work, we study the problem of learning a fixed-structure Gaussian Bayesian network up to a bounded error in total variation distance. We analyze the commonly used node-wise least squares regression (LeastSquares) and prove that it has a near-optimal sample complexity. We also study a couple of new algorithms for the problem: - BatchAvgLeastSquares takes the average of several batches of least squares solutions at each node, so that one can interpolate between the batch size and the number of batches. We show that BatchAvgLeastSquares also has near-optimal sample complexity. - CauchyEst takes the median of solutions to several batches of linear systems at each node. We show that the algorithm specialized to polytrees, CauchyEstTree, has near-optimal sample complexity. Experimentally, we show that for uncontaminated, realizable data, the LeastSquares algorithm performs best, but in the presence of contamination or DAG misspecification, CauchyEst/CauchyEstTree and BatchAvgLeastSquares respectively perform better.

Causal discovery from observational data is a challenging task to which an exact solution cannot always be identified. Under assumptions about the data-generative process, the causal graph can often be identified up to an equivalence class. Proposing new realistic assumptions to circumscribe such equivalence classes is an active field of research. In this work, we propose a new set of assumptions that constrain possible causal relationships based on the nature of the variables. We thus introduce typed directed acyclic graphs, in which variable types are used to determine the validity of causal relationships. We demonstrate, both theoretically and empirically, that the proposed assumptions can result in significant gains in the identification of the causal graph.

Learning the structure of a causal graphical model using both observational and interventional data is a fundamental problem in many scientific fields. A promising direction is continuous optimization for score-based methods, which efficiently learn the causal graph in a data-driven manner. However, to date, those methods require constrained optimization to enforce acyclicity or lack convergence guarantees. In this paper, we present ENCO, an efficient structure learning method for directed, acyclic causal graphs leveraging observational and interventional data. ENCO formulates the graph search as an optimization of independent edge likelihoods, with the edge orientation being modeled as a separate parameter. Consequently, we can provide convergence guarantees of ENCO under mild conditions without constraining the score function with respect to acyclicity. In experiments, we show that ENCO can efficiently recover graphs with hundreds of nodes, an order of magnitude larger than what was previously possible, while handling deterministic variables and latent confounders.

Rough Set based concepts of Span and Spanning Sets were recently proposed to deal with uncertainties in data. Here, this paper, presents novel concepts for generic decision-making process using Rough Set based span for a decision table. Majority of problems in Artificial Intelligence deal with decision making. This paper provides real life applications of proposed Rough Set based span for decision tables. Here, novel concept of span for a decision table is proposed, illustrated with real life example of flood relief and rescue team assignment. Its uses, applications and properties are explored. The key contribution of paper is primarily to study decision making using Rough Set based Span for a decision tables, as against an information system in prior works. Here, the main contribution is that decision classes are automatically learned by the technique of Rough Set based span, for a particular problem, hence automating the decision-making process. These decision-making tools based on span can guide an expert in taking decisions in tough and time-bound situations.

The multi-period dynamics of energy storage (ES), intermittent renewable generation and uncontrollable power loads, make the optimization of power system operation (PSO) challenging. A multi-period optimal PSO under uncertainty is formulated using the chance-constrained optimization (CCO) modeling paradigm, where the constraints include the nonlinear energy storage and AC power flow models. Based on the emerging scenario optimization method which does not rely on pre-known probability distribution functions, this paper develops a novel solution method for this challenging CCO problem. The proposed meth-od is computationally effective for mainly two reasons. First, the original AC power flow constraints are approximated by a set of learning-assisted quadratic convex inequalities based on a generalized least absolute shrinkage and selection operator. Second, considering the physical patterns of data and motived by learning-based sampling, the strategic sampling method is developed to significantly reduce the required number of scenarios through different sampling strategies. The simulation results on IEEE standard systems indicate that 1) the proposed strategic sampling significantly improves the computational efficiency of the scenario-based approach for solving the chance-constrained optimal PSO problem, 2) the data-driven convex approximation of power flow can be promising alternatives of nonlinear and nonconvex AC power flow.