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The digital age is changing the role of educators and pushing for a paradigm shift in the education system as a whole. Growing demand for general and specialized education inside and outside classrooms is at the heart of this rising trend. In modern, heterogeneous learning environments, the one-size-fits-all approach is proven to be fundamentally flawed. Individualization through adaptivity is, therefore, crucial to nurture individual potential and address accessibility needs and neurodiversity. By formalizing a learning framework that takes into account all these different aspects, we aim to define and implement an open, content-agnostic, and extensible eTutoring platform to design and consume adaptive and gamified learning experiences. Adaptive technology supplementing teaching can extend the reach of every teacher, making it possible to scale 1-1 learning experiences. There are many successful existing technologies available but they come with fixed environments that are not always suitable for the targeted audiences of the course material. This paper presents PolyGloT, a system able to help teachers to design and implement a gamified and adaptive learning paths. Through it we address some important issues including the engagement, fairness, and effectiveness of learning environments. We do not only propose an innovative platform that could foster the learning process of different disciplines, but it could also help teachers and instructors in organizing learning material in an easy-access repository, Comment: 6 pages; 5 figures

We generalize Pearl's back-door criterion for directed acyclic graphs (DAGs) to more general types of graphs that describe Markov equivalence classes of DAGs and/or allow for arbitrarily many hidden variables. We also give easily checkable necessary and sufficient graphical criteria for the existence of a set of variables that satisfies our generalized back-door criterion, when considering a single intervention and a single outcome variable. Moreover, if such a set exists, we provide an explicit set that fulfills the criterion. We illustrate the results in several examples. R-code is available in the R-package pcalg., Comment: Published at http://dx.doi.org/10.1214/14-AOS1295 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider constraint-based methods for causal structure learning, such as the PC-, FCI-, RFCI- and CCD- algorithms (Spirtes et al. (2000, 1993), Richardson (1996), Colombo et al. (2012), Claassen et al. (2013)). The first step of all these algorithms consists of the PC-algorithm. This algorithm is known to be order-dependent, in the sense that the output can depend on the order in which the variables are given. This order-dependence is a minor issue in low-dimensional settings. We show, however, that it can be very pronounced in high-dimensional settings, where it can lead to highly variable results. We propose several modifications of the PC-algorithm (and hence also of the other algorithms) that remove part or all of this order-dependence. All proposed modifications are consistent in high-dimensional settings under the same conditions as their original counterparts. We compare the PC-, FCI-, and RFCI-algorithms and their modifications in simulation studies and on a yeast gene expression data set. We show that our modifications yield similar performance in low-dimensional settings and improved performance in high-dimensional settings. All software is implemented in the R-package pcalg.

We consider the problem of learning causal information between random variables in directed acyclic graphs (DAGs) when allowing arbitrarily many latent and selection variables. The FCI (Fast Causal Inference) algorithm has been explicitly designed to infer conditional independence and causal information in such settings. However, FCI is computationally infeasible for large graphs. We therefore propose the new RFCI algorithm, which is much faster than FCI. In some situations the output of RFCI is slightly less informative, in particular with respect to conditional independence information. However, we prove that any causal information in the output of RFCI is correct in the asymptotic limit. We also define a class of graphs on which the outputs of FCI and RFCI are identical. We prove consistency of FCI and RFCI in sparse high-dimensional settings, and demonstrate in simulations that the estimation performances of the algorithms are very similar. All software is implemented in the R-package pcalg., Comment: Published in at http://dx.doi.org/10.1214/11-AOS940 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)