We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic information theory we can obtain SCI, a highly robust, effective and computationally efficient test for conditional independence---and show it outperforms the state of the art when applied in constraint-based inference methods such as stable PC. Second, building upon on SCI, we show how to tell apart the parents and children of a given node based on the algorithmic Markov condition. We give the Climb algorithm to efficiently discover the directed, causal Markov blanket---and show it is at least as accurate as inferring the global network, while being much more efficient. Last, but not least, we detail how we can use the Climb score to direct those edges that state of the art causal discovery algorithms based on PC or GES leave undirected---and show this improves their precision, recall and F1 scores by up to 20 .
Many methods for discovering Markov blankets and causal networks rely on conditional independence tests @cite_15 @cite_4 ---which, by conditioning on the empty set, can also be used to measure association. For discrete data, the state of the art commonly relies on either the @math test or conditional mutual information (CMI) @cite_15 @cite_29 @cite_14 . Both measures, however, have drawbacks. The @math test is biased to independence when only limited data is available, whereas CMI has the opposite problem. In particular, it holds that @math for any @math , even if @math . For both @math and CMI it is necessary to define a significance threshold or cut-off, which can be arbitrary. Our proposed method based on algorithmic independence overcomes these limitations.
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We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Abstract Irregular sampling and “stable sampling” of band-limited functions have been studied by H.J. Landau [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52]. We prove that quasicrystals are sets of stable sampling. To cite this article: B. Matei, Y. Meyer, C. R. Acad. Sci. Paris, Ser. I 346 (2008). Many machine learning tasks, such as learning with invariance and policy evaluation in reinforcement learning, can be characterized as problems of learning from conditional distributions. In such problems, each sample @math itself is associated with a conditional distribution @math represented by samples @math , and the goal is to learn a function @math that links these conditional distributions to target values @math . These learning problems become very challenging when we only have limited samples or in the extreme case only one sample from each conditional distribution. Commonly used approaches either assume that @math is independent of @math , or require an overwhelmingly large samples from each conditional distribution. To address these challenges, we propose a novel approach which employs a new min-max reformulation of the learning from conditional distribution problem. With such new reformulation, we only need to deal with the joint distribution @math . We also design an efficient learning algorithm, Embedding-SGD, and establish theoretical sample complexity for such problems. Finally, our numerical experiments on both synthetic and real-world datasets show that the proposed approach can significantly improve over the existing algorithms. We consider a large class of random geometric graphs constructed from samples @math of independent, identically distributed observations of an underlying probability measure @math on a bounded domain @math . The popular modularity' clustering method specifies a partition @math of the set @math as the solution of an optimization problem. In this paper, under conditions on @math and @math , we derive scaling limits of the modularity clustering on random geometric graphs. Among other results, we show a geometric form of consistency: When the number of clusters is a priori bounded above, the discrete optimal partitions @math converge in a certain sense to a continuum partition @math of the underlying domain @math , characterized as the solution of a type of Kelvin's shape optimization problem.