Information-theoretic inference of common ancestors

Steudel, Bastian; Ay, Nihat

doi:10.48550/arxiv.1010.5720

Cited by 9 publications

(24 citation statements)

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“…See appendix for details of how this curves are computed. causal inference, namely the inference of latent common ancestors [14,44]. As we will show next, the topology alone of these quantum networks imply non-trivial constraints on the correlations that can be obtained between the different parties.…”

Section: Quantum Networkmentioning

confidence: 99%

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Information–theoretic implications of quantum causal structures

2015

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It is a relatively new insight of classical statistics that empirical data can contain information about causation rather than mere correlation. First algorithms have been proposed that are capable of testing whether a presumed causal relationship is compatible with an observed distribution. However, no systematic method is known for treating such problems in a way that generalizes to quantum systems. Here, we describe a general algorithm for computing information-theoretic constraints on the correlations that can arise from a given causal structure, where we allow for quantum systems as well as classical random variables. The general technique is applied to two relevant cases: first, we show that the principle of information causality appears naturally in our framework and go on to generalize and strengthen it. Second, we derive bounds on the correlations that can occur in a networked architecture, where a set of few-body quantum systems is distributed among some parties.

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Section: Quantum Networkmentioning

confidence: 99%

“…Given some observed correlations between them, the basic question is then: Can the correlations between these n variables be explained by (hidden) common ancestors connecting at most 2 of them? The simplest of such common ancestors scenarios (n = 3), the so called triangle scenario [5,44,45], is illustrated in Fig. 1.…”

Section: Quantum Networkmentioning

confidence: 99%

Information–theoretic implications of quantum causal structures

2015

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“…The triangle scenario, shown in fig. 4, has already received some interest in quantum foundations [8,18,11] and the causality literature [38]. Branciard et al initially introduced the scenario with definitions matching our C and Q [8].…”

Section: The Trianglementioning

confidence: 99%

Theory-independent limits on correlations from generalized Bayesian networks

Henson¹,

Lal²,

Pusey³

2014

New J. Phys.

198

391

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Bayesian networks provide a powerful tool for reasoning about probabilistic causation, used in many areas of science. They are, however, intrinsically classical. In particular, Bayesian networks naturally yield the Bell inequalities. Inspired by this connection, we generalize the formalism of classical Bayesian networks in order to investigate non-classical correlations in arbitrary causal structures. Our framework of 'generalized Bayesian networks' replaces latent variables with the resources of any generalized probabilistic theory, most importantly quantum theory, but also, for example, Popescu-Rohrlich boxes. We obtain three main sets of results. Firstly, we prove that all of the observable conditional independences required by the classical theory also hold in our generalization; to obtain this, we extend the classical d-separation theorem to our setting. Secondly, we find that the theory-independent constraints on probabilities can go beyond these conditional independences. For example we find that no probabilistic theory predicts perfect correlation between three parties using only bipartite common causes. Finally, we begin a classification of those causal structures, such as the Bell scenario, that may yield a separation between classical, quantum and general-probabilistic correlations.

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“…Methods to automatically size representations to optimize this trade-off will be explored in future work. Other intriguing directions include using the bounds presented to characterize RBMs and auto-encoders [1], and exploring connections to the information bottleneck [21,22], multivariate information measures [13,14,17], EM [23,24], and "recognition models" [25].…”

Section: Discussionmentioning

confidence: 99%

“…The name is motivated by results that show that if AI α (X; Y ) is positive for some α, it implies the existence of common ancestors for some (α-dependent) set of X i 's in any DAG that describes X [17]. We do not make use of those results, but the overlap in expressions is suggestive.…”

Section: A Single Latent Factormentioning

confidence: 99%

Maximally Informative Hierarchical Representations of High-Dimensional Data

Steeg,

Galstyan

2014

Preprint

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We consider a set of probabilistic functions of some input variables as a representation of the inputs. We present bounds on how informative a representation is about input data. We extend these bounds to hierarchical representations so that we can quantify the contribution of each layer towards capturing the information in the original data. The special form of these bounds leads to a simple, bottom-up optimization procedure to construct hierarchical representations that are also maximally informative about the data. This optimization has linear computational complexity and constant sample complexity in the number of variables. These results establish a new approach to unsupervised learning of deep representations that is both principled and practical. We demonstrate the usefulness of the approach on both synthetic and real-world data. This paper considers the problem of unsupervised learning of hierarchical representations from highdimensional data. Deep representations are becoming increasingly indispensable for solving the greatest challenges in machine learning including problems in image recognition, speech, and language [1]. Theoretical foundations have not kept pace, making it difficult to understand why existing methods fail in some domains and succeed in others. Here, we start from the abstract point of view that any probabilistic functions of some input variables constitute a representation. The usefulness of a representation depends on (unknown) details about the data distribution. Instead of mak-

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Information-theoretic inference of common ancestors

Cited by 9 publications

References 0 publications

Information–theoretic implications of quantum causal structures

Information–theoretic implications of quantum causal structures

Theory-independent limits on correlations from generalized Bayesian networks

Maximally Informative Hierarchical Representations of High-Dimensional Data

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