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Register a new userQuantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information
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Added by
Pedro Mediano
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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This article introduces a model-agnostic approach to study statistical synergy, a form of emergence in which patterns at large scales are not traceable from lower scales. Our framework leverages various multivariate extensions of Shannons mutual information, and introduces the O-information as a metric capable of characterising synergy- and redundancy-dominated systems. We develop key analytical properties of the O-information, and study how it relates to other metrics of high-order interactions from the statistical mechanics and neuroscience literature. Finally, as a proof of concept, we use the proposed framework to explore the relevance of statistical synergy in Baroque music scores.
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Besides mimicking bio-chemical and multi-scale communication mechanisms, molecular communication forms a theoretical framework for virus infection processes. Towards this goal, aerosol and droplet transmission has recently been modeled as a multiuser scenario. In this letter, the infection performance is evaluated by means of a mutual information analysis, and by an even simpler probabilistic performance measure which is closely related to absorbed viruses. The so-called infection rate depends on the distribution of the channel input events as well as on the transition probabilities between channel input and output events. The infection rate is investigated analytically for five basic discrete memoryless channel models. Numerical results for the transition probabilities are obtained by Monte Carlo simulations for pathogen-laden particle transmission in four typical indoor environments: two-person office, corridor, classroom, and bus. Particle transfer contributed significantly to infectious diseases like SARS-CoV-2 and influenza.
The mutual information between two jointly distributed random variables $X$ and $Y$ is a functional of the joint distribution $P_{XY},$ which is sometimes difficult to handle or estimate. A coarser description of the statistical behavior of $(X,Y)$ is given by the marginal distributions $P_X, P_Y$ and the adjacency relation induced by the joint distribution, where $x$ and $y$ are adjacent if $P(x,y)>0$. We derive a lower bound on the mutual information in terms of these entities. The bound is obtained by viewing the channel from $X$ to $Y$ as a probability distribution on a set of possible actions, where an action determines the output for any possible input, and is independently drawn. We also provide an alternative proof based on convex optimization, that yields a generally tighter bound. Finally, we derive an upper bound on the mutual information in terms of adjacency events between the action and the pair $(X,Y)$, where in this case an action $a$ and a pair $(x,y)$ are adjacent if $y=a(x)$. As an example, we apply our bounds to the binary deletion channel and show that for the special case of an i.i.d. input distribution and a range of deletion probabilities, our lower and upper bounds both outperform the best known bounds for the mutual information.
Machine learning models are known to memorize the unique properties of individual data points in a training set. This memorization capability can be exploited by several types of attacks to infer information about the training data, most notably, membership inference attacks. In this paper, we propose an approach based on information leakage for guaranteeing membership privacy. Specifically, we propose to use a conditional form of the notion of maximal leakage to quantify the information leaking about individual data entries in a dataset, i.e., the entrywise information leakage. We apply our privacy analysis to the Private Aggregation of Teacher Ensembles (PATE) framework for privacy-preserving classification of sensitive data and prove that the entrywise information leakage of its aggregation mechanism is Schur-concave when the injected noise has a log-concave probability density. The Schur-concavity of this leakage implies that increased consensus among teachers in labeling a query reduces its associated privacy cost. Finally, we derive upper bounds on the entrywise information leakage when the aggregation mechanism uses Laplace distributed noise.
High-order, beyond-pairwise interdependencies are at the core of biological, economic, and social complex systems, and their adequate analysis is paramount to understand, engineer, and control such systems. This paper presents a framework to measure high-order interdependence that disentangles their effect on each individual pattern exhibited by a multivariate system. The approach is centred on the local O-information, a new measure that assesses the balance between synergistic and redundant interdependencies at each pattern. To illustrate the potential of this framework, we present a detailed analysis of music scores from J.S. Bach, which reveals how high-order interdependence is deeply connected with highly non-trivial aspects of the musical discourse. Our results place the local O-information as a promising tool of wide applicability, which opens new perspectives for analysing high-order relationships in the patterns exhibited by complex systems.
To provide an efficient approach to characterize the input-output mutual information (MI) under additive white Gaussian noise (AWGN) channel, this short report fits the curves of exact MI under multilevel quadrature amplitude modulation (M-QAM) signal inputs via multi-exponential decay curve fitting (M-EDCF). Even though the definition expression for instanious MI versus Signal to Noise Ratio (SNR) is complex and the containing integral is intractable, our new developed fitting formula holds a neat and compact form, which possesses high precision as well as low complexity. Generally speaking, this approximation formula of MI can promote the research of performance analysis in practical communication system under discrete inputs.
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