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Register a new userAn operational information decomposition via synergistic disclosure
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Added by
Fernando Rosas
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Multivariate information decompositions hold promise to yield insight into complex systems, and stand out for their ability to identify synergistic phenomena. However, the adoption of these approaches has been hindered by there being multiple possible decompositions, and no precise guidance for preferring one over the others. At the heart of this disagreement lies the absence of a clear operational interpretation of what synergistic information is. Here we fill this gap by proposing a new information decomposition based on a novel operationalisation of informational synergy, which leverages recent developments in the literature of data privacy. Our decomposition is defined for any number of information sources, and its atoms can be calculated using elementary optimisation techniques. The decomposition provides a natural coarse-graining that scales gracefully with the systems size, and is applicable in a wide range of scenarios of practical interest.
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Given two random variables $X$ and $Y$, an operational approach is undertaken to quantify the ``leakage of information from $X$ to $Y$. The resulting measure $mathcal{L}(X !! to !! Y)$ is called emph{maximal leakage}, and is defined as the multiplicative increase, upon observing $Y$, of the probability of correctly guessing a randomized function of $X$, maximized over all such randomized functions. A closed-form expression for $mathcal{L}(X !! to !! Y)$ is given for discrete $X$ and $Y$, and it is subsequently generalized to handle a large class of random variables. The resulting properties are shown to be consistent with an axiomatic view of a leakage measure, and the definition is shown to be robust to variations in the setup. Moreover, a variant of the Shannon cipher system is studied, in which performance of an encryption scheme is measured using maximal leakage. A single-letter characterization of the optimal limit of (normalized) maximal leakage is derived and asymptotically-optimal encryption schemes are demonstrated. Furthermore, the sample complexity of estimating maximal leakage from data is characterized up to subpolynomial factors. Finally, the emph{guessing} framework used to define maximal leakage is used to give operational interpretations of commonly used leakage measures, such as Shannon capacity, maximal correlation, and local differential privacy.
The information that two random variables $Y$, $Z$ contain about a third random variable $X$ can have aspects of shared information (contained in both $Y$ and $Z$), of complementary information (only available from $(Y,Z)$ together) and of unique information (contained exclusively in either $Y$ or $Z$). Here, we study measures $widetilde{SI}$ of shared, $widetilde{UI}$ unique and $widetilde{CI}$ complementary information introduced by Bertschinger et al., which are motivated from a decision theoretic perspective. We find that in most cases the intuitive rule that more variables contain more information applies, with the exception that $widetilde{SI}$ and $widetilde{CI}$ information are not monotone in the target variable $X$. Additionally, we show that it is not possible to extend the bivariate information decomposition into $widetilde{SI}$, $widetilde{UI}$ and $widetilde{CI}$ to a non-negative decomposition on the partial information lattice of Williams and Beer. Nevertheless, the quantities $widetilde{UI}$, $widetilde{SI}$ and $widetilde{CI}$ have a well-defined interpretation, even in the multivariate setting.
The characterisation of information processing is an important task in complex systems science. Information dynamics is a quantitative methodology for modelling the intrinsic information processing conducted by a process represented as a time series, but to date has only been formulated in discrete time. Building on previous work which demonstrated how to formulate transfer entropy in continuous time, we give a total account of information processing in this setting, incorporating information storage. We find that a convergent rate of predictive capacity, comprised of the transfer entropy and active information storage, does not exist, arising through divergent rates of active information storage. We identify that active information storage can be decomposed into two separate quantities that characterise predictive capacity stored in a process: active memory utilisation and instantaneous predictive capacity. The latter involves prediction related to path regularity and so solely inherits the divergent properties of the active information storage, whilst the former permits definitions of pathwise and rate quantities. We formulate measures of memory utilisation for jump and neural spiking processes and illustrate measures of information processing in synthetic neural spiking models and coupled Ornstein-Uhlenbeck models. The application to synthetic neural spiking models demonstrates that active memory utilisation for point processes consists of discontinuous jump contributions (at spikes) interrupting a continuously varying contribution (relating to waiting times between spikes), complementing the behaviour previously demonstrated for transfer entropy in these processes.
We offer a new approach to the information decomposition problem in information theory: given a target random variable co-distributed with multiple source variables, how can we decompose the mutual information into a sum of non-negative terms that quantify the contributions of each random variable, not only individually but also in combination? We derive our composition from cooperative game theory. It can be seen as assigning a fair share of the mutual information to each combination of the source variables. Our decomposition is based on a different lattice from the usual partial information decomposition (PID) approach, and as a consequence our decomposition has a smaller number of terms: it has analogs of the synergy and unique information terms, but lacks terms corresponding to redundancy. Because of this, it is able to obey equivalents of the axioms known as local positivity and identity, which cannot be simultaneously satisfied by a PID measure.
This work constructs a discrete random variable that, when conditioned upon, ensures information stability of quasi-images. Using this construction, a new methodology is derived to obtain information theoretic necessary conditions directly from operational requirements. In particular, this methodology is used to derive new necessary conditions for keyed authentication over discrete memoryless channels and to establish the capacity region of the wiretap channel, subject to finite leakage and finite error, under two different secrecy metrics. These examples establish the usefulness of the proposed methodology.
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