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Register a new userInvariant components of synergy, redundancy, and unique information among three variables
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Added by
Giuseppe Pica
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In a system of three stochastic variables, the Partial Information Decomposition (PID) of Williams and Beer dissects the information that two variables (sources) carry about a third variable (target) into nonnegative information atoms that describe redundant, unique, and synergistic modes of dependencies among the variables. However, the classification of the three variables into two sources and one target limits the dependency modes that can be quantitatively resolved, and does not naturally suit all systems. Here, we extend the PID to describe trivariate modes of dependencies in full generality, without introducing additional decomposition axioms or making assumptions about the target/source nature of the variables. By comparing different PID lattices of the same system, we unveil a finer PID structure made of seven nonnegative information subatoms that are invariant to different target/source classifications and that are sufficient to construct any PID lattice. This finer structure naturally splits redundant information into two nonnegative components: the source redundancy, which arises from the pairwise correlations between the source variables, and the non-source redundancy, which does not, and relates to the synergistic information the sources carry about the target. The invariant structure is also sufficient to construct the systems entropy, hence it characterizes completely all the interdependencies in the system.
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We consider the problem of decomposing the total mutual information conveyed by a pair of predictor random variables about a target random variable into redundant, unique and synergistic contributions. We focus on the relationship between redundant information and the more familiar information-theoretic notions of common information. Our main contribution is an impossibility result. We show that for independent predictor random variables, any common information based measure of redundancy cannot induce a nonnegative decomposition of the total mutual information. Interestingly, this entails that any reasonable measure of redundant information cannot be derived by optimization over a single random variable.
The unique information ($UI$) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the $UI$, which implies that the $UI$ is never greater than one of the best known upper bounds on the two-way secret key rate. We conjecture that the $UI$ lower bounds the two-way rate and discuss implications of the conjecture.
The partial information decomposition (PID) is a promising framework for decomposing a joint random variable into the amount of influence each source variable Xi has on a target variable Y, relative to the other sources. For two sources, influence breaks down into the information that both X0 and X1 redundantly share with Y, what X0 uniquely shares with Y, what X1 uniquely shares with Y, and finally what X0 and X1 synergistically share with Y. Unfortunately, considerable disagreement has arisen as to how these four components should be quantified. Drawing from cryptography, we consider the secret key agreement rate as an operational method of quantifying unique informations. Secret key agreement rate comes in several forms, depending upon which parties are permitted to communicate. We demonstrate that three of these four forms are inconsistent with the PID. The remaining form implies certain interpretations as to the PIDs meaning---interpretations not present in PIDs definition but that, we argue, need to be explicit. These reveal an inconsistency between third-order connected information, two-way secret key agreement rate, and synergy. Similar difficulties arise with a popular PID measure in light the results here as well as from a maximum entropy viewpoint. We close by reviewing the challenges facing the PID.
Selecting a minimal feature set that is maximally informative about a target variable is a central task in machine learning and statistics. Information theory provides a powerful framework for formulating feature selection algorithms -- yet, a rigorous, information-theoretic definition of feature relevancy, which accounts for feature interactions such as redundant and synergistic contributions, is still missing. We argue that this lack is inherent to classical information theory which does not provide measures to decompose the information a set of variables provides about a target into unique, redundant, and synergistic contributions. Such a decomposition has been introduced only recently by the partial information decomposition (PID) framework. Using PID, we clarify why feature selection is a conceptually difficult problem when approached using information theory and provide a novel definition of feature relevancy and redundancy in PID terms. From this definition, we show that the conditional mutual information (CMI) maximizes relevancy while minimizing redundancy and propose an iterative, CMI-based algorithm for practical feature selection. We demonstrate the power of our CMI-based algorithm in comparison to the unconditional mutual information on benchmark examples and provide corresponding PID estimates to highlight how PID allows to quantify information contribution of features and their interactions in feature-selection problems.
The maximum possible throughput (or the rate of job completion) of a multi-server system is typically the sum of the service rates of individual servers. Recent work shows that launching multiple replicas of a job and canceling them as soon as one copy finishes can boost the throughput, especially when the service time distribution has high variability. This means that redundancy can, in fact, create synergy among servers such that their overall throughput is greater than the sum of individual servers. This work seeks to find the fundamental limit of the throughput boost achieved by job replication and the optimal replication policy to achieve it. While most previous works consider upfront replication policies, we expand the set of possible policies to delayed launch of replicas. The search for the optimal adaptive replication policy can be formulated as a Markov Decision Process, using which we propose two myopic replication policies, MaxRate and AdaRep, to adaptively replicate jobs. In order to quantify the optimality gap of these and other policies, we derive upper bounds on the service capacity, which provide fundamental limits on the throughput of queueing systems with redundancy.
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