A Perspective on Unique Information: Directionality, Intuitions, and Secret Key Agreement

Recently, the partial information decomposition emerged as a promising framework for identifying the meaningful components of the information contained in a joint distribution. Its adoption and practical application, however, have been stymied by the lack of a generally-accepted method of quantifying its components. Here, we briefly discuss the bivariate (two-source) partial information decomposition and two implicitly directional interpretations used to intuitively motivate alternative component definitions. Drawing parallels with secret key agreement rates from information-theoretic cryptography, we demonstrate that these intuitions are mutually incompatible and suggest that this underlies the persistence of competing definitions and interpretations. Having highlighted this hitherto unacknowledged issue, we outline several possible solutions.Comment: 5 pages, 3 tables; http://csc.ucdavis.edu/~cmg/compmech/pubs/pid_intuition.ht

Unique Information via Dependency Constraints

The partial information decomposition (PID) is perhaps the leading proposal for resolving information shared between a set of sources and a target into redundant, synergistic, and unique constituents. Unfortunately, the PID framework has been hindered by a lack of a generally agreed-upon, multivariate method of quantifying the constituents. Here, we take a step toward rectifying this by developing a decomposition based on a new method that quantifies unique information. We first develop a broadly applicable method---the dependency decomposition---that delineates how statistical dependencies influence the structure of a joint distribution. The dependency decomposition then allows us to define a measure of the information about a target that can be uniquely attributed to a particular source as the least amount which the source-target statistical dependency can influence the information shared between the sources and the target. The result is the first measure that satisfies the core axioms of the PID framework while not satisfying the Blackwell relation, which depends on a particular interpretation of how the variables are related. This makes a key step forward to a practical PID.Comment: 15 pages, 7 figures, 2 tables, 3 appendices; http://csc.ucdavis.edu/~cmg/compmech/pubs/idep.ht

Unique Information and Secret Key Agreement

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 PID's meaning---interpretations not present in PID's 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.Comment: 9 pages, 3 figures, 4 tables; http://csc.ucdavis.edu/~cmg/compmech/pubs/pid_skar.htm. arXiv admin note: text overlap with arXiv:1808.0860

Locally Homogeneous Spaces, Induced Killing Vector Fields and Applications to Bianchi Prototypes

An answer to the question: Can, in general, the adoption of a given symmetry induce a further symmetry, which might be hidden at a first level? has been attempted in the context of differential geometry of locally homogeneous spaces. Based on E. Cartan's theory of moving frames, a methodology for finding all symmetries for any n dimensional locally homogeneous space is provided. The analysis is applied to 3 dimensional spaces, whereby the embedding of them into a 4 dimensional Lorentzian manifold is examined and special solutions to Einstein's field equations are recovered. The analysis is mainly of local character, since the interest is focused on local structures based on differential equations (and their symmetries), rather than on the implications of, e.g., the analytic continuation of their solution(s) and their dynamics in the large.Comment: 27 pages, no figues, no tables, one reference added, spelling and punctuation issues correcte

Modes of Information Flow

Information flow between components of a system takes many forms and is key to understanding the organization and functioning of large-scale, complex systems. We demonstrate three modalities of information flow from time series X to time series Y. Intrinsic information flow exists when the past of X is individually predictive of the present of Y, independent of Y's past; this is most commonly considered information flow. Shared information flow exists when X's past is predictive of Y's present in the same manner as Y's past; this occurs due to synchronization or common driving, for example. Finally, synergistic information flow occurs when neither X's nor Y's pasts are predictive of Y's present on their own, but taken together they are. The two most broadly-employed information-theoretic methods of quantifying information flow---time-delayed mutual information and transfer entropy---are both sensitive to a pair of these modalities: time-delayed mutual information to both intrinsic and shared flow, and transfer entropy to both intrinsic and synergistic flow. To quantify each mode individually we introduce our cryptographic flow ansatz, positing that intrinsic flow is synonymous with secret key agreement between X and Y. Based on this, we employ an easily-computed secret-key-agreement bound---intrinsic mutual information&mdashto quantify the three flow modalities in a variety of systems including asymmetric flows and financial markets.Comment: 11 pages; 10 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/ite.ht

Intersection Information based on Common Randomness

The introduction of the partial information decomposition generated a flurry of proposals for defining an intersection information that quantifies how much of "the same information" two or more random variables specify about a target random variable. As of yet, none is wholly satisfactory. A palatable measure of intersection information would provide a principled way to quantify slippery concepts, such as synergy. Here, we introduce an intersection information measure based on the G\'acs-K\"orner common random variable that is the first to satisfy the coveted target monotonicity property. Our measure is imperfect, too, and we suggest directions for improvement.Comment: 19 pages, 5 figure

Phytoplankton Community and Algal Toxicity at a Recurring Bloom in Sullivan Bay, Kabetogama Lake, Minnesota, USA

Kabetogama Lake in Voyageurs National Park, Minnesota, USA suffers from recurring late summer algal blooms that often contain toxin-producing cyanobacteria. Previous research identified the toxin microcystin in blooms, but we wanted to better understand how the algal and cyanobacterial community changed throughout an open water season and how changes in community structure were related to toxin production. Therefore, we sampled one recurring bloom location throughout the entire open water season. The uniqueness of this study is the absence of urban and agricultural nutrient sources, the remote location, and the collection of samples before any visible blooms were present. Through quantitative polymerase chain reaction (qPCR), we discovered that toxin-forming cyanobacteria were present before visible blooms and toxins not previously detected in this region (anatoxin-a and saxitoxin) were present, indicating that sampling for additional toxins and sampling earlier in the season may be necessary to assess ecosystems and human health risk

Antiphase dynamics in a multimode semiconductor laser with optical injection

A detailed experimental study of antiphase dynamics in a two-mode semiconductor laser with optical injection is presented. The device is a specially designed Fabry-Perot laser that supports two primary modes with a THz frequency spacing. Injection in one of the primary modes of the device leads to a rich variety of single and two-mode dynamical scenarios, which are reproduced with remarkable accuracy by a four dimensional rate equation model. Numerical bifurcation analysis reveals the importance of torus bifurcations in mediating transitions to antiphase dynamics and of saddle-node of limit cycle bifurcations in switching of the dynamics between single and two-mode regimes.Comment: 7 pages, 9 figure