We define a measure of redundant information based on projections in the space of probability distributions. Redundant information between random variables is information that is shared between those variables. But in contrast to mutual information, redundant information denotes information that is shared about the outcome of a third variable. Formalizing this concept, and being able to measure it, is required for the non-negative decomposition of mutual information into redundant and synergistic information. Previous attempts to formalize redundant or synergistic information struggle to capture some desired properties. We introduce a new formalism for redundant information and prove that it satisfies all the properties necessary outlined in earlier work, as well as an additional criterion that we propose to be necessary to capture redundancy. We also demonstrate the behaviour of this new measure for several examples, compare it to previous measures and apply it to the decomposition of transfer entropy.

We use a notion of causal independence based on intervention, which is a fundamental concept of the theory of causal networks, to define a measure for the strength of a causal effect. We call this measure "information flow" and compare it with known information flow measures such as transfer entropy.

The perception-action cycle is often defined as "the circular flow of information between an organism and its environment in the course of a sensory guided sequence of actions towards a goal" (Fuster 2001(Fuster , 2006. The question we address in this paper is in what sense this "flow of information" can be described by Shannon's measures of information introduced in his mathematical theory of communication. We provide an affirmative answer to this question using an intriguing analogy between Shannon's classical model of communication and the Perception-Action-Cycle. In particular, decision and action sequences turn out to be directly analogous to codes in communication, and their complexity -the minimal number of (binary) decisions required for reaching a goal -directly bounded by information measures, as in communication. This analogy allows us to extend the standard Reinforcement Learning framework. The latter considers the future expected reward in the course of a behaviour sequence towards a goal (value-to-go). Here, we additionally incorporate a measure of information associated with this sequence: the cumulated information processing cost or bandwidth required to specify the future decision and action sequence (information-to-go).Using a graphical model, we derive a recursive Bellman optimality equation for information measures, in analogy to Reinforcement Learning; from this, we obtain new algorithms for calculating the optimal trade-off between the value-to-go and the required information-to-go, unifying the ideas behind the Bellman and the Blahut-Arimoto iterations. This trade-off between value-to-go and information-togo provides a complete analogy with the compression-distortion trade-off in source coding. The present new formulation connects seemingly unrelated optimization problems. The algorithm is demonstrated on grid world examples.

Abstract-The classical approach to using utility functions suffers from the drawback of having to design and tweak the functions on a case by case basis. Inspired by examples from the animal kingdom, social sciences and games we propose empowerment, a rather universal function, defined as the information-theoretic capacity of an agent's actuation channel. The concept applies to any sensorimotor apparatus. Empowerment as a measure reflects the properties of the apparatus as long as they are observable due to the coupling of sensors and actuators via the environment. Using two simple experiments we also demonstrate how empowerment influences sensor-actuator evolution.

The central resource processed by the sensorimotor system of an organism is information. We propose an information-based quantity that allows one to characterize the efficiency of the perception-action loop of an abstract organism model. It measures the potential of the organism to imprint information on the environment via its actuators in a way that can be recaptured by its sensors, essentially quantifying the options available and visible to the organism. Various scenarios suggest that such a quantity could identify the preferred direction of evolution or adaptation of the sensorimotor loop of organisms.