Robustness and Conditional Independence Ideals

We study notions of robustness of Markov kernels and probability distribution of a system that is described by $n$ input random variables and one output random variable. Markov kernels can be expanded in a series of potentials that allow to describe the system's behaviour after knockouts. Robustness imposes structural constraints on these potentials. Robustness of probability distributions is defined via conditional independence statements. These statements can be studied algebraically. The corresponding conditional independence ideals are related to binary edge ideals. The set of robust probability distributions lies on an algebraic variety. We compute a Gr\"obner basis of this ideal and study the irreducible decomposition of the variety. These algebraic results allow to parametrize the set of all robust probability distributions.Comment: 16 page

Adaptive Dynamics for Interacting Markovian Processes

Dynamics of information flow in adaptively interacting stochastic processes is studied. We give an extended form of game dynamics for Markovian processes and study its behavior to observe information flow through the system. Examples of the adaptive dynamics for two stochastic processes interacting through matching pennies game interaction are exhibited along with underlying causal structure

Quantifying Morphological Computation

The field of embodied intelligence emphasises the importance of the morphology and environment with respect to the behaviour of a cognitive system. The contribution of the morphology to the behaviour, commonly known as morphological computation, is well-recognised in this community. We believe that the field would benefit from a formalisation of this concept as we would like to ask how much the morphology and the environment contribute to an embodied agent's behaviour, or how an embodied agent can maximise the exploitation of its morphology within its environment. In this work we derive two concepts of measuring morphological computation, and we discuss their relation to the Information Bottleneck Method. The first concepts asks how much the world contributes to the overall behaviour and the second concept asks how much the agent's action contributes to a behaviour. Various measures are derived from the concepts and validated in two experiments which highlight their strengths and weaknesses

Information-theoretic inference of common ancestors

A directed acyclic graph (DAG) partially represents the conditional independence structure among observations of a system if the local Markov condition holds, that is, if every variable is independent of its non-descendants given its parents. In general, there is a whole class of DAGs that represents a given set of conditional independence relations. We are interested in properties of this class that can be derived from observations of a subsystem only. To this end, we prove an information theoretic inequality that allows for the inference of common ancestors of observed parts in any DAG representing some unknown larger system. More explicitly, we show that a large amount of dependence in terms of mutual information among the observations implies the existence of a common ancestor that distributes this information. Within the causal interpretation of DAGs our result can be seen as a quantitative extension of Reichenbach's Principle of Common Cause to more than two variables. Our conclusions are valid also for non-probabilistic observations such as binary strings, since we state the proof for an axiomatized notion of mutual information that includes the stochastic as well as the algorithmic version.Comment: 18 pages, 4 figure

Refinements of Universal Approximation Results for Deep Belief Networks and Restricted Boltzmann Machines

We improve recently published results about resources of Restricted Boltzmann Machines (RBM) and Deep Belief Networks (DBN) required to make them Universal Approximators. We show that any distribution p on the set of binary vectors of length n can be arbitrarily well approximated by an RBM with k-1 hidden units, where k is the minimal number of pairs of binary vectors differing in only one entry such that their union contains the support set of p. In important cases this number is half of the cardinality of the support set of p. We construct a DBN with 2^n/2(n-b), b ~ log(n), hidden layers of width n that is capable of approximating any distribution on {0,1}^n arbitrarily well. This confirms a conjecture presented by Le Roux and Bengio 2010