Johannes Rauh

Johannes Rauh, Nihat Ay

We study a notion of robustness of a Markov kernel that describes a system of several input random variables and one output random variable. Robustness requires that the behaviour of the system does not change if one or several of the input variables are knocked out. If the system is required to be robust against too many knockouts, then the output variable cannot distinguish reliably between input states and must be independent of the input. We study how many input states the output variable can distinguish as a function of the required level of robustness. Gibbs potentials allow a mechanistic description of the behaviour of the system after knockouts. Robustness imposes structural constraints on these potentials. We show that interaction families of Gibbs potentials allow to describe robust systems. Given a distribution of the input random variables and the Markov kernel describing the system, we obtain a joint probability distribution. Robustness implies a number of conditional independence statements for this joint distribution. The set of all probability distributions corresponding to robust systems can be decomposed into a finite union of components, and we find parametrizations of the components. The decomposition corresponds to a primary decomposition of the conditional independence ideal and can be derived from more general results about generalized binomial edge ideals.

Johannes Rauh, Pradeep Kr. Banerjee, Eckehard Olbrich et al.

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.

Pradeep Kr. Banerjee, Eckehard Olbrich, Jürgen Jost et al.

Given two channels that convey information about the same random variable, we introduce two measures of the unique information of one channel with respect to the other. The two quantities are based on the notion of generalized weighted Le Cam deficiencies and differ on whether one channel can approximate the other by a randomization at either its input or output. We relate the proposed quantities to an existing measure of unique information which we call the minimum-synergy unique information. We give an operational interpretation of the latter in terms of an upper bound on the one-way secret key rate and discuss the role of the unique informations in the context of nonnegative mutual information decompositions into unique, redundant and synergistic components.

Guido Montufar, Johannes Rauh

We investigate the geometry of optimal memoryless time independent decision making in relation to the amount of information that the acting agent has about the state of the system. We show that the expected long term reward, discounted or per time step, is maximized by policies that randomize among at most $k$ actions whenever at most $k$ world states are consistent with the agent's observation. Moreover, we show that the expected reward per time step can be studied in terms of the expected discounted reward. Our main tool is a geometric version of the policy improvement lemma, which identifies a polyhedral cone of policy changes in which the state value function increases for all states.

Guido Montufar, Johannes Rauh

We investigate the representation of hierarchical models in terms of marginals of other hierarchical models with smaller interactions. We focus on binary variables and marginals of pairwise interaction models whose hidden variables are conditionally independent given the visible variables. In this case the problem is equivalent to the representation of linear subspaces of polynomials by feedforward neural networks with soft-plus computational units. We show that every hidden variable can freely model multiple interactions among the visible variables, which allows us to generalize and improve previous results. In particular, we show that a restricted Boltzmann machine with less than $[ 2(\log(v)+1) / (v+1) ] 2^v-1$ hidden binary variables can approximate every distribution of $v$ visible binary variables arbitrarily well, compared to $2^{v-1}-1$ from the best previously known result.

Keyan Ghazi-Zahedi, Johannes Rauh

The question how an agent is affected by its embodiment has attracted growing attention in recent years. A new field of artificial intelligence has emerged, which is based on the idea that intelligence cannot be understood without taking into account embodiment. We believe that a formal approach to quantifying the embodiment's effect on the agent's behaviour is beneficial to the fields of artificial life and artificial intelligence. The contribution of an agent's body and environment to its behaviour is also known as morphological computation. Therefore, in this work, we propose a quantification of morphological computation, which is based on an information decomposition of the sensorimotor loop into shared, unique and synergistic information. In numerical simulation based on a formal representation of the sensorimotor loop, we show that the unique information of the body and environment is a good measure for morphological computation. The results are compared to our previously derived quantification of morphological computation.

Guido Montufar, Johannes Rauh, Nihat Ay

We present explicit classes of probability distributions that can be learned by Restricted Boltzmann Machines (RBMs) depending on the number of units that they contain, and which are representative for the expressive power of the model. We use this to show that the maximal Kullback-Leibler divergence to the RBM model with $n$ visible and $m$ hidden units is bounded from above by $n - \left\lfloor \log(m+1) \right\rfloor - \frac{m+1}{2^{\left\lfloor\log(m+1)\right\rfloor}} \approx (n -1) - \log(m+1)$. In this way we can specify the number of hidden units that guarantees a sufficiently rich model containing different classes of distributions and respecting a given error tolerance.

Guido Montufar, Johannes Rauh, Nihat Ay

We review recent results about the maximal values of the Kullback-Leibler information divergence from statistical models defined by neural networks, including naive Bayes models, restricted Boltzmann machines, deep belief networks, and various classes of exponential families. We illustrate approaches to compute the maximal divergence from a given model starting from simple sub- or super-models. We give a new result for deep and narrow belief networks with finite-valued units.

Guido F. Montufar, Johannes Rauh

We compute the expected value of the Kullback-Leibler divergence to various fundamental statistical models with respect to canonical priors on the probability simplex. We obtain closed formulas for the expected model approximation errors, depending on the dimension of the models and the cardinalities of their sample spaces. For the uniform prior, the expected divergence from any model containing the uniform distribution is bounded by a constant $1-γ$, and for the models that we consider, this bound is approached if the state space is very large and the models' dimension does not grow too fast. For Dirichlet priors the expected divergence is bounded in a similar way, if the concentration parameters take reasonable values. These results serve as reference values for more complicated statistical models.