  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.
