  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.
