arXiv subjects

Johannes Rauh

Publications and source records attributed to Johannes Rauh.

41 records · Page 3Linked to original sources

arXiv2011

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 Gröbner 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.

math.AC↗

arXiv2011

Support Sets in Exponential Families and Oriented Matroid Theory

The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.

math.ST↗

arXiv2010

Knockouts, Robustness and Cell Cycles

The response to a knockout of a node is a characteristic feature of a networked dynamical system. Knockout resilience in the dynamics of the remaining nodes is a sign of robustness. Here we study the effect of knockouts for binary state sequences and their implementations in terms of Boolean threshold networks. Beside random sequences with biologically plausible constraints, we analyze the cell cycle sequence of the species Saccharomyces cerevisiae and the Boolean networks implementing it. Comparing with an appropriate null model we do not find evidence that the yeast wildtype network is optimized for high knockout resilience. Our notion of knockout resilience weakly correlates with the size of the basin of attraction, which has also been considered a measure of robustness.

q-bio.MN↗

arXiv2009

Finding the Maximizers of the Information Divergence from an Exponential Family

This paper investigates maximizers of the information divergence from an exponential family $E$. It is shown that the $rI$-projection of a maximizer $P$ to $E$ is a convex combination of $P$ and a probability measure $P_-$ with disjoint support and the same value of the sufficient statistics $A$. This observation can be used to transform the original problem of maximizing $D(\cdot||E)$ over the set of all probability measures into the maximization of a function $\Dbar$ over a convex subset of $\ker A$. The global maximizers of both problems correspond to each other. Furthermore, finding all local maximizers of $\Dbar$ yields all local maximizers of $D(\cdot||E)$. This paper also proposes two algorithms to find the maximizers of $\Dbar$ and applies them to two examples, where the maximizers of $D(\cdot||E)$ were not known before.

cs.IT↗

arXiv2009

Binomial edge ideals and conditional independence statements

We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation

math.AC↗