We study how to lift Markov bases and Grobner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber pr
oduct of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.
The information that two random variables $Y$, $Z$ contain about a third random variable $X$ can have aspects of shared information (contained in both $Y$ and $Z$), of complementary information (only available from $(Y,Z)$ together) and of unique inf
ormation (contained exclusively in either $Y$ or $Z$). Here, we study measures $widetilde{SI}$ of shared, $widetilde{UI}$ unique and $widetilde{CI}$ complementary information introduced by Bertschinger et al., which are motivated from a decision theoretic perspective. We find that in most cases the intuitive rule that more variables contain more information applies, with the exception that $widetilde{SI}$ and $widetilde{CI}$ information are not monotone in the target variable $X$. Additionally, we show that it is not possible to extend the bivariate information decomposition into $widetilde{SI}$, $widetilde{UI}$ and $widetilde{CI}$ to a non-negative decomposition on the partial information lattice of Williams and Beer. Nevertheless, the quantities $widetilde{UI}$, $widetilde{SI}$ and $widetilde{CI}$ have a well-defined interpretation, even in the multivariate setting.
Blackwells theorem shows the equivalence of two preorders on the set of information channels. Here, we restate, and slightly generalize, his result in terms of random variables. Furthermore, we prove that the corresponding partial order is not a latt
ice; that is, least upper bounds and greatest lower bounds do not exist.
This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Grobner basis can be computed by studying paths in
the graph. Since these Grobner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.
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.
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 t
he 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.
