120 research outputs found
Robustness and Conditional Independence Ideals
We study notions of robustness of Markov kernels and probability distribution
of a system that is described by 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.Comment: 16 page
Secret Sharing and Shared Information
Secret sharing is a cryptographic discipline in which the goal is to
distribute information about a secret over a set of participants in such a way
that only specific authorized combinations of participants together can
reconstruct the secret. Thus, secret sharing schemes are systems of variables
in which it is very clearly specified which subsets have information about the
secret. As such, they provide perfect model systems for information
decompositions. However, following this intuition too far leads to an
information decomposition with negative partial information terms, which are
difficult to interpret. One possible explanation is that the partial
information lattice proposed by Williams and Beer is incomplete and has to be
extended to incorporate terms corresponding to higher order redundancy. These
results put bounds on information decompositions that follow the partial
information framework, and they hint at where the partial information lattice
needs to be improved.Comment: 9 pages, 1 figure. The material was presented at a Workshop on
information decompositions at FIAS, Frankfurt, in 12/2016. The revision
includes changes in the definition of combinations of secret sharing schemes.
Section 3 and Appendix now discusses in how far existing measures satisfy the
proposed properties. The concluding section is considerably revise
Finding the Maximizers of the Information Divergence from an Exponential Family
This paper investigates maximizers of the information divergence from an
exponential family . It is shown that the -projection of a maximizer
to is a convex combination of and a probability measure with
disjoint support and the same value of the sufficient statistics . This
observation can be used to transform the original problem of maximizing
over the set of all probability measures into the maximization of
a function \Dbar over a convex subset of . The global maximizers of
both problems correspond to each other. Furthermore, finding all local
maximizers of \Dbar yields all local maximizers of .
This paper also proposes two algorithms to find the maximizers of \Dbar and
applies them to two examples, where the maximizers of were not
known before.Comment: 25 page
Generalized Binomial Edge Ideals
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 Gr\"obner basis can be computed
by studying paths in the graph. Since these Gr\"obner 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.Comment: 6 pages. arXiv admin note: substantial text overlap with
arXiv:1110.133
The Blackwell relation defines no lattice
Blackwell's 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 lattice; that is, least upper bounds and greatest lower bounds
do not exist.Comment: 5 pages, 1 figur
Hierarchical Models as Marginals of Hierarchical Models
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 hidden binary variables can approximate
every distribution of visible binary variables arbitrarily well, compared
to from the best previously known result.Comment: 18 pages, 4 figures, 2 tables, WUPES'1
- β¦