4,182 research outputs found
Discovery of statistical equivalence classes using computer algebra
Discrete statistical models supported on labelled event trees can be
specified using so-called interpolating polynomials which are generalizations
of generating functions. These admit a nested representation. A new algorithm
exploits the primary decomposition of monomial ideals associated with an
interpolating polynomial to quickly compute all nested representations of that
polynomial. It hereby determines an important subclass of all trees
representing the same statistical model. To illustrate this method we analyze
the full polynomial equivalence class of a staged tree representing the best
fitting model inferred from a real-world dataset.Comment: 26 pages, 9 figure
Relations between cumulants in noncommutative probability
We express classical, free, Boolean and monotone cumulants in terms of each
other, using combinatorics of heaps, pyramids, Tutte polynomials and
permutations. We completely determine the coefficients of these formulas with
the exception of the formula for classical cumulants in terms of monotone
cumulants whose coefficients are only partially computed.Comment: 27 pages, 7 figures, AMS LaTe
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes
We prove that there exist bipartite Ramanujan graphs of every degree and
every number of vertices. The proof is based on analyzing the expected
characteristic polynomial of a union of random perfect matchings, and involves
three ingredients: (1) a formula for the expected characteristic polynomial of
the sum of a regular graph with a random permutation of another regular graph,
(2) a proof that this expected polynomial is real rooted and that the family of
polynomials considered in this sum is an interlacing family, and (3) strong
bounds on the roots of the expected characteristic polynomial of a union of
random perfect matchings, established using the framework of finite free
convolutions we recently introduced
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