9,405 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
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Proof-Pattern Recognition and Lemma Discovery in ACL2
We present a novel technique for combining statistical machine learning for
proof-pattern recognition with symbolic methods for lemma discovery. The
resulting tool, ACL2(ml), gathers proof statistics and uses statistical
pattern-recognition to pre-processes data from libraries, and then suggests
auxiliary lemmas in new proofs by analogy with already seen examples. This
paper presents the implementation of ACL2(ml) alongside theoretical
descriptions of the proof-pattern recognition and lemma discovery methods
involved in it
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
Equations defining probability tree models
Coloured probability tree models are statistical models coding conditional
independence between events depicted in a tree graph. They are more general
than the very important class of context-specific Bayesian networks. In this
paper, we study the algebraic properties of their ideal of model invariants.
The generators of this ideal can be easily read from the tree graph and have a
straightforward interpretation in terms of the underlying model: they are
differences of odds ratios coming from conditional probabilities. One of the
key findings in this analysis is that the tree is a convenient tool for
understanding the exact algebraic way in which the sum-to-1 conditions on the
parameter space translate into the sum-to-one conditions on the joint
probabilities of the statistical model. This enables us to identify necessary
and sufficient graphical conditions for a staged tree model to be a toric
variety intersected with a probability simplex.Comment: 22 pages, 4 figure
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