333 research outputs found
Scaling Limits for Minimal and Random Spanning Trees in Two Dimensions
A general formulation is presented for continuum scaling limits of stochastic
spanning trees. A spanning tree is expressed in this limit through a consistent
collection of subtrees, which includes a tree for every finite set of endpoints
in . Tightness of the distribution, as , is established for
the following two-dimensional examples: the uniformly random spanning tree on
, the minimal spanning tree on (with random edge
lengths), and the Euclidean minimal spanning tree on a Poisson process of
points in with density . In each case, sample trees are
proven to have the following properties, with probability one with respect to
any of the limiting measures: i) there is a single route to infinity (as was
known for ), ii) the tree branches are given by curves which are
regular in the sense of H\"older continuity, iii) the branches are also rough,
in the sense that their Hausdorff dimension exceeds one, iv) there is a random
dense subset of , of dimension strictly between one and two, on the
complement of which (and only there) the spanning subtrees are unique with
continuous dependence on the endpoints, v) branching occurs at countably many
points in , and vi) the branching numbers are uniformly bounded. The
results include tightness for the loop erased random walk (LERW) in two
dimensions. The proofs proceed through the derivation of scale-invariant power
bounds on the probabilities of repeated crossings of annuli.Comment: Revised; 54 pages, 6 figures (LaTex
Conjugate Projective Limits
We characterize conjugate nonparametric Bayesian models as projective limits
of conjugate, finite-dimensional Bayesian models. In particular, we identify a
large class of nonparametric models representable as infinite-dimensional
analogues of exponential family distributions and their canonical conjugate
priors. This class contains most models studied in the literature, including
Dirichlet processes and Gaussian process regression models. To derive these
results, we introduce a representation of infinite-dimensional Bayesian models
by projective limits of regular conditional probabilities. We show under which
conditions the nonparametric model itself, its sufficient statistics, and -- if
they exist -- conjugate updates of the posterior are projective limits of their
respective finite-dimensional counterparts. We illustrate our results both by
application to existing nonparametric models and by construction of a model on
infinite permutations.Comment: 49 pages; improved version: revised proof of theorem 3 (results
unchanged), discussion added, exposition revise
The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems
In literature on imprecise probability little attention is paid to the fact
that imprecise probabilities are precise on some events. We call these sets
system of precision. We show that, under mild assumptions, the system of
precision of a lower and upper probability form a so-called
(pre-)Dynkin-system. Interestingly, there are several settings, ranging from
machine learning on partial data over frequential probability theory to quantum
probability theory and decision making under uncertainty, in which a priori the
probabilities are only desired to be precise on a specific underlying set
system. At the core of all of these settings lies the observation that precise
beliefs, probabilities or frequencies on two events do not necessarily imply
this precision to hold for the intersection of those events. Here,
(pre-)Dynkin-systems have been adopted as systems of precision, too. We show
that, under extendability conditions, those pre-Dynkin-systems equipped with
probabilities can be embedded into algebras of sets. Surprisingly, the
extendability conditions elaborated in a strand of work in quantum physics are
equivalent to coherence in the sense of Walley (1991, Statistical reasoning
with imprecise probabilities, p. 84). Thus, literature on probabilities on
pre-Dynkin-systems gets linked to the literature on imprecise probability.
Finally, we spell out a lattice duality which rigorously relates the system of
precision to credal sets of probabilities. In particular, we provide a hitherto
undescribed, parametrized family of coherent imprecise probabilities
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