189 research outputs found
Partition MCMC for inference on acyclic digraphs
Acyclic digraphs are the underlying representation of Bayesian networks, a
widely used class of probabilistic graphical models. Learning the underlying
graph from data is a way of gaining insights about the structural properties of
a domain. Structure learning forms one of the inference challenges of
statistical graphical models.
MCMC methods, notably structure MCMC, to sample graphs from the posterior
distribution given the data are probably the only viable option for Bayesian
model averaging. Score modularity and restrictions on the number of parents of
each node allow the graphs to be grouped into larger collections, which can be
scored as a whole to improve the chain's convergence. Current examples of
algorithms taking advantage of grouping are the biased order MCMC, which acts
on the alternative space of permuted triangular matrices, and non ergodic edge
reversal moves.
Here we propose a novel algorithm, which employs the underlying combinatorial
structure of DAGs to define a new grouping. As a result convergence is improved
compared to structure MCMC, while still retaining the property of producing an
unbiased sample. Finally the method can be combined with edge reversal moves to
improve the sampler further.Comment: Revised version. 34 pages, 16 figures. R code available at
https://github.com/annlia/partitionMCM
The semiclassical relation between open trajectories and periodic orbits for the Wigner time delay
The Wigner time delay of a classically chaotic quantum system can be
expressed semiclassically either in terms of pairs of scattering trajectories
that enter and leave the system or in terms of the periodic orbits trapped
inside the system. We show how these two pictures are related on the
semiclassical level. We start from the semiclassical formula with the
scattering trajectories and derive from it all terms in the periodic orbit
formula for the time delay. The main ingredient in this calculation is a new
type of correlation between scattering trajectories which is due to
trajectories that approach the trapped periodic orbits closely. The equivalence
between the two pictures is also demonstrated by considering correlation
functions of the time delay. A corresponding calculation for the conductance
gives no periodic orbit contributions in leading order.Comment: 21 pages, 5 figure
Sequential Monte Carlo EM for multivariate probit models
Multivariate probit models (MPM) have the appealing feature of capturing some
of the dependence structure between the components of multidimensional binary
responses. The key for the dependence modelling is the covariance matrix of an
underlying latent multivariate Gaussian. Most approaches to MLE in multivariate
probit regression rely on MCEM algorithms to avoid computationally intensive
evaluations of multivariate normal orthant probabilities. As an alternative to
the much used Gibbs sampler a new SMC sampler for truncated multivariate
normals is proposed. The algorithm proceeds in two stages where samples are
first drawn from truncated multivariate Student distributions and then
further evolved towards a Gaussian. The sampler is then embedded in a MCEM
algorithm. The sequential nature of SMC methods can be exploited to design a
fully sequential version of the EM, where the samples are simply updated from
one iteration to the next rather than resampled from scratch. Recycling the
samples in this manner significantly reduces the computational cost. An
alternative view of the standard conditional maximisation step provides the
basis for an iterative procedure to fully perform the maximisation needed in
the EM algorithm. The identifiability of MPM is also thoroughly discussed. In
particular, the likelihood invariance can be embedded in the EM algorithm to
ensure that constrained and unconstrained maximisation are equivalent. A simple
iterative procedure is then derived for either maximisation which takes
effectively no computational time. The method is validated by applying it to
the widely analysed Six Cities dataset and on a higher dimensional simulated
example. Previous approaches to the Six Cities overly restrict the parameter
space but, by considering the correct invariance, the maximum likelihood is
quite naturally improved when treating the full unrestricted model.Comment: 26 pages, 2 figures. In press, Computational Statistics & Data
Analysi
Moments of the Wigner delay times
The Wigner time delay is a measure of the time spent by a particle inside the
scattering region of an open system. For chaotic systems, the statistics of the
individual delay times (whose average is the Wigner time delay) are thought to
be well described by random matrix theory. Here we present a semiclassical
derivation showing the validity of random matrix results. In order to simplify
the semiclassical treatment, we express the moments of the delay times in terms
of correlation functions of scattering matrices at different energies. In the
semiclassical approximation, the elements of the scattering matrix are given in
terms of the classical scattering trajectories, requiring one to study
correlations between sets of such trajectories. We describe the structure of
correlated sets of trajectories and formulate the rules for their evaluation to
the leading order in inverse channel number. This allows us to derive a
polynomial equation satisfied by the generating function of the moments. Along
with showing the agreement of our semiclassical results with the moments
predicted by random matrix theory, we infer that the scattering matrix is
unitary to all orders in the semiclassical approximation.Comment: Refereed version. 18 pages, 5 figure
Quantum graphs whose spectra mimic the zeros of the Riemann zeta function
One of the most famous problems in mathematics is the Riemann hypothesis:
that the non-trivial zeros of the Riemann zeta function lie on a line in the
complex plane. One way to prove the hypothesis would be to identify the zeros
as eigenvalues of a Hermitian operator, many of whose properties can be derived
through the analogy to quantum chaos. Using this, we construct a set of quantum
graphs that have the same oscillating part of the density of states as the
Riemann zeros, offering an explanation of the overall minus sign. The smooth
part is completely different, and hence also the spectrum, but the graphs pick
out the low-lying zeros.Comment: 8 pages, 8 pdf figure
Semiclassics for chaotic systems with tunnel barriers
The addition of tunnel barriers to open chaotic systems, as well as
representing more general physical systems, leads to much richer semiclassical
dynamics. In particular, we present here a complete semiclassical treatment for
these systems, in the regime where Ehrenfest time effects are negligible and
for times shorter than the Heisenberg time. To start we explore the trajectory
structures which contribute to the survival probability, and find results that
are also in agreement with random matrix theory. Then we progress to the
treatment of the probability current density and are able to show, using
recursion relation arguments, that the continuity equation connecting the
current density to the survival probability is satisfied to all orders in the
semiclassical approximation. Following on, we also consider a correlation
function of the scattering matrix, for which we have to treat a new set of
possible trajectory diagrams. By simplifying the contributions of these
diagrams, we show that the results obtained here are consistent with known
properties of the scattering matrix. The correlation function can be trivially
connected to the ac and dc conductances, quantities of particular interest for
which finally we present a semiclassical expansion.Comment: Refereed version. 26 pages, 3 figures in 6 part
Transport moments beyond the leading order
For chaotic cavities with scattering leads attached, transport properties can
be approximated in terms of the classical trajectories which enter and exit the
system. With a semiclassical treatment involving fine correlations between such
trajectories we develop a diagrammatic technique to calculate the moments of
various transport quantities. Namely, we find the moments of the transmission
and reflection eigenvalues for systems with and without time reversal symmetry.
We also derive related quantities involving an energy dependence: the moments
of the Wigner delay times and the density of states of chaotic Andreev
billiards, where we find that the gap in the density persists when subleading
corrections are included. Finally, we show how to adapt our techniques to
non-linear statistics by calculating the correlation between transport moments.
In each setting, the answer for the -th moment is obtained for arbitrary
(in the form of a moment generating function) and for up to the three leading
orders in terms of the inverse channel number. Our results suggest patterns
which should hold for further corrections and by matching with the low order
moments available from random matrix theory we derive likely higher order
generating functions.Comment: Refereed version. 43 pages, 10 figure
The Variance of Causal Effect Estimators for Binary V-structures
Adjusting for covariates is a well established method to estimate the total
causal effect of an exposure variable on an outcome of interest. Depending on
the causal structure of the mechanism under study there may be different
adjustment sets, equally valid from a theoretical perspective, leading to
identical causal effects. However, in practice, with finite data, estimators
built on different sets may display different precision. To investigate the
extent of this variability we consider the simplest non-trivial non-linear
model of a v-structure on three nodes for binary data. We explicitly compute
and compare the variance of the two possible different causal estimators.
Further, by going beyond leading order asymptotics we show that there are
parameter regimes where the set with the asymptotically optimal variance does
depend on the edge coefficients, a result which is not captured by the recent
leading order developments for general causal models.Comment: 14 pages, 2 figure
Multiparticle correlations in mesoscopic scattering: boson sampling, birthday paradox, and Hong-Ou-Mandel profiles
The interplay between single-particle interference and quantum
indistinguishability leads to signature correlations in many-body scattering.
We uncover these with a semiclassical calculation of the transmission
probabilities through mesoscopic cavities for systems of non-interacting
particles. For chaotic cavities we provide the universal form of the first two
moments of the transmission probabilities over ensembles of random unitary
matrices, including weak localization and dephasing effects. If the incoming
many-body state consists of two macroscopically occupied wavepackets, their
time delay drives a quantum-classical transition along a boundary determined by
the bosonic birthday paradox. Mesoscopic chaotic scattering of Bose-Einstein
condensates is then a realistic candidate to build a boson sampler and to
observe the macroscopic Hong-Ou-Mandel effect.Comment: 6+11 pages, 3+3 figure
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