4,399 research outputs found
How to Couple from the Past Using a Read-Once Source of Randomness
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure
Particle Metropolis-Hastings using gradient and Hessian information
Particle Metropolis-Hastings (PMH) allows for Bayesian parameter inference in
nonlinear state space models by combining Markov chain Monte Carlo (MCMC) and
particle filtering. The latter is used to estimate the intractable likelihood.
In its original formulation, PMH makes use of a marginal MCMC proposal for the
parameters, typically a Gaussian random walk. However, this can lead to a poor
exploration of the parameter space and an inefficient use of the generated
particles.
We propose a number of alternative versions of PMH that incorporate gradient
and Hessian information about the posterior into the proposal. This information
is more or less obtained as a byproduct of the likelihood estimation. Indeed,
we show how to estimate the required information using a fixed-lag particle
smoother, with a computational cost growing linearly in the number of
particles. We conclude that the proposed methods can: (i) decrease the length
of the burn-in phase, (ii) increase the mixing of the Markov chain at the
stationary phase, and (iii) make the proposal distribution scale invariant
which simplifies tuning.Comment: 27 pages, 5 figures, 2 tables. The final publication is available at
Springer via: http://dx.doi.org/10.1007/s11222-014-9510-
Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix
Within a random-matrix-theory approach, we use the nearest-neighbor energy
level spacing distribution and the entropic eigenfunction localization
length to study spectral and eigenfunction properties (of adjacency
matrices) of weighted random--geometric and random--rectangular graphs. A
random--geometric graph (RGG) considers a set of vertices uniformly and
independently distributed on the unit square, while for a random--rectangular
graph (RRG) the embedding geometry is a rectangle. The RRG model depends on
three parameters: The rectangle side lengths and , the connection
radius , and the number of vertices . We then study in detail the case
which corresponds to weighted RGGs and explore weighted RRGs
characterized by , i.e.~two-dimensional geometries, but also approach
the limit of quasi-one-dimensional wires when . In general we look for
the scaling properties of and as a function of , and .
We find that the ratio , with , fixes the
properties of both RGGs and RRGs. Moreover, when we show that
spectral and eigenfunction properties of weighted RRGs are universal for the
fixed ratio , with .Comment: 8 pages, 6 figure
Earthquake Size Distribution: Power-Law with Exponent Beta = 1/2?
We propose that the widely observed and universal Gutenberg-Richter relation
is a mathematical consequence of the critical branching nature of earthquake
process in a brittle fracture environment. These arguments, though preliminary,
are confirmed by recent investigations of the seismic moment distribution in
global earthquake catalogs and by the results on the distribution in crystals
of dislocation avalanche sizes. We consider possible systematic and random
errors in determining earthquake size, especially its seismic moment. These
effects increase the estimate of the parameter beta of the power-law
distribution of earthquake sizes. In particular, we find that estimated
beta-values may be inflated by 1-3% because relative moment uncertainties
decrease with increasing earthquake size. Moreover, earthquake clustering
greatly influences the beta-parameter. If clusters (aftershock sequences) are
taken as the entity to be studied, then the exponent value for their size
distribution would decrease by 5-10%. The complexity of any earthquake source
also inflates the estimated beta-value by at least 3-7%. The centroid depth
distribution also should influence the beta-value, an approximate calculation
suggests that the exponent value may be increased by 2-6%. Taking all these
effects into account, we propose that the recently obtained beta-value of 0.63
could be reduced to about 0.52--0.56: near the universal constant value (1/2)
predicted by theoretical arguments. We also consider possible consequences of
the universal beta-value and its relevance for theoretical and practical
understanding of earthquake occurrence in various tectonic and Earth structure
environments. Using comparative crystal deformation results may help us
understand the generation of seismic tremors and slow earthquakes and
illuminate the transition from brittle fracture to plastic flow.Comment: 46 pages, 2 tables, 11 figures 53 pages, 2 tables, 12 figure
Opinion fluctuations and disagreement in social networks
We study a tractable opinion dynamics model that generates long-run
disagreements and persistent opinion fluctuations. Our model involves an
inhomogeneous stochastic gossip process of continuous opinion dynamics in a
society consisting of two types of agents: regular agents, who update their
beliefs according to information that they receive from their social neighbors;
and stubborn agents, who never update their opinions. When the society contains
stubborn agents with different opinions, the belief dynamics never lead to a
consensus (among the regular agents). Instead, beliefs in the society fail to
converge almost surely, the belief profile keeps on fluctuating in an ergodic
fashion, and it converges in law to a non-degenerate random vector. The
structure of the network and the location of the stubborn agents within it
shape the opinion dynamics. The expected belief vector evolves according to an
ordinary differential equation coinciding with the Kolmogorov backward equation
of a continuous-time Markov chain with absorbing states corresponding to the
stubborn agents and converges to a harmonic vector, with every regular agent's
value being the weighted average of its neighbors' values, and boundary
conditions corresponding to the stubborn agents'. Expected cross-products of
the agents' beliefs allow for a similar characterization in terms of coupled
Markov chains on the network. We prove that, in large-scale societies which are
highly fluid, meaning that the product of the mixing time of the Markov chain
on the graph describing the social network and the relative size of the
linkages to stubborn agents vanishes as the population size grows large, a
condition of \emph{homogeneous influence} emerges, whereby the stationary
beliefs' marginal distributions of most of the regular agents have
approximately equal first and second moments.Comment: 33 pages, accepted for publication in Mathematics of Operation
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