59,631 research outputs found

### Sequential Monte Carlo with Highly Informative Observations

We propose sequential Monte Carlo (SMC) methods for sampling the posterior
distribution of state-space models under highly informative observation
regimes, a situation in which standard SMC methods can perform poorly. A
special case is simulating bridges between given initial and final values. The
basic idea is to introduce a schedule of intermediate weighting and resampling
times between observation times, which guide particles towards the final state.
This can always be done for continuous-time models, and may be done for
discrete-time models under sparse observation regimes; our main focus is on
continuous-time diffusion processes. The methods are broadly applicable in that
they support multivariate models with partial observation, do not require
simulation of the backward transition (which is often unavailable), and, where
possible, avoid pointwise evaluation of the forward transition. When simulating
bridges, the last cannot be avoided entirely without concessions, and we
suggest an epsilon-ball approach (reminiscent of Approximate Bayesian
Computation) as a workaround. Compared to the bootstrap particle filter, the
new methods deliver substantially reduced mean squared error in normalising
constant estimates, even after accounting for execution time. The methods are
demonstrated for state estimation with two toy examples, and for parameter
estimation (within a particle marginal Metropolis--Hastings sampler) with three
applied examples in econometrics, epidemiology and marine biogeochemistry.Comment: 25 pages, 11 figure

### A two-stage approach to relaxation in billiard systems of locally confined hard spheres

We consider the three-dimensional dynamics of systems of many interacting
hard spheres, each individually confined to a dispersive environment, and show
that the macroscopic limit of such systems is characterized by a coefficient of
heat conduction whose value reduces to a dimensional formula in the limit of
vanishingly small rate of interaction. It is argued that this limit arises from
an effective loss of memory. Similarities with the diffusion of a tagged
particle in binary mixtures are emphasized.Comment: Submitted to Chaos, special issue "Statistical Mechanics and
Billiard-Type Dynamical Systems

### Tight Bounds for Consensus Systems Convergence

We analyze the asymptotic convergence of all infinite products of matrices
taken in a given finite set, by looking only at finite or periodic products. It
is known that when the matrices of the set have a common nonincreasing
polyhedral norm, all infinite products converge to zero if and only if all
infinite periodic products with period smaller than a certain value converge to
zero, and bounds exist on that value.
We provide a stronger bound holding for both polyhedral norms and polyhedral
seminorms. In the latter case, the matrix products do not necessarily converge
to 0, but all trajectories of the associated system converge to a common
invariant space. We prove our bound to be tight, in the sense that for any
polyhedral seminorm, there is a set of matrices such that not all infinite
products converge, but every periodic product with period smaller than our
bound does converge.
Our technique is based on an analysis of the combinatorial structure of the
face lattice of the unit ball of the nonincreasing seminorm. The bound we
obtain is equal to half the size of the largest antichain in this lattice.
Explicitly evaluating this quantity may be challenging in some cases. We
therefore link our problem with the Sperner property: the property that, for
some graded posets, -- in this case the face lattice of the unit ball -- the
size of the largest antichain is equal to the size of the largest rank level.
We show that some sets of matrices with invariant polyhedral seminorms lead
to posets that do not have that Sperner property. However, this property holds
for the polyhedron obtained when treating sets of stochastic matrices, and our
bound can then be easily evaluated in that case. In particular, we show that
for the dimension of the space $n \geq 8$, our bound is smaller than the
previously known bound by a multiplicative factor of $\frac{3}{2 \sqrt{\pi
n}}$

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