62 research outputs found
Partially observed Markov random fields are variable neighborhood random fields
The present paper has two goals. First to present a natural example of a new
class of random fields which are the variable neighborhood random fields. The
example we consider is a partially observed nearest neighbor binary Markov
random field. The second goal is to establish sufficient conditions ensuring
that the variable neighborhoods are almost surely finite. We discuss the
relationship between the almost sure finiteness of the interaction
neighborhoods and the presence/absence of phase transition of the underlying
Markov random field. In the case where the underlying random field has no phase
transition we show that the finiteness of neighborhoods depends on a specific
relation between the noise level and the minimum values of the one-point
specification of the Markov random field. The case in which there is phase
transition is addressed in the frame of the ferromagnetic Ising model. We prove
that the existence of infinite interaction neighborhoods depends on the phase.Comment: To appear in Journal of Statistical Physic
Perfect simulation of a coupling achieving the -distance between ordered pairs of binary chains of infinite order
We explicitly construct a coupling attaining Ornstein's -distance
between ordered pairs of binary chains of infinite order. Our main tool is a
representation of the transition probabilities of the coupled bivariate chain
of infinite order as a countable mixture of Markov transition probabilities of
increasing order. Under suitable conditions on the loss of memory of the
chains, this representation implies that the coupled chain can be represented
as a concatenation of iid sequence of bivariate finite random strings of
symbols. The perfect simulation algorithm is based on the fact that we can
identify the first regeneration point to the left of the origin almost surely.Comment: Typos corrected. The final publication is available at
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Perfect simulation of infinite range Gibbs measures and coupling with their finite range approximations
In this paper we address the questions of perfectly sampling a Gibbs measure
with infinite range interactions and of perfectly sampling the measure together
with its finite range approximations. We solve these questions by introducing a
perfect simulation algorithm for the measure and for the coupled measures. The
algorithm works for general Gibbsian interaction under requirements on the
tails of the interaction. As a consequence we obtain an upper bound for the
error we make when sampling from a finite range approximation instead of the
true infinite range measure
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