  We present a sequential sampling methodology for weakly structural Markov
laws, arising naturally in a Bayesian structure learning context for
decomposable graphical models. As a key component of our suggested approach, we
show that the problem of graph estimation, which in general lacks natural
sequential interpretation, can be recast into a sequential setting by proposing
a recursive Feynman-Kac model that generates a flow of junction tree
distributions over a space of increasing dimensions. We focus on particle McMC
methods to provide samples on this space, in particular on particle Gibbs (PG),
as it allows for generating McMC chains with global moves on an underlying
space of decomposable graphs. To further improve the PG mixing properties, we
incorporate a systematic refreshment step implemented through direct sampling
from a backward kernel. The theoretical properties of the algorithm are
investigated, showing that the proposed refreshment step improves the
performance in terms of asymptotic variance of the estimated distribution. The
suggested sampling methodology is illustrated through a collection of numerical
examples demonstrating high accuracy in Bayesian graph structure learning in
both discrete and continuous graphical models.
