  In this paper, we study the problem of sampling from a graphical model when
the model itself is changing dynamically with time. This problem derives its
interest from a variety of inference, learning, and sampling settings in
machine learning, computer vision, statistical physics, and theoretical
computer science. While the problem of sampling from a static graphical model
has received considerable attention, theoretical works for its dynamic variants
have been largely lacking. The main contribution of this paper is an algorithm
that can sample dynamically from a broad class of graphical models over
discrete random variables. Our algorithm is parallel and Las Vegas: it knows
when to stop and it outputs samples from the exact distribution. We also
provide sufficient conditions under which this algorithm runs in time
proportional to the size of the update, on general graphical models as well as
well-studied specific spin systems. In particular we obtain, for the Ising
model (ferromagnetic or anti-ferromagnetic) and for the hardcore model the
first dynamic sampling algorithms that can handle both edge and vertex updates
(addition, deletion, change of functions), both efficient within regimes that
are close to the respective uniqueness regimes, beyond which, even for the
static and approximate sampling, no local algorithms were known or the problem
itself is intractable. Our dynamic sampling algorithm relies on a local
resampling algorithm and a new "equilibrium" property that is shown to be
satisfied by our algorithm at each step, and enables us to prove its
correctness. This equilibrium property is robust enough to guarantee the
correctness of our algorithm, helps us improve bounds on fast convergence on
specific models, and should be of independent interest.
