Post-Monte Carlo D: MCMC Sampling on Constraint Networks– From Fixed Topology to Structurally Adaptive Inference Dynamics

MCMCmethods have become core computational tools for Bayesian statistical
inference and probabilistic modeling. In the standard paradigm of probabilistic
graphical models and constraint satisfaction problems, MCMC sampling is typi
cally performed on a constraint network with a predetermined, fixed topological
structure—the set of variables and the dependency relations among them are com
pletely specified before sampling begins. However, in numerous practical applica
tion scenarios, the structure of the constraint network itself is precisely the central
object of inference: from structure learning of gene regulatory networks to relation
discovery in social networks, from knowledge base construction in statistical rela
tional learning to constraint solving in hardware verification, dependency relations
among variables often need to be simultaneously inferred during the sampling pro
cess. When the topological structure of the constraint network becomes the object
of inference, traditional MCMC methods operating on a fixed graph structure face
fundamental methodological challenges. This paper systematically examines four
progressive levels of MCMC sampling on constraint networks: constraint-aware
sampling under a fixed topology, Bayesian structure learning where the topological
structure is the object of inference, the propagation mechanism of structural uncer
tainty in assignment inference, and joint sampling where the constraint network and
variable assignments co-evolve over time. By analyzing the slice sampling mecha
nism of the MC-SAT algorithm under probabilistic-logical hybrid constraints, the
cross-dimensional exploration strategies of reversible jump MCMC in graph space,
the implementation of Bayesian model averaging within a dual MCMC framework,
and MCMC adaptation schemes in mixed Boolean/integer constraint spaces, this
paper reveals the profound requirements that the variability of constraint network
topology imposes on MCMC convergence theory and algorithm design.