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There are many ways to do this, and the diagram below illustrates one of them (the red points on the lattice are one block, the blue points are another): The red blocks form a tree-structured UGM conditioned on the blue blocks, and vice versa.
However, when the system is subject to constraints, it is difficult to find irreducible proposal distributions.
We suggest a “block-wise” approach for constrained sampling and optimisation.
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Markov chain Monte Carlo methods are widely used to study highly structured stochastic systems.
The approximate inference methods from the previous demo correspond to the special case where each variable forms its own block.
By conditioning on all variables outside the block, it is straightforward to do exact calculations within the block.