Mean field approximation for CRF inference

CRF Inference Problem CRF over variables: CRF distribution: MAP inference: MPM (maximum posterior marginals) inference:

Other notation Unnormalized distribution Variational distribution Expectation Entropy

Variational Inference Inference => minimize KL-divergence General Objective Function

Mean field approximation Variational distribution => product of independent marginals: Expectations: Entropy:

Mean field objective Objective

Local optimality conditions Lagrangian Setting derivatives to 0 gives conditions for local optimality

Coordinate ascent Sequential coordinate ascent Initialize Q_i’s to uniform distribution For i = 1...N, update vector Q_i by summing expectations over all cliques involving X_i (while fixing all Q_j, j!=i) Parallel updates algorithm As above, but perform updates in step 2 for all Q_i’s in parrallel (i.e. Generating Q^1, Q^2...)

Comparison with belief propagation Objective Factored energy functional Local polytope

Comparison with belief propagation Message updates: Extracting beliefs (after convergence):

Comparison with belief propagation - = => Bethe free energy for pairwise graphs Bethe cluster graphs: General: Pairwise:

Mean field updates Updates in dense CRF (Krahenbuhl NIPS ’11) Evaluate using filtering =

Higher-order potentials Pattern-based potentials P^n-Potts potentials

Higher-order potentials Co-occurrence potentials L(X) = set of labels present in X {Y_1,...Y_L} = set of binary latent variables