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