1. Mean field approximation for CRF inference

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

3. Other notation Unnormalized distribution Variational distribution
Expectation
Entropy

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

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

6. Mean field objective
Objective

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

8. 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...)

9. Comparison with belief propagation
Objective
Factored energy functional
Local polytope

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

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

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

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

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