1. 6.8 Maximizer of the Posterior Marginals 6.9 Iterated Conditional Modes of the Posterior Distribution Jang, HaYoung

2. Optimal Bayesian Estimation Posterior margianl distribution  Sum of all posterior distributions Energy Function

3. Optimal Bayesian Estimation Optimal Bayesian estimator f a* with respect to the energy Optimal Bayesian estimator f a* for all s ≠ q over the finite set of possible values Q i

4. Segmentation and Reconstruction Error: the number of elements that are classified incorrectly Optimal segmentation estimate

5. Segmentation and Reconstruction Smoothness enforcing energy through quadratic form Mean Optimal restoration estimate (Thresholded posterior mean)

6. The MPM Algorithm Posterior marginals  Counting the number of times each label occurs at a site Mean field Metropolis-based MPM algorithm 1. Initialize the lattice system by choosing the pixel value at each site that maximizes the likelihood. 2. For each site in the lattice choose a pixel value at random subject to the symmetry requirement. 3. Calculate the energy change ΔU. 4. If ΔU < 0, accept the move. 5. Or else accept the move with probability exp{- ΔU }. 6. Repeat steps 2 through 5 n times, saving configurations f (k+1) to f (n). 7. Form p i ( q | g ) according to posterior marginals. 8. For each site in the lattice, choose the pixel value q satisfying the condition for all pixel values s.

7. Quenching and ICM Iterated conditional modes of the posterior distribution  Choose pixel values that maximizes the posterior distribution rather than selecting pixel values based on the Gibbs sampler or Metropolis algorithm  Depend on the prior distribution on the pixel values in the neighborhood of the site of interset  Replace the prior probabilites p(f i ) with the conditionanl probabilites p(f i |f ∂i )

8. Quenching and ICM

9. The ICM Algorithm Mode of the posterior distribution κ=2ασ 2 ICM Algorithm 1. Initialize the lattice by choosing the values at each lattice site that maximize the likelihood. 2. For each site in the lattice update the pixel value by selecting the quantity that maximizes the posterior energy, by finding the minimum energy according to mode of the posterior distribution or by using an equivalent expression appropriate for a different potential, or in some other manner. 3. Iterate, by repeating step 2 until finished.