. Author manuscript; available in PMC: 2016 Apr 29.

Published in final edited form as: Ann Appl Stat. 2015 Nov 2;9(3):1571–1600. doi: 10.1214/15-AOAS844

Algorithm 2.

HMRF parameter estimation

1. Initialize the states of node I_i = 1if Z_i > Z_thres and 0 otherwise.

2. For t = 1, . . ., T

(a) Update (b̂^(t), ĉ^(t)) by maximizing the pseudo likelihood

\prod_{i} \frac{\exp {b I_{i} + c I_{i} Ω_{i} . I}}{\exp {b I_{i} + c I_{i} Ω_{i} . I} + \exp {b (1 - I_{i}) + c (1 - I_{i}) Ω_{i} . I}}

(b) Apply a single cycle of the iterative conditional mode [ICM, Besag (1986)] algorithm to update I. Specifically, we obtain a new

{\hat{I}}_{j}^{(t)}

based on

P (I_{j} ∣ Z; {\hat{I}}_{- i}, {\hat{b}}^{(t)}, {\hat{c}}^{(t)}) \propto f (z_{i} ∣ {\hat{I}}_{i}) P (I_{i} ∣ {\hat{I}}_{- i}, {\hat{b}}^{(t)}, {\hat{c}}^{(t)})

({\hat{μ}}^{(t - 1)}, {\hat{σ}}_{0}^{2 (t - 1)}, {\hat{σ}}_{1}^{2 (t - 1)})

({\hat{μ}}^{(t)}, {\hat{σ}}_{0}^{2 (t)}, {\hat{σ}}_{1}^{2 (t)})

{\hat{μ}}^{(t)} = \frac{\sum_{i} P (I_{i} = 1 ∣ Z, {\hat{b}}^{(t)}; {\hat{c}}^{(t)}) Z_{i}}{\sum_{i} P (I_{i} = 1 ∣ Z, {\hat{b}}^{(t)}; {\hat{c}}^{(t)})}

{\hat{σ}}_{0}^{2 (t)} = \frac{\sum_{i} P (I_{i} = 0 ∣ Z, {\hat{b}}^{(t)}; {\hat{c}}^{(t)}) Z_{i}^{2}}{\sum_{i} P (I_{i} = 0 ∣ Z, {\hat{b}}^{(t)}; {\hat{c}}^{(t)})}

{\hat{σ}}_{1}^{2 (t)} = \frac{\sum_{i} P (I_{i} = 1 ∣ Z, {\hat{b}}^{(t)}; {\hat{c}}^{(t)}) {(Z_{i} - {\hat{μ}}^{(t)})}^{2}}{\sum_{i} P (I_{i} = 1 ∣ Z, {\hat{b}}^{(t)}; {\hat{c}}^{(t)})}

3. Return

(\hat{b}, \hat{c}, \hat{μ}, {\hat{σ}}_{0}^{2}, {\hat{σ}}_{1}^{2}) = ({\hat{b}}^{(T)}, {\hat{c}}^{(T)}, {\hat{μ}}^{(T)}, {\hat{σ}}_{0}^{2 (T)}, {\hat{σ}}_{1}^{2 (T)})