Graphical Models for Ordinal Data

. Author manuscript; available in PMC: 2016 Mar 31.

Published in final edited form as: J Comput Graph Stat. 2014 Mar 13;24(1):183–204. doi: 10.1080/10618600.2014.889023


Algorithm 1 The EM Algorithm for estimating Ω

1:	Initialize $E (z_{i, j} ∣ x_{i}; \hat{Θ}, \hat{Ω}) \approx E (z_{i, j} ∣ x_{i, j}; \hat{Θ})$ , $E (z_{i, j}^{2} ∣ x_{i}; \hat{Θ}, \hat{Ω}) \approx E (z_{i, j}^{2} ∣ x_{i, j}; \hat{Θ})$ and $E (z_{i, j} z_{i, j^{'}} ∣ x_{i}; \hat{Θ}, \hat{Ω}) \approx E (z_{i, j} ∣ x_{i, j}; \hat{Θ}) E (z_{i, j^{'}} ∣ x_{i, j^{'}}; \hat{Θ})$ for i = 1, …, n and j,j′ = 1, …,p;
2:	Initialize s_j,j′ for 1 ≤ j,j′ ≤ p using the Line 1 above, and then estimate $\hat{Ω}$ by maximizing criterion (7); {Start outer loop}
3:	repeat
4:	E-step: estimate S in (6); {Start inner loop}
5:	repeat
6:	for i = 1 to n do
7:	if j = j′ then
8:	Update $E (z_{i, j}^{2} ∣ x_{i}; \hat{Θ}, \hat{Ω})$ using RHS of equation (18) for j = 1, …, p;
9:	else
10:	Update $E (z_{i, j} ∣ x_{i}; \hat{Θ}, \hat{Ω})$ using RHS of equation (17) for j = 1, …, p and then set $E (z_{i, j} z_{i, j^{'}} ∣ x_{i}; \hat{Θ}, \hat{Ω}) = E (z_{i, j} ∣ x_{i}; \hat{Θ}, \hat{Ω}) E (z_{i, j^{'}} ∣ x_{i}; \hat{Θ}, \hat{Ω})$ for 1 ≤ j ≠ j′ ≤ p;
11:	end if
12:	end for
13:	Update $s_{j, j^{'}} = 1 ∕ n \sum_{i = 1}^{n} E (z_{i, j} z_{i, j^{'}} ∣ x_{i}; \hat{Θ}, \hat{Ω})$ for 1 ≤ j,j′ ≤ p;
14:	until The inner loop converges;
15:	M-step: update $\hat{Ω}$ by maximizing criterion (7);
16:	until The outer loop converges.