It contradicts with equation (11) due to the fact that two different marginal distributions of item j ∗ leads to the two different joint distributions. Hence, each column of T$_{I}$$_{i}$ is precisely one column of P $_{i}$. T$_{I}$$_{i}$ is of full column rank with rank M as a result. Then M + M + M ≥ 2 M + 2 whenever M ≥ 2. We apply Lemma 1 and use the same argument as in the proof of Theorem 1. 

Proof of Theorem 3. There exist three non-overlapp subsets of items I$_{1}$, I$_{2}$ , and I$_{3}$ such that I$_{1}$ ⋃ I$_{2}$ ⋃ I$_{3}$ = { 1 , . . . , J } . We write the three-way array W = [ T$_{I}$$_{1}$ Λ , T$_{I}$$_{2}$, T$_{I}$$_{3}$ ], where T$_{I}$$_{1}$, T$_{I}$$_{2}$ , and T$_{I}$$_{3}$ are the T -matrices of subsets I$_{1}$, I$_{2}$ , and I$_{3}$ respectively and Λ is a ∏ J j $_{=1}$k$_{j}$ by ∏ J j $_{=1}$k$_{j}$ diagonal matrix with α -th diagonal element being π α . Thus W is a ∏ $^{j}$∈$_{I$_{1}$}$k$_{j}$ by $^{j}$∈$_{I$_{2}$}$k$_{j}$ by $^{j}$∈$_{I$_{3}$}$k$_{j}$ array. It is not hard to see that W ( y$_{1}$ , y$_{2}$ , y$_{3}$ ) = ∑ α t 1 y$_{1}$ $_{α}$t 2 y$_{2}$ $_{α}$t 3 y$_{3}$ $_{α}$, where t $^{i}$y$_{i}$ α is the ( y$_{i}$ , α )-th element of matrix T$_{I}$$_{i}$ . In other words, W ( y$_{1}$ , y$_{2}$ , y$_{3}$ ) = P ( Y = ( y$_{1}$ , y$_{2}$ , y$_{3}$ )). 

Suppose that there exists another set of parameters of the model giving the same distribution; that is, another decomposition of W = [ T ′ $_{I$_{1}$}$Λ $^{′}$, T ′ $_{I$_{2}$}$, T ′ $_{I$_{3}$}$]. Because T$_{I}$$_{i}$ are all full column rank. By applying the Lemma 1, we have that T$_{I}$$_{1}$ Λ = T ′ $_{I$_{1}$}$Λ $^{′}$PA , T$_{I}$$_{2}$ = T ′ $_{I$_{2}$}$PB , and T$_{I}$$_{3}$ = T ′ $_{I$_{3}$}$PC . Here, A, B , and C are all diagonal matrix with ABC = I and I is an identity and P is a column permutation matrix. 

The sum of each column of T$_{I}$$_{2}$, T$_{I}$$_{3}$, T ′ $_{I$_{2}$}$, and T ′ I$_{3}$ equals 1. Then, B and C must be both identity matrices. As a result, A is identity too. Due to the same reason that the sum of each column of T$_{I}$$_{1}$ and T ′ I$_{1}$ is 1, we have Λ = P $^{′}$Λ$^{′}$P and T$_{I}$$_{1}$ = T ′ $_{I$_{1}$}$P . Besides, T$_{I}$$_{2}$ = T ′ $_{I$_{2}$}$P and T$_{I}$$_{3}$ = T ′ $_{I$_{3}$}$P . We conclude that all parameters are identifiable up to a permutation of the columns. 

Proof of Theorem 4. According to Theorem 3, it is sufficient to find three non-overlap subsets of items I$_{1}$, I$_{2}$ , and I$_{3}$ such that I$_{1}$ ⋃ I$_{2}$ ⋃ I$_{3}$ = { 1 , . . . , J } and their corresponding T -matrices T$_{I}$$_{1}$, T$_{I}$$_{2}$, T$_{I}$$_{3}$ are all full column rank. 

We construct the three subsets as follows: I$_{1}$ = ⋃ K k $_{=1}$I$_{1}$$_{,k}$ , I$_{2}$ = ⋃ K k $_{=1}$I$_{2}$$_{,k}$ , I$_{3}$ = J \ ( I$_{1}$ ⋃ I$_{2}$ ). Then we need to show that I$_{1}$, I$_{2}$, I$_{3}$ are non-overlap and their T -matrix T$_{I}$$_{1}$, T$_{I}$$_{2}$, T$_{I}$$_{3}$ are of full column rank. 

