TABLE 1.

Six possible data scenarios for the kth interval in the partition given by $\mathit{\tau }$ (see Section 2)

$\left({\mathit{Y}}_{\mathbf{1}\mathit{i},\mathit{k}-\mathbf{1}},{\mathit{Y}}_{\mathbf{2}\mathit{i},\mathit{k}-\mathbf{1}}\right)$	$\left({\mathit{Y}}_{\mathbf{1}\mathit{i},\mathit{k}},{\mathit{Y}}_{\mathbf{2}\mathit{i},\mathit{k}}\right)$	Likelihood contribution
(0, 0)	(0, 0)	$1-{\pi }_{1i,k}-{\pi }_{2i,k}(0)+{\pi }_{12i,k}$
(0, 0)	(1, 0)	${\pi }_{1i,k}-{\pi }_{12i,k}$
(0, 0)	(0, 1)	${\pi }_{2i,k}(0)-{\pi }_{12i,k}$
(0, 0)	(1, 1)	${\pi }_{12i,k}$
(1, 0)	(1, 0)	$1-{\pi }_{2i,k}(1)$
(1, 0)	(1, 1)	${\pi }_{2i,k}(1)$