the random variable $\mathbf{X}_{(D)}$ conditioned on $D=k$. Given $D=k$, the values $\{x_j:j \ge n_{(k)}\}$ are unobserved and hence do not contribute to the density; indeed, they do not belong to the adaptation-rule driven sample space, and consequently, they do not belong to a $\sigma$-field defined on the experiment’s sample space, measurable space, probability space, and hence, to the statistical experiment as a whole. Some researchers view $\{x_j:j \ge n_{(k)}\}$ as missing data, but in this paper, by analogy with structural zeroes in contingency tables, these values are excluded from the sample space.

Now the *joint* density for this GSD can be written as $$\begin{aligned}
 \label{densjoint}
f_{\mathbf{X}_{(D)}}\left(\textbf{\textit{x}}_{(D)}\vert \theta \right) :=
\prod_{k=1}^{K} [f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta)]^{I\left(D=k\right)}
=\prod_{k=1}^{K}f_{\mathbf{X}_{(k)}}^{sub}
,\end{aligned}$$ where $f_{\mathbf{X}_{(k)}}^{sub}:=[f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta)]^{I\left(D=k\right)}$ denotes *sub-densities* with support defined by $D=k$, $k=1,\ldots,K$. The joint density (\[densjoint\]) is a mixture with its components defined on non-overlapping regions of the density’s support. Considering the density conditional on the observed data, $\left(k,\textbf{\textit{x}}_{(k)}\right)$, the (unconditional) likelihood is $$\begin{aligned}
 \label{LKjoint}
{\mathcal{L}}\left(\theta \vert k,\textbf{\textit{x}}_{(k)} \right) =
f_{\mathbf{X}_{(k)}}^{sub}.\end{aligned}$$ In contrast, the density of observations conditional on stopping at stage $k$, i.e., the density of $ \mathbf{X}_{(k)}=\mathbf{X}_{(D)}\vert \{D=k\}$, is $$\begin{aligned}
\label{densK}
f^c_{\mathbf{X}_{(k)}} 
&:= f_{\mathbf{X}_{(D)}} \left(\textbf{\textit{x}}_{(D)}\vert D=k, \theta \right) 
= \frac{I\left(D=k\right)}{\text{Pr}_{\theta}\left(D=k\right)} f_{\mathbf{X}_{(k)}} \left(\textbf{\textit{x}}_{(k)}\vert  \theta \right)
= I\left(D=k\right)\frac{f_{\mathbf{X}_{(k)}}^{sub}}{\text{Pr}_{\theta}\left(D=k\right)}\end{aligned}$$ with 