associated likelihood $$\begin{aligned}
 \label{LKcon}
{\mathcal{L}}^c\left(\theta \vert k,\textbf{\textit{x}}_{(k)}  \right) =
 I\left(D=k\right)\frac{f_{\mathbf{X}_{(k)}}^{sub}}{\text{Pr}_{\theta}\left(D=k\right)}.\end{aligned}$$ The indicator function in and emphasizes that support for the random variables is reduced by the conditioning (see Figure \[support\_white\]). Note also in Figure \[support\_white\] that the support conditional on stopping at one stage is disjoint from the support conditional on stopping at another stage. If the stopping rule is not random and the experiment stops with $n_{(k)}$ observations, $\text{Pr}_{\theta}(D=k)\equiv 1$ and $$\begin{aligned}
 \label{Lfix}
{\mathcal{L}}^{fix}\left(\theta \vert k,\textbf{\textit{x}}_{(k)}  \right) =
f_{\mathbf{X}_{(k)}}.\end{aligned}$$

\

Both ${\cal{L}}$ and ${\cal{L}}^c$ are functions of $\theta$ and the observed data $(k, \textbf{\textit{x}}_{(k)})$, but ${\cal{L}}$ and ${\cal{L}}^c$ are continuous functions only of $\theta$ and have discontinuities in $(k, \textbf{\textit{x}}_{(k)})$ since the distribution of $\mathbf{X}_{(D)}$ is a mixture. These discontinuities are inherited by unconditional and conditional MLEs $\widehat\theta$ and $\widehat\theta^c$, respectively, that are defined below. Adaptation can change distributions of the MLEs and their information measures.

Conditional on $D=k$, MLEs maximizing ${\mathcal{L}}$ and ${\mathcal{L}}^c$, respectively, are $$\begin{aligned}
\widehat\theta_{(k)} &= \arg \max_{\theta} f_{\mathbf{X}_{(k)}}^{sub} \quad \textrm{ and } \quad 
\widehat\theta^{c}_{(k)} = \arg \max_{\theta} \left[I\left(D=k\right) \frac{f_{\mathbf{X}_{(k)}}^{sub}}{\text{Pr}_{\theta}\left(D=k\right)}\right].\end{aligned}$$ The estimator $\widehat\theta^{c}_{(k)}$ was suggested by @koopmeiners2012. Overall, the MLEs derived from and , respectively, can be written as $$\widehat\theta = \sum_{k=1}^K I(D=k) \widehat\theta_{(k)}\quad\textrm { and 