}\quad
\widehat\theta^c = \sum_{k=1}^K I(D=k) \widehat\theta_{(k)}^c.$$

For every observed pair $\left(k,\textbf{\textit{x}}_{(k)}\right)$, $f_{\mathbf{X}_{(k)}}^{sub} =f_{\mathbf{X}_{(k)}}$ for all $\theta$ and, consequently, $$\widehat\theta_{(k)}^{fix} := \arg \max_{\theta} f_{\mathbf{X}_{(k)}} = \arg \max_{\theta} f_{\mathbf{X}_{(k)}}^{sub} = \widehat\theta_{(k)}$$ and $$\widehat\theta^{fix}=\sum_{k=1}^K I(D=k) \widehat\theta_{(k)}^{fix} 
= \sum_{k=1}^K I(D=k) \widehat\theta_{(k)}=
\widehat\theta,$$ which means that MLEs stay unchanged for maximization of either ${\mathcal{L}}$ and ${\mathcal{L}}^{fix}$.

Conditional and Unconditional Information
-----------------------------------------

The superscript *obs* is used to denote the observed information (and quantities obtained from it); and the superscript *fix* is used to denote the quantities derived assuming that study stops at a pre-determined fixed sample size (no adaptation). As previously, the superscript *c* denotes quantities obtained from the conditional likelihood and the stopping stage is referenced, when needed, in subscripts.

With this notation, the *observed information measures* derived from ${\cal{L}}$ , ${\cal{L}}^c$, and ${\cal{L}}^{fix}$, respectively, are $$\begin{aligned}
\label{obsI}
{\cal{I}}^{obs}_{(k)} 
&=-\frac{\partial^2}{\partial\theta^2} \log  f_{\mathbf{X}_{(k)}}^{sub}(\textbf{\textit{x}}_{(k)}|\theta)
=-\frac{\partial^2}{\partial\theta^2}  I(D=k)\log  f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta),
\\
{\cal{I}}^{obs,c}_{(k)}  
&={\cal{I}}^{obs}_{(k)} +\frac{\partial^2}{\partial\theta^2} \log\text{Pr}_{\theta}\left(D=k\right) \quad\textrm{ and }\\
{\cal{I}}^{obs,fix}_{(k)}&=-\frac{\partial^2}{\partial\theta^2} \log  f_{\mathbf{X}_{(k)}}.\end{aligned}$$ Observed information matrices evaluated at MLEs are positive definite. Therefore,\
$\frac{\partial^2}{\partial\theta^2} \log\text{Pr}_{\theta}\left(D=k\right)\vert_{\theta=\widehat\theta}<0$ and ${\cal{I}}^{obs,c}_{(k)}\vert_{\theta=\widehat\theta} < {\cal{I}}^{obs}_{(k)}\vert_{\theta=\widehat\theta}$.

*Stopped expected information measures* are $$\begin{aligned}
 {\cal{I}}_{(k)}&:= \text{E}_{\textbf{X}_{(k)}}\left[{\cal{I}}^{obs}_{(k)}\right]
=\int \cdots \int {\cal{I}}^{obs}_{(k)}
\frac{f_{\mathbf{X}_{(k)}}^{sub}(\textbf{\textit{x}}_{(k)}|\theta)}{\text{Pr}_{\theta}(D=k)}\prod_{i=1}^{n_{(k)}} dx_i \quad\textrm{ and }\\
{\cal{I}}^c_{(k)}&:= \text{E}_{\textbf{X}_{(k)}}\left[{\cal{I}}^{obs,c}_{(k)}\right] =\int \cdots \int \left[ {\cal{I}}^{obs}_{(k)} +\frac{\partial^2}{\partial\theta^2} \log\text{Pr}_{\theta}\left(D=k\right)\right]
 \frac{f_{\mathbf{X}_{(k)}}^{sub}(\textbf{\textit{x}}_{(k)}|\theta)}{\text{Pr}_{\theta}(D=k)}\prod_{i=1}^{n_{(k)}} dx_i.\end{aligned}$$ Under $\text{Pr}_{\theta}(D=k)\equiv 1$, ${\cal{I}}_{(k)}$ 