and ${\cal{I}}_{(k)}^c$ reduce to Fisher information from $n_{(k)}$ observations: $$\begin{aligned}
\label{I_(k)2}
 {\cal{I}}_{(k)}^{fix}&=\int \cdots \int \left[-\frac{\partial^2}{\partial\theta^2} \log  f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta) \right]
f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta)\prod_{i=1}^{n_{(k)}} dx_i.\end{aligned}$$Overall, the expected (Fisher) information measures are $$\begin{aligned}
{\cal{I}} &= \text{E}\left[{\cal{I}}^{obs}_{(D)}\right] 
=
 \text{E}_{D}\left[{\cal{I}}_{(k)}\right] 
=\sum_{k=1}^D \text{Pr}_{\theta}(D=k) \text{E}_{\textbf{X}_{(k)}}\left[{\cal{I}}^{obs}_{(k)}\right] \quad\textrm{ and } \notag\\
{\cal{I}}^c &=\text{E}\left[{\cal{I}}^{obs,c}_{(D)}\right]
=\text{E}_D\left[{\cal{I}}^{c}_{(k)}\right]
= \sum_{k=1}^D \text{Pr}_{\theta}(D=k) \text{E}_{\textbf{X}_{(k)}}\left[{\cal{I}}^{obs,c}_{(k)}\right]\notag \\
&= {\cal{I}} - \text{E}_D\left[-\frac{\partial^2}{\partial\theta^2} \log  \text{Pr}_{\theta}\left(D\right)\right].\label{e6}\end{aligned}$$ Note that the information loss incurred by conditioning is equal to the amount of Fisher information about $\theta$ in the early stopping rule, namely $\text{E}_D\left[-\frac{\partial^2}{\partial\theta^2} \log  \text{Pr}_{\theta}\left(D\right)\right]$. If ${\text{Pr}}_{\theta}\left(D\right)$ does not depend on $\theta$, ${\cal{I}}={\cal{I}}^{c}$. If the fact that ${\cal{I}}_{(k)}^{fix}$ assumes $\text{Pr}_{\theta}(D=k)\equiv 1$ is ignored, the use of ${\cal{I}}_{(k)}^{fix}$ with every $D=k$ leads to $${\cal{I}}^{fix} := \sum_{k=1}^K \text{Pr}_{\theta}\left(D=k\right)  {\cal{I}}_{(k)}^{fix}.$$ Since $f_{\mathbf{X}_{(k)}}^{sub}(\textbf{\textit{x}}_{(k)}|\theta)$ can be positive only when $D=k$, and $f_{\mathbf{X}_{(k)}}^{sub}(\textbf{\textit{x}}_{(k)}|\theta) = f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta)$ when $D=k$, ${\cal{I}}_{(k)}$ is re-written as $$\begin{aligned}
{\cal{I}}_{(k)} 
&=& \underset{I(D=k)}{\int \cdots \int} \left[-\frac{\partial^2}{\partial\theta^2} \log  f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta) \right]
\frac{f_{\mathbf{X}_{(k)}}(\textbf{\textit{x}}_{(k)}|\theta)}{\text{Pr}_{\theta}(D=k)}\prod_{i=1}^{n_{(k)}} dx_i\label{e7}, \end{aligned}$$ From (\[I\_(k)2\]) and (\[e7\]), ${\cal{I}}_{(k)} \ne {\cal{I}}_{(k)}^{fix}$ under GSDs. At $\theta=\widehat\theta$, $\frac{\partial^2}{\partial\theta^2}\log  f_{\mathbf{X}_{(k)}}\left(\textbf{\textit{x}}_{(k)}|\theta=\widehat\theta\right) \le 0$ and ${\cal{I}}_{(k)} \le {\cal{I}}_{(k)}^{fix}$ and ${\cal{I}} \le {\cal{I}}^{fix}$.

Example: Pocock One-Sided Two-Group Sequential Z-test {#Pocock_example}
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Pocock’s design is a simple two-stage study design for testing $H_0: \theta = 0$ versus $H_1: \theta = \theta_1$ with $X_i \sim N(\theta,1)$. Under $n_1=n_2=100$,$c_1=2.18$ is 