I'm reading the paper by Sundaresan A. et al. (2007). On the page 5, I have confused with the variables definition in the target function of an optimization problem.
C. Optimal Threshold for Local Sensors
In Section III-A, we assumed that $\tau_1$ and $\tau_2$ are individual sensor thresholds. It can be seen that $P_D$ and $P_{FA}$ given by Eq.(18) and Eq.(19) are functions of $\tau_1$ and $\tau_2$. Constraining $P_{FA}=\alpha$, $P_D$ can be written as $P_D(\tau_1, \tau_2)=Q(\frac{\sigma_0(\tau_1, \tau_2)Q^{-1}(α) +\mu_0(\tau_1, \tau_2)−\mu_1(\tau_1, \tau_2)}{\sigma_1(\tau_1, \tau_2)}).\tag{30}$
The sensor thresholds are chosen to maximize $P_D$ at a particular value of $P_{FA}$. Hence the optimal sensor thresholds are given by $(\tau_1^*, \tau_2^*) = \arg \max_{\tau_1,\tau_2} P_D(\tau_1,\tau_2). \tag{31}$
First time, the mean $\mu_0$, $\mu_1$ introduced in Eq.(14), (16), on page 3: $\mu_0=N[C_1q_1+C_2q_2+C_3q_3], \tag{14}$
$\mu_1=N[C_1p_1+C_2p_2+C_3p_3], \tag{16}$ where $N$ is constant equal to $100$, $C_1, C_2, C_3$ are constants, and $p_1, p_2, p_3$, $q_1, q_2, q_3$ are probabilities.
Later, the authors assumed that $C_3=0$. I can suppose, that I can rewrite Eq.(14), (16) without the last terms: $\mu_0=N[C_1q_1+C_2q_2], \tag{14'}$
$\mu_1=N[C_1p_1+C_2p_2]. \tag{16'}$
Finally, I can denote $\mu_0=\mu_0(q_1,q_2)$, $\mu_1=\mu_1(p_1,p_2)$, i.e. we have two functions and four variables.
Question. How to derive the function $P_D$ in Eq. (30) such that the function of two variable $\tau_1$, $\tau_2$ only?
Remark. Should I use the relations: $p_1=1-p_2$, $q_1=1-q_2$ because
$p_1$, $p_2$ and $q_1$, $q_2$ the success probabilities under H1, H0 respectively in the Bernoulli scheme.
I have found the answer on my question. I read again and found that in Eq.(5)-(8), the probabilities $p_1$, $p_2$, $q_1$ and $q_2$ are the functions of $\tau_1, \tau_2$, therefore, I can write $\mu_0=\mu_0(p_1, p_2)=\mu_0(p_1(\tau_1), p_2(\tau_2))=\mu_0(\tau_1, \tau_2)$.