probability of observing $z_i$-score, given stimulus $c$, is $P_i'(c- \sigma_i z_i)$. Thus $$\begin{aligned}
p(\vec{z}|c) = P_1'(c-x_1)\cdot\ldots\cdot P_n'(c-x_n).\label{eq:pzc}\end{aligned}$$ Let us assume that the displayed stimulus has a uniform distribution, i.e., that $p(c)$ is constant (not going into mathematical nuances). To define the decision function, we need to know when $p(c>0|\vec{z}) \geq 1/2$ or, in other words, when the probability that the second answer is correct is greater than $1/2$. As is a Gaussian function of $c$, finding its maximum leads to the condition $$\begin{aligned}
\frac{x_1}{\sigma_1^2}+\ldots+\frac{x_n}{\sigma_n^2}&\geq 0,\end{aligned}$$ or equivalently, using the slope parameter, $$\begin{aligned}
s_1 z_1+\ldots+s_n z_n&\geq 0.\end{aligned}$$ Thus, when the condition holds, choosing the second option is the Bayes optimal choice. Unfortunately, in this model we only have access to values of $\vec{z}$, not to individual performances. To obtain the precise answer, we need to know the whole distribution of $\sigma_i$ (or $s_i$). Instead, we can use the approximate condition for the choice of the second option, $$\begin{aligned}
z_1+\ldots+z_n&\geq 0\label{eq:zsum},\end{aligned}$$ to obtain a lower bound on the performance. The condition is exact for participants with equal performances (and should be close to the optimal if the values of $\sigma_{i}$ do not vary much). This equation can be seen as a type 