**5. Extension for simultaneous search for the optimal sensor rules and fusion rule**

In this section, we extend the Monte Carlo method to search for the optimal sensor rules and the optimal fusion rule simultaneously. Firstly, the necessary condition is generalized to search for the optimal sensor rules and the optimal fusion rule simultaneously. Secondly, we describe a generalized Monte Carlo Gauss-Seidel iterative algorithm. We also give the convergence of the iterative algorithm.

## **5.1 A necessary condition for the optimal sensor rules and the optimal fusion rule**

Note that (15) can be rewritten as follows:

$$\begin{split} \mathsf{C\_{MC}}(\operatorname{I}\_{1}(\operatorname{y}\_{1}),\ldots,\operatorname{I}\_{L}(\operatorname{y}\_{L});\mathcal{F}^{0},\mathcal{F}^{\epsilon\operatorname{0}},\mathcal{P}^{\epsilon\operatorname{1}},N) \\ = &c+\frac{1}{N}\sum\_{i=1}^{N}\sum\_{k'=1}^{2^{L}}\sum\_{k=1}^{2^{L}}[1-\mathcal{F}^{0}(\mathsf{s}\_{k'})]P(\mathsf{s}\_{k'}|\mathsf{s}\_{k})\cdot P\_{\mathsf{s}\_{k}}(\operatorname{I}(\mathbf{Y}\_{i})\frac{\hat{L}(\mathbf{Y}\_{i})}{\operatorname{g}(\mathbf{Y}\_{i})} \\ = &c+\frac{1}{N}\sum\_{k'=1}^{2^{L}}[1-\mathcal{F}^{0}(\mathsf{s}\_{k'})]\sum\_{i=1}^{N}\sum\_{k=1}^{2^{L}}P(\mathsf{s}\_{k'}|\mathsf{s}\_{k})\cdot P\_{\mathsf{s}\_{k}}(\operatorname{I}(\mathbf{Y}\_{i})\frac{\hat{L}(\mathbf{Y}\_{i})}{\operatorname{g}(\mathbf{Y}\_{i})},\end{split} \tag{42}$$

where *Psk* ð Þ **<sup>I</sup>**ð Þ *Yi* <sup>≜</sup> <sup>Q</sup>*<sup>L</sup> <sup>j</sup>*¼<sup>1</sup> *sk*ð Þ*<sup>j</sup> Ij Yij* � � <sup>þ</sup> ð Þ <sup>1</sup> � *sk*ð Þ*<sup>j</sup>* <sup>1</sup> � *Ij Yij* � � � � � � and **<sup>I</sup>**ð Þ¼ *Yi* ð Þ *I*1ð Þ *Yi*<sup>1</sup> , *I*2ð Þ *Yi*<sup>2</sup> , … , *IL*ð Þ *YiL* . Since *Psk* ð**I**ð Þ¼ *Yi* 1 if and only if *Ij* ¼ *sk*ð Þ*j* for all *j* ¼ 1, … , *L*, (39) can be simplified as follows:

$$\begin{aligned} &\mathbf{C}\_{\text{MC}}(\mathbf{I}\_{1}(\mathbf{y}\_{1}),\ldots,\mathbf{I}\_{L}(\mathbf{y}\_{L});\mathbf{F}^{0},\mathbf{;}P^{\epsilon \mathbf{0}},\mathbf{P}^{\epsilon \mathbf{1}},\mathbf{N}) \\ &=\mathbf{c} + \frac{\mathbf{1}}{N} \sum\_{k'=1}^{2^{L}} [\mathbf{1} - F^{0}(\mathbf{s}\_{k'})] \cdot \sum\_{i=1}^{N} \mathbf{P}(\mathbf{s}\_{k'} | (I\_{1}(\mathbf{Y}\_{i1}),\ldots,I\_{L}(\mathbf{Y}\_{i\bar{L}}))) \frac{\hat{L}(\mathbf{Y}\_{i})}{\mathbf{g}(\mathbf{Y}\_{i})},\end{aligned} \tag{43}$$

where the terms *Psk* ð Þ¼ **I**ð Þ *Yi* 0 are eliminated.

Remark 6: According to (20) and (40), the necessary condition for the optimal sensor rules is similar to Lemma 2 and the necessary condition for the optimal fusion rule is given by

$$F^0(s\_{k'}) = I \left[ \sum\_{i=1}^N P(s\_{k'} | (I\_1(Y\_{i1}), \dots, I\_L(Y\_{iL}))) \cdot \frac{\hat{L}(Y\_i)}{\mathbf{g}(Y\_i)} \right] \tag{44}$$

for *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> 1, … , 2*<sup>L</sup>*. The proofs are similar to Lemma 2.
