**1. Introduction**

Distributed detection has been an active research area in the past decades [1–7]. It involves the design of decision rules for the sensors<sup>1</sup> and the fusion rule [8]. Early work on distributed detection mainly focused on conditionally independent sensor observations, such as [2, 4, 9, 10], and the resulting optimal sensor decision rules, as well as the fusion rule, were likelihood ratio tests (LRTs). Details on distributed detection with conditionally independent sensor observations can be seen in [1, 6, 7] and references therein.

<sup>1</sup> In the rest of the paper, the term "sensor rules" refers to the "decision rules at the sensors."

In this chapter, we focus on conditionally dependent observations in sensor networks. In [5], the computational difficulty of obtaining the optimal sensor rules was shown by a rigorous mathematical approach. Some early progress was made on the derivation of sensor rules for the dependent observation case such as in [11–15]. More recently, a hierarchical conditional independence model was provided that was applicable to some specific classes of multisensor detection problems with dependent observations [16]. Copula-based distributed decision fusion methods have been proposed to deal with dependent observations in sensor networks, such as [17–19] and references therein. Given a fusion rule, Monte Carlo methods were proposed to reduce the computational complexity of deriving sensor decision rules with ideal channels in [20, 21], and the optimal sensor rules were obtained analytically for the *K*-out-of-*L* fusion rule in [20].

Some works on the derivation of optimal fusion rules can be seen in [15, 22–24]. For some specific parallel network decision systems, a unified fusion rule was presented in [15]. Some further results on the problem are available in [25, 26]. In [27], the authors provided methods that search for the sensor rules and the fusion rule simultaneously by combining the methods of [2] and [15] in order to attain nearoptimal system performance.

The works discussed thus far assumed the availability of ideal channels in sensor networks. However, channel errors between the sensors and the fusion center are omnipresent in practical multisensor detection networks, and, therefore, studies on multisensor detection in the presence of nonideal channels have attracted some recent interest, such as in [8, 28–33]. Under the Neyman-Pearson criterion, the design of sensor rules in the presence of nonideal channels was addressed in [32]. The parallel fusion structure was extended by incorporating the fading channel layer and two alternative fusion schemes were presented based on fixed sensor rules in [28]. It was shown that the optimal sensor decision rule that minimizes the error probability at the fusion center is equivalent to a local LRT for independent sensor observations in [29]. Under Neyman-Pearson and Bayesian criteria, the work was generalized to dependent and noisy channels, respectively, in [8]. In [31], the authors considered the optimal sensor rules with channel errors *via* Riemann sum approximation under a given fusion rule for general dependent sensor observations. Although the method based on the Riemann sum approximation has been developed for dependent observations with channel errors, it is too computationally expensive to be of practical use in large-scale sensor networks.

In this chapter, a Monte Carlo importance sampling method is provided to reduce the computational complexity of multisensor detection fusion with channel errors. Based on the strong law of large numbers, the Bayesian cost function is approximated by its empirical average through the Monte Carlo importance sampling method. The main contributions of this chapter are listed below:


3.The Monte Carlo Gauss-Seidel optimization algorithm is extended to simultaneously search for the optimal sensor rules and the optimal fusion rule.

Numerical examples show the effectiveness of the new algorithms for large-scale sensor networks with dependent observations and channel errors.

The rest of this chapter is organized as follows: In Section 2, the parallel binary Bayesian detection network with channel errors is formulated and the Monte Carlo cost function is introduced. In Section 3, the necessary condition for the optimal sensor rules is presented. For the *K*-out-of-*L* fusion rule, the analytical form for the optimal sensor rules is provided. In Section 4, the Monte Carlo Gauss-Seidel iterative algorithm and its convergence analysis are presented. The extension to search for the optimal sensor rules and the optimal fusion rule are simultaneously described in Section 5. Simulation results are provided in Section 6. Conclusions are contained in Section 7.
