## 6 Extension to Scenarios with Fusion: CEO Problem
### Fusion Rule  
Denote as  
\[\boldsymbol{\mathcal{L}}_{i}=[\mathcal{L}_{i}^{(1)},\ldots,\mathcal{L}_{i}^{(n )}]\qquad\qquad i=0,\ldots,L-1\]  
the vector of LLRs, relative to \(\boldsymbol{x}_{i}\), at the output of the \(n\) decoders. In order to estimate \(b_{i}\), we consider the following maximum a posteriori (MAP) fusion rule:  
\[\hat{b}_{i}\triangleq\operatorname*{argmax}_{b_{i}=0,1}P(b_{i}|\boldsymbol{ \mathcal{L}}_{i}) \tag{29}\]  
The MAP strategy (29) can be rewritten, by using the total probability theorem, as  
\[\hat{b}_{i}=\operatorname*{argmax}_{b_{i}=0,1}\sum_{\{\boldsymbol{x}_{i}\}}P (b_{i}|\boldsymbol{\mathcal{L}}_{i},\boldsymbol{x}_{i})P(\boldsymbol{x}_{i}| \boldsymbol{\mathcal{L}}_{i}) \tag{30}\]  
where \(\boldsymbol{x}_{i}=[x_{i}^{(1)},\ldots,x_{i}^{(n)}]\) are the noisy binary observations, relative to the \(i\)-th information symbol \(b_{i}\), at the inputs of the sensors and the sum in (30) is carried out over all possible \(2^{L}\) configurations for \(\boldsymbol{x}_{i}\)  
Figure 13: CEO scenario: multiple access scheme followed by fusion. Using the definition of conditional probability and the chain rule, the first probability at the right-hand side of (30) can be written as  
\[P(b_{i}|\boldsymbol{\mathcal{L}}_{i},\boldsymbol{x}_{i})=\frac{P(b_{i},\boldsymbol {\mathcal{L}}_{i},\boldsymbol{x}_{i})}{P(\boldsymbol{\mathcal{L}}_{i}, \boldsymbol{x}_{i})}=\frac{P(\boldsymbol{\mathcal{L}}_{i}|\boldsymbol{x}_{i},b _{i})P(\boldsymbol{x}_{i}|b_{i})P(b_{i})}{P(\boldsymbol{\mathcal{L}}_{i}| \boldsymbol{x}_{i})P(\boldsymbol{x}_{i})} \tag{31}\]  
Since \(\boldsymbol{\mathcal{L}}_{i}\) depend only on \(\boldsymbol{x}_{i}\), owing to the considered JCD scheme, it follows that  
\[P(\boldsymbol{\mathcal{L}}_{i}|\boldsymbol{x}_{i},b_{i})=P(\boldsymbol{ \mathcal{L}}_{i}|\boldsymbol{x}_{i})\]  
Moreover, since \(P(b_{i}=0)=P(b_{i}=1)=1/2\), from (31) one obtains  
\[P(b_{i}|\boldsymbol{\mathcal{L}}_{i},\boldsymbol{x}_{i})=\frac{P(\boldsymbol{ x}_{i}|b_{i})P(b_{i})}{P(\boldsymbol{x}_{i})}=\frac{P(\boldsymbol{x}_{i}|b_{i})}{ \sum_{b^{*}=0,1}P(\boldsymbol{x}_{i}|b^{*})}\]  
At this point, we assume that the second probability at the right-hand side of (30) can be expressed as  
\[P(\boldsymbol{x}_{i}|\boldsymbol{\mathcal{L}}_{i})=\prod_{j=1}^{n}P(x_{i}^{(j )}|\mathcal{L}_{i}^{(j)}) \tag{32}\]  
Equation (32) corresponds to assuming that the a posteriori probability of the \(i\)-th symbol at the input of the \(j\)-th source, i.e., \(x_{i}^{(j)}\), depends only on the LLR, at the same epoch, generated by the corresponding decoder, i.e., \(\mathcal{L}_{i}^{(j)}\). In other words, we assume that the a posteriori probability of \(x_{i}^{(j)}\) does not depend on the other LLRs. This is reasonable, since the proposed JCD scheme exploits the existing correlation between the sources. Therefore, the LLR \(\mathcal{L}_{i}^{(j)}\) already \"embeds\" the contribution from the other decoders. We remark that, in the presence of a different (e.g., non-iterative) JCD scheme, this assumption might have to be reconsidered. Finally, the fusion rule (30) becomes  
\[\hat{b}_{i}=\underset{b_{i}=0,1}{\mathrm{argmax}}\sum_{\{\boldsymbol{x}_{i} \}}\underbrace{\frac{P(\boldsymbol{x}_{i}|b_{i})}{\sum_{b^{*}=0,1}P( \boldsymbol{x}_{i}|b^{*})}}_{\text{from the correlation model}}\underbrace{\prod_{j=1}^{n}P(x_{i}^{(j )}|\mathcal{L}_{i}^{(j)})}_{\text{from LLRs}} \tag{33}\]  
where we have highlighted that each addendum of the outer sum can be expressed as a product of two terms: the first one depends only on the correlation between the observations \(\{x_{i}^{(j)}\}\), whereas the second one depends only on the LLRs, i.e., on the iterative JCD scheme. The probability of decision error on a single bit can then be written as  
\[P_{\mathrm{e}}=\frac{1}{2}\left[P(\hat{b}_{i}=0|b_{i}=1)+P(\hat{b}_{i}=1|b_{i }=0)\right] \tag{34}\]  
The evaluation of the _average_ probability of decision error can be carried out through simulations, by averaging out over all transmitted packets.