## 2 Problem Formulation
### Distributed Bayesian Filter Formulation  
Consider the following discrete-time stochastic non-linear dynamics,  
\[x_{k+1}=f(x_{k})+w_{k}, \tag{1}\]  
where \(x_{k}\in\mathbb{R}^{n}\) is the state that needs to be estimated and \(w_{k}\) is zero mean Gaussian noise with \(\mathbb{E}\{w_{k}w_{k}^{\top}\}=Q_{k}\). Dynamics (1) describes the state transition \(p(x_{k}|x_{k-1})\). Assume that the state \(x_{k}\) is observed by a network of \(N\) sensors, whose measurement model is given as:  
\[z_{i,k}=h(x_{k})+v_{i,k},\;i=1,2,\ldots,N, \tag{2}\]  
where \(z_{i,k}\in\mathbb{R}^{m_{i}}\) is the measurement by sensor \(i\) at time \(k\). \(v_{i,k}\sim\mathbb{N}(\mathbf{0},R_{i,k})\) is the measurement noise of sensor \(i\), where \(\mathbf{0}\in\mathbb{R}_{i}^{m}\) denotes the zero vector. Measurement Equation (2) describes the measurement likelihood function \(p(z_{i,k}|x_{k}),\;i=1,2,\ldots,N\). In this paper, we assume that \(w_{k}\) and \(v_{i,k}\forall i\in\mathcal{V}\) are independent, and all \(z_{i,k},\forall i\in\mathcal{V}\) are conditionally independent of \(x_{k}\). The communication of the network is modeled by an undirected graph \(\mathcal{G}_{k}=(\mathcal{V},\mathcal{E}_{k},\mathcal{A}_{k})\), which consists of the set of sensors \(\mathcal{V}=\{1,2,\ldots,N\}\), the set of edges \(\mathcal{E}_{k}\subseteq\mathcal{V}\times\mathcal{V}\) and the weighted adjacent matrix \(\mathcal{A}_{k}=[a_{i,k}]\). In the weighted adjacent matrix \(\mathcal{A}_{k}\), all the elements are nonnegative, row stochastic, and the diagonal elements are all positive, i.e., \(a_{ii,k}>0,a_{ii,k}\geq 0,\sum_{j\in\mathcal{V}}a_{ii,k}=1\). If \(a_{ijk}>0,j\
eq i\), there is an edge \((i,j)\in\mathcal{E}_{k}\), which means nodes \(i\) and \(j\) can directly communicate, and node \(j\) is called the neighbor of node \(i\). The degree matrix is defined as \(D^{(\mathcal{G})}=diag\{d_{1},d_{2},\ldots,d_{N}\}\in R^{N\times N}\), where the diagonal element \(d_{i}\) is the number of nodes connected to node \(i\). All the neighbors of node \(i\) including itself can be represented by the set \(\{j\in\mathcal{V}|(i,j)\in\mathcal{E}_{k}\}\bigcup\{i\}\triangleq\mathcal{N} _{i,k}\), whose size is denoted as \(|\mathcal{N}_{i,k}|\). In this paper, we assume that the undirected graph \(\mathcal{G}_{k}\) is connected for all \(k\). The adjacent matrix \(\mathcal{A}_{k}\) represents the weights of nodes. Note that for a certain graph, there exists an infinite number of associated adjacency matrices. To ensure the double stochastic nature of adjacent matrix \(\mathcal{A}_{k}\), a possible choice of the weights [32] is:  
\[a_{ij,k} =\frac{1}{\max\{|\mathcal{N}_{i,k}|,|\mathcal{N}_{j,k}|\}},\;j \in\mathcal{N}_{i,k},i\
eq j, \tag{3}\] \[a_{ii,k} =1-\sum_{j\in\mathcal{N}_{i,k},j\
eq i}a_{ij,k}. \tag{4}\]Denote the global measurement as \(z_{k}=[z_{1,k}^{\top},z_{2,k}^{\top},\ldots,z_{N,k}^{\top}]^{\top}\). The relationship between \(z_{k}\) and \(x_{k}\) can be given as global likelihood function \(p(z_{k}|x_{k})\), and the relationship between \(x_{k}\) and measurement \(z_{i,k}\) can be described as local likelihood function \(p(z_{i,k}|x_{k})\). Under the assumption that the agents are independent of each other, the global likelihood function can be expressed as a product of local likelihood functions,  
\[p(z_{k}|x_{k})=\prod_{i=1}^{N}p(z_{i,k}|x_{k}). \tag{5}\]  
In this paper, we assume that the state \(x_{k}\) is conditional independent with all past measurements \(z_{1:k-1}\), i.e., \(p(x_{k}|z_{1:k})=p(x_{k}|z_{k})\). Sensor \(i\) knows the dynamics (1) and local likelihood function \(p(z_{i,k}|x_{k})\) and does not know the global likelihood function \(p(z_{k}|x_{k})\). Sensor \(i\) can only communicate with its neighbors. The aim of the Bayesian filter is to compute posterior distribution \(p(x_{k}|z_{k})\). The recursive solution to compute \(p(x_{k}|z_{k})\) consists of prediction and update steps.