## 2 Problem Formulation
### Consensus of Probability Densities  
In this section, we will describe the consensus of probability density functions, which will be used to solve the distributed Bayesian filter problem in Section 3. The traditional average consensus problem is defined under Euclidean space [32]. However, the measure under Euclidean space is not suitable for the probability distribution. For example, two normal distributions \(\mathbb{N}(0,10,000)\) and \(\mathbb{N}(10,10,000)\) are almost indistinguishable, the Euclidean distance between the parameter is 10. In contrast, the distributions \(\mathbb{N}(0,0.01)\) and \(\mathbb{N}(0.1,0.01)\) barely overlap, but this is not reflected in the Euclidean distance, which is only 0.1. A more natural measure between two densities is Kullback-Leibler divergence rather than Euclidean distance. K-L divergence between two PDFs \(p(\cdot\,)\) and \(q(\cdot\,)\) is defined as:  
\[D_{KL}(p\|q)=\int p(x)\log\frac{p(x)}{q(x)}dx. \tag{8}\]Following [19; 33], the centroid depending on K-L divergence (CKL) is considered, which describes the centroid form initial PDFs,  
\[\bar{p}=\arg\inf\sum_{i=1}^{N}a_{i}D_{KL}(p\|p_{i}). \tag{9}\]  
where \(a_{i}\geq 0,\forall i\) are weights and should satisfy \(\sum_{i\in\mathcal{N}}a_{i}=1\). The centroid in (9) turns out to be [19],  
\[\bar{p}(x)=\frac{\prod_{i=1}^{N}[p_{i}(x)]^{a_{i}}}{\int\prod_{i=1}^{N}[p_{i}(x )]^{a_{i}}dx}. \tag{10}\]  
It is worth noting that the CKL can be seen as an example of Bregman information as the mean of Bregman divergence [34; 35]. An important feature of (10) is that it is suitable for distributed computation. Namely, the CKL can be achieved by some consensus algorithms, which requires that the data are only transmitted between agents and their neighbors at each step. Thus, CKL can be computed by the consensus algorithm as follows,  
\[p_{i}^{(t)}(x)=\frac{\prod_{j\in\mathcal{N}_{i}}[p_{j}^{(t-1)}(x)]^{a_{ijk}}}{ \int\prod_{j\in\mathcal{N}_{i}}[p_{j}^{(t-1)}(x)]^{a_{ijk}}dx} \tag{11}\]  
where \(t=1,2,\ldots\) refer to the \(t\)-th step and \(a_{ijk}\) is the weights between agent \(i\) and \(j\). When the distribution is given, the consensus of PDFs will be achieved by manipulating the corresponding parameters. The following lemma shows how to compute consensus on the exponential family iteratively, which can be found in [36]. **Lemma 1**. : _Consider the network \(\mathcal{G}_{k}=(\mathcal{V},\mathcal{E}_{k},\mathcal{A}_{k})\). Let a PDF \(p_{i}^{(t)}(x)=f(x;\lambda_{i}^{(t)}),\forall i\in\mathcal{N}\) be exponential distribution families, where \(\lambda\) is a natural parameter. Then, iteratively update (11) given by:_  
\[\lambda_{i}^{(t+1)}=\sum_{j\in\mathcal{N}_{i}}a_{ijk}\lambda_{j}^{(t)} \tag{12}\]  
**Remark 1**. : _An exponential family can be expressed in the following form [37],_  
\[p(\theta)=h(\theta)\exp\{\lambda^{\top}u(\theta)-A_{g}(\lambda)\}, \tag{13}\]  
_where \(\lambda\) is a natural parameter, \(A_{g}(\lambda)\) is a log-normalizer and \(h(\theta)\) is a carrier measure [38]. The exponential families include many of the most common distributions, e.g., Gaussian, Poisson, Bernoulli, Wishart, and many other. Namely, those distributions can be written in the form of exponential families (13)._  
**Remark 2**. : _It should be noted that K-L divergence defined in (8) is not symmetric. In [33], the authors discussed sided Bregman centroids, i.e., right-sided centroid \(\bar{p}^{R}=\arg\inf\sum_{i=1}^{N}a_{i}D_{KL}(p_{i}\|p)\) and left-sided centroid \(\bar{p}^{L}=\arg\inf\sum_{i=1}^{N}a_{i}D_{KL}(p\|p_{i})\). As shown in [33] (Theorem 3.1 in [33]), \(\bar{p}^{R}\) can be expressed as a convex combination of PFDs, i.e., \(\bar{p}^{R}=\sum_{i=1}^{N}a_{i}p_{i}\), which always is the center of mass. Notice that \(\bar{p}^{R}\) is hard to obtain, if \(p_{i}\) and \(p_{i}\) are correlated. However, for the distributed estimation problem, \(p_{i}\) from different nodes need to be fused at each time \(k\), and consequently, they are correlated (see [17]).