# Distributed Diffusion Multi-Distribution Filter with IMM for Heavy-Tailed Noise
## 3 Distribution Diffusion Multi-Distribution Filtering of IMM
### ATC-Diffusion Multi-Distribution Filtering  
Consider the following Bayes filtering process of single-node \(i\) with measurement sequence \(Z^{k}=\{z_{k^{\prime}}^{1},z_{k^{\prime}}^{2},z_{k^{\prime}}^{3}...z_{k}^{n}\},n=| \mathcal{N}|\), and \(Z_{k}=\{z_{1}^{i},z_{2}^{i},...,z_{k^{\prime}}^{i},i\in\mathcal{N}\}\) which is the measurement sequence gained up to instant k, including prior and posterior PDFs of multi-distribution:  
\[p_{k|k-1}^{i}(x_{k}|Z_{k},\mathcal{H}_{h})\] \[=\int\left[p^{i}(x_{k}|x_{k-1},\mathcal{H}_{h})p_{k-1|k-1}^{i}(x_ {k-1}|Z_{k-1})dx_{k-1}\right. \tag{6}\]  
\[p_{k|k}^{i}(x_{k}|Z_{k},\mathcal{H}_{h})\propto p(z_{k}^{i}|x_{k})p_{k|k-1}^{i }(x_{k}|Z_{k},\mathcal{H}_{h}) \tag{7}\]  
\[p_{k|k}^{i}(x_{k}|Z_{k})\propto\sum_{h\in\mathcal{H}}p_{k|k}^{i}(x_{k}|Z_{k}, \mathcal{H}_{h}) \tag{8}\]  
The mixed PDFs of single node will be used as a single node in the distributed network for subsequent fusion. A diffusion-based consensus strategy is employed to develop a distributed untraceable Kalman filter. This approach, characterized by localized information exchange between neighboring nodes and iterative diffusion of fused results, offers advantages in implementation simplicity, data consistency, high availability, and rapid response to read operations. The diffusion algorithm dictates the information exchange rules, enabling agreement on state estimates. This agreement is achieved by leveraging the total probability theorem and Bayes' law to fuse local information with values received from neighboring nodes, under probability density functions (PDFs) or probability mass functions (PMFs), facilitating convergence through diffusion [26]. A diffusion filtering with distribution network requires two steps: adapt means single node filtering and combine means network consensus strategy. Compared to consensus algorithm, which require many iterations, diffusion algorithm only need one iteration, greatly reducing the communication burden. In addition, consensus algorithm calculates the geometric mean of the posterior estimates of all neighboring nodes, while diffusion algorithm first calculate the local posterior estimates. This means that nodes using consensus algorithm are more affected by neighboring nodes, thereby reducing their robustness to abnormal nodes. Assume that there is no delay in communication between neighboring nodes in the network. Let \(\odot\) denote the inner product between functions \(f(\cdot)\) and \(g(\cdot)\) such that  
\[f\odot g\triangleq\int f(x)g(x)dx \tag{9}\]  
and, \(\bar{p}_{k|k}^{i}(x_{k}|Z_{k})\) denotes the PDFs after diffusion, then the Bayesian process of network can be shown as follows:  
\[p_{k|k-1}^{i}(x_{k}|Z_{k})=\bar{p}_{k-1|k-1}^{i}(x_{k-1}|Z_{k-1})\odot p(x_{ k}|x_{k-1},Z_{k}) \tag{10}\]  
\[p_{k|k}^{j}(x_{k}|Z_{k})\propto\prod_{i\in\mathcal{N}_{i}}p(z_{k}^{i}|x_{k},Z _{k})]\odot p_{k|k-1}^{i}(x_{k}|Z_{k}) \tag{11}\]  
where \(Z_{k}\) means the set of the measurement set \(Z^{k}\) up to instant k. Following the above local update, the combine step replaces \(p_{k|k}^{j}(x_{k})\) at each node i with a merged PDFs:  
\[\bar{p}_{k|k}^{j}(x_{k}|Z_{k})\propto\prod_{i\in\mathcal{N}_{i}}\left[p_{k|k}^{ i}(x_{k}|Z_{k})\right]^{k_{k}^{i}} \tag{12}\]where the real exponents \(\kappa_{k}^{i}\) are such that  
\[\sum_{i\in N_{i}}\kappa_{k}^{i}=1\]  
\(\bar{p}_{k|k}^{i}(x_{k})\) is the merged pdf that minimizes, at instant k and at node i, the weighted average KL divergence  
\[\sum_{i\in\mathcal{N}}\kappa_{k}^{i}D_{KL}(p^{*}||p_{k|k}^{i}) \tag{13}\]  
over all possible PDFs \(p^{*}\). Following [26], as the iteration count increases without bound, the merged probability density function approaches the function \(p^{*}\) which minimizes the average Kullback-Leibler divergence across the network.