Decentralised possibilistic inference with applications to target tracking

1. Problem Statement

The paper addresses the challenge of decentralised sensor fusion for target tracking in networks where multiple agents must share information to estimate the state and existence of a target. Key difficulties include:

• Data Association: Determining which observations correspond to the target versus false alarms (clutter).
• Target Birth/Death: The target may appear or disappear at unknown times.
• Decentralisation Constraints: Sensors communicate over a network (graph) without a central fusion node.
• Correlation and Independence: In probabilistic approaches, fusing estimates from different sensors is difficult when cross-correlations are unknown. Standard methods (like arithmetic or geometric averaging) often rely on ad-hoc assumptions that can lead to information loss, over-confidence, or inconsistency with the optimal centralised solution.
• Weakly Informative Observations: Scenarios where individual sensors cannot fully observe the target state (e.g., observing only position or only velocity) require robust handling of uncertainty.

The authors argue that Probability Theory struggles with these specific fusion tasks because it lacks a foundational rule for fusing distributions that strictly preserves source independence without ad-hoc approximations.

2. Methodology

The proposed solution is a Decentralised Possibilistic Bayesian Inference framework based on Possibility Theory.

Core Theoretical Framework

• Possibility Functions (p.f.): Instead of probability density functions (p.d.f.), the system uses possibility functions $f(x)$ where $\sup f(x) = 1$. These represent degrees of belief or credibility rather than frequency-based probabilities.
• Independence Definition: In this framework, independence is defined as the absence of a relation between the information held by agents, not necessarily the variables themselves. This allows for a principled fusion rule.
• Fusion Rule: For $n$ independent sources $f_i$, the fused possibility function is:
  $$\hat{f}(x) = \frac{\prod_{i=1}^n f_i(x)}{\sup_{x'} \prod_{i=1}^n f_i(x')}$$
  This rule is proven to be asymptotically exact, meaning that under specific conditions, repeated decentralised fusion recovers the exact posterior of the optimal centralised ("oracle") solution.

The Possibilistic Bernoulli Filter

The authors apply this framework to the Bernoulli Filter, which jointly infers:

1. Existence: The probability/possibility that a target exists (birth/death process).
2. State: The target's state (position, velocity, etc.).

Key Modifications for Decentralisation:

• Discounting: To preserve independence during fusion, the prior information at each node is "discounted" (raised to a power $\omega_i$) before being shared.
• Markov Transition Discounting: The state transition model is also discounted to prevent double-counting of dynamics information.
• Handling Gaussian Mixtures: Since the implementation uses Gaussian max-mixtures, the authors leverage the property that Gaussian p.f.s are closed under exponentiation. This allows for closed-form updates and fusion, unlike probabilistic geometric averaging which lacks closed-form solutions for the power of a mixture.
• Hybrid Likelihoods: The framework accommodates scenarios where observations are partially probabilistic (noise) and partially possibilistic (unknown correlations), using a "latent error" model to bound the dependence conservatively.

Algorithm

The decentralised algorithm operates in three steps per time step:

1. Prediction: Nodes predict the state using a discounted Markov transition.
2. Local Update: Nodes update their local posterior using their specific observations.
3. Fusion (Iterative): Nodes exchange their local posteriors with neighbors. They combine information using a weighted product rule (based on a weight matrix $\Gamma$) iteratively ($L$ times) until convergence.

3. Key Contributions

1. Principled Fusion Rule: Introduction of a decentralised possibilistic inference method proven to be algebraically equivalent to the centralised solution under specific conditions (Assumptions A1 and A2 regarding weight distribution and network connectivity).
2. Bernoulli Filter Implementation: Detailed derivation of the possibilistic Bernoulli filter with Gaussian max-mixtures, demonstrating applicability beyond linear-Gaussian settings.
3. Handling Unknown Dependence: A novel method to handle partially unknown dependencies between sensors by isolating correlated components and using conservative upper bounds, avoiding the need for explicit correlation estimation.
4. Theoretical Guarantees: Proof that the decentralised approach preserves the independence of local posteriors and converges to the centralised optimum, a property often lacking in probabilistic averaging baselines.

4. Results

The authors evaluated the method against Arithmetic Average (AA) and Geometric Average (GA) probabilistic baselines, as well as a Probabilistic Oracle (centralised), across three scenarios:

• Scenario 1: Standard Linear Tracking

  • Metrics: Cardinality error (target count), Localisation error, OSPA distance, and Entropy.
  • Findings: The possibilistic method significantly outperformed AA and GA in cardinality estimation (correctly detecting target birth/death). While localisation errors were similar, the possibilistic method maintained better uncertainty estimates (lower entropy inflation) compared to GA, which tended to be overly conservative. The decentralised method closely matched the Oracle performance even with limited communication iterations ($L=2$).

• Scenario 2: Weakly Informative Observations

  • Setup: Sensors observed only partial state components (e.g., position vs. velocity), making individual observations non-informative.
  • Findings: AA and GA failed to converge effectively, resulting in high localisation errors and poor OSPA scores. The possibilistic method successfully fused the partial information, closely tracking the Oracle's performance. This highlights the method's robustness when individual sensors lack full state visibility.

• Scenario 3: Non-linear Observations with Partial Dependence

  • Setup: Range-bearing (non-linear) observations with a shared latent error (correlated noise) across sensors.
  • Findings: The proposed method was robust to varying levels of correlation ($c$). The baselines, which do not explicitly model the correlation structure in their fusion rules, showed degraded performance as correlation increased. The possibilistic approach maintained high credibility of target presence and low localisation error.

Computational Cost: The proposed method had a computational cost comparable to the baselines (approx. 1.1x of GA), making it feasible for real-time applications.

5. Significance

• Theoretical Breakthrough: The paper provides a rigorous mathematical foundation for decentralised fusion that avoids the "double counting" of information often found in probabilistic consensus algorithms. It proves that decentralised fusion can be exact (recovering the centralised result) without requiring a central node.
• Superior Performance in Complex Scenarios: The method demonstrates that Possibility Theory is particularly well-suited for problems involving uncertainty in existence (birth/death) and partial observability, outperforming standard probabilistic averaging techniques in these domains.
• Practical Applicability: By utilizing Gaussian max-mixtures and providing closed-form updates, the approach is implementable in real-world sensor networks. It offers a robust alternative for multi-target tracking where sensor correlations are unknown or difficult to model.
• Future Impact: The framework lays the groundwork for more advanced multi-target tracking (e.g., labelled multi-Bernoulli filters) and robust inference in environments with calibration errors or unknown sensor dependencies.