Formal Definitions of Conservative PDFs

Under ideal conditions, the probability density function (PDF) of a random variable, such as a sensor measurement, would be well known and amenable to computation and communication tasks. However, this is often not the case, so the user looks for some other PDF that approximates the true but intractable PDF. Conservativeness is a commonly sought property of this approximating PDF, especially in distributed or unstructured data systems where the data being fused may contain un-known correlations. Roughly, a conservative approximation is one that overestimates the uncertainty of a system. While prior work has introduced some definitions of conservativeness, these definitions either apply only to normal distributions or violate some of the intuitive appeal of (Gaussian) conservative definitions. This work provides a general and intuitive definition of conservativeness that is applicable to any probability distribution, including multi-modal and uniform distributions. Unfortunately, we show that this \emph{strong} definition of conservative cannot be used to evaluate data fusion techniques. Therefore, we also describe a weaker definition of conservative and show it is preserved through common data fusion methods such as the linear and log-linear opinion pool, and homogeneous functionals. In addition, we show that after fusion, weak conservativeness is preserved by Bayesian updates. These strong and weak definitions of conservativeness can help design and evaluate potential correlation-agnostic data fusion techniques

Arithmetic Average Density Fusion -- Part I: Some Statistic and Information-theoretic Results

Finite mixture such as the Gaussian mixture is a flexible and powerful probabilistic modeling tool for representing the multimodal distribution widely involved in many estimation and learning problems. The core of it is representing the target distribution by the arithmetic average (AA) of a finite number of sub-distributions which constitute a mixture. While the mixture has been widely used for single sensor filter design, it is only recent that the AA fusion demonstrates compelling performance for multi-sensor filter design. In this paper, some statistic and information-theoretic results are given on the covariance consistency, mean square error, mode-preservation capacity, and the information divergence of the AA fusion approach. In particular, based on the concept of conservative fusion, the relationship of the AA fusion with the existing conservative fusion approaches such as covariance union and covariance intersection is exposed. A suboptimal weighting approach has been proposed, which jointly with the best mixture-fit property of the AA fusion leads to a max-min optimization problem. Linear Gaussian models are considered for algorithm illustration and simulation comparison, resulting in the first-ever AA fusion-based multi-sensor Kalman filter.Comment: 30 pages, 14 figures, 3 tables. Information Fusion, 202

Minimum information loss fusion in distributed sensor networks

A key assumption of distributed data fusion is that individual nodes have no knowledge of the global network topology and use only information which is available locally. This paper considers the weighted exponential product (WEP) rule as a methodology for conservatively fusing estimates with an unknown degree of correlation between them. We provide a preliminary investigation into how the methodology for selecting the mixing parameter can be used to minimize the information loss in the fused covariance as opposed to reducing the Shannon entropy, and hence maximize the information of the fused covariance. Our results suggest that selecting a mixing parameter which minimizes the information loss ensures that information which is exclusive to the estimates from one source is not lost during the fusion process. These results indicate that minimizing the information loss provides a robust technique for selecting the mixing parameter in WEP fusion

A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

A Bayesian Framework to Constrain the Photon Mass with a Catalog of Fast Radio Bursts

A hypothetical photon mass, $m_\gamma$, gives an energy-dependent light speed in a Lorentz-invariant theory. Such a modification causes an additional time delay between photons of different energies when they travel through a fixed distance. Fast radio bursts (FRBs), with their short time duration and cosmological propagation distance, are excellent astrophysical objects to constrain $m_\gamma$. Here for the first time we develop a Bayesian framework to study this problem with a catalog of FRBs. Those FRBs with and without redshift measurement are both useful in this framework, and can be combined in a Bayesian way. A catalog of 21 FRBs (including 20 FRBs without redshift measurement, and one, FRB 121102, with a measured redshift $z=0.19273 \pm 0.00008$) give a combined limit $m_\gamma \leq 8.7 \times 10^{-51}\, {\rm kg}$, or equivalently $m_\gamma \leq 4.9 \times 10^{-15}\, {\rm eV}/c^2$ ($m_\gamma \leq 1.5\times10^{-50} \, {\rm kg}$, or equivalently $m_\gamma \leq 8.4 \times 10^{-15} \,{\rm eV}/c^2$) at 68% (95%) confidence level, which represents the best limit that comes purely from kinematics. The framework proposed here will be valuable when FRBs are observed daily in the future. Increment in the number of FRBs, and refinement in the knowledge about the electron distributions in the Milky Way, the host galaxies of FRBs, and the intergalactic median, will further tighten the constraint.Comment: 10 pages, 6 figures; Physical Review D, in pres

Probabilistic Traversability Model for Risk-Aware Motion Planning in Off-Road Environments

A key challenge in off-road navigation is that even visually similar terrains or ones from the same semantic class may have substantially different traction properties. Existing work typically assumes no wheel slip or uses the expected traction for motion planning, where the predicted trajectories provide a poor indication of the actual performance if the terrain traction has high uncertainty. In contrast, this work proposes to analyze terrain traversability with the empirical distribution of traction parameters in unicycle dynamics, which can be learned by a neural network in a self-supervised fashion. The probabilistic traction model leads to two risk-aware cost formulations that account for the worst-case expected cost and traction. To help the learned model generalize to unseen environment, terrains with features that lead to unreliable predictions are detected via a density estimator fit to the trained network's latent space and avoided via auxiliary penalties during planning. Simulation results demonstrate that the proposed approach outperforms existing work that assumes no slip or uses the expected traction in both navigation success rate and completion time. Furthermore, avoiding terrains with low density-based confidence score achieves up to 30% improvement in success rate when the learned traction model is used in a novel environment.Comment: To appear in IROS23. Video and code: https://github.com/mit-acl/mppi_numb