The Epistemic Abductive Filter: Closed-Form Possibilistic Estimation, Its Optimality, and Its Kalman Form

The Kalman filter is the simple operationalization of Bayesian inference for linear systems. This paper gives the corresponding filter for the Theory of Epistemic Abductive Geometry (TEAG): the Epistemic Abductive Filter (EAF), a possibilistic filter. Its state is one ellipsoid, the support of the states that the declared bounds and the evidence haven’t ruled out, and each update has a closed form. The prediction carries the support through the dynamics and adds the declared unmodeled motion at a fixed tolerance; the update is the least-volume member of Schweppe’s family enclosing the intersection of the support with the measurement’s cut, among the members that widen no direction by more than that tolerance whatever the datum; an empty intersection, decided exactly by one scalar, is answered by widening through the dynamics; and the committed state is the center of the support. While the declarations hold, the support contains every state they and the data admit, the truth included, so an empty intersection proves that at least one declaration is false. The emptiness test is exact. The update leaves the least log-volume of any admissible member. The center is the minimax estimate over the support in every norm, and it minimizes the possibilistic upper expected loss, a Choquet integral, for every symmetric convex loss, as Sherman’s theorem makes the Kalman mean optimal for every such loss under a Gaussian. Each prediction and update is TEAG’s operation on the impossibility field with the exact field replaced by a quadratic minorant, and, while no epoch needs widening, the EAF’s state is sufficient: it reproduces exactly the weighted batch conjunction of all declarations and data with the weights it chose, which encloses the exact conjunction, as the Kalman filter’s state reproduces batch least squares. The center update has exactly the Kalman form, with the measurement covariance replaced by a scale the geometry chooses, and the posterior shape is the Kalman covariance scaled by a factor that depends on the datum: the support contracts by what each measurement left admissible, where the Kalman covariance contracts by the same amount whatever is measured. Applied to orbit determination by linearizing about the center, with every constant taken from the general ESPF, the EAF holds the truth inside its support at every epoch of a nominal low Earth orbit arc, of stressed arcs with an undisclosed station bias of 2 to 20 m and an unmodeled maneuver, and of a real geostationary arc, with a narrower support and a smaller final error than the general filter’s on the low Earth orbit arcs. Run with one configuration across station biases from 2 to 50 m, five burns, twenty noise records and offset starts, it loses the truth only while its gate learns a bias of fifty resolutions and in the passes right after a burn on two runs, where the general ESPF’s tested commitment holds it; a linearization indicator computed from the support marks where the linear form fails.