# Selection-Induced Contraction of Innovation Statistics in Gated Kalman Filters

arXiv: https://arxiv.org/abs/2512.18508 · HTML: https://mlsanity.com/paper/2512.18508/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 20 Dec 2025.

## Builds on (most cited first, 5 of 5)

- [Adaptive Adjustment of Noise Covariance in Kalman Filter for Dynamic State Estimation](https://mlsanity.com/paper/1702.00884/) (2017)
- [A Hybrid Model and Learning-Based Adaptive Navigation Filter](https://mlsanity.com/paper/2207.12082/) (2022)
- [Learning Vehicle Trajectory Uncertainty](https://mlsanity.com/paper/2206.04409/) (2022)
- [Kalman Filter Auto-tuning through Enforcing Chi-Squared Normalized Error Distributions with Bayesian Optimization](https://mlsanity.com/paper/2306.07225/) (2023)
- [Matrix-Valued Measures and Wishart Statistics for Target Tracking Applications](https://mlsanity.com/paper/2406.00861/) (2024)

## Abstract (arXiv)

Validation gating is a fundamental component of classical Kalman-based tracking systems. Only measurements whose normalized innovation squared (NIS) falls below a prescribed threshold are considered for state update. While this procedure is statistically motivated by the chi-square distribution, it implicitly replaces the unconditional innovation process with a conditionally observed one, restricted to the validation event. This paper shows that innovation statistics computed after gating converge to gate-conditioned rather than nominal quantities. Under classical linear--Gaussian assumptions, we derive exact expressions for the first- and second-order moments of the innovation conditioned on ellipsoidal gating, and show that gating induces a deterministic, dimension-dependent contraction of the innovation covariance. The analysis is extended to NN association, which is shown to act as an additional statistical selection operator. We prove that selecting the minimum-norm innovation among multiple in-gate measurements introduces an unavoidable energy contraction, implying that nominal innovation statistics cannot be preserved under nontrivial gating and association. Closed-form results in the two-dimensional case quantify the combined effects and illustrate their practical significance.