Gaussian Rejection Filtering

• Gaussian Rejection Filtering is a robust method for nonlinear state-space models that down-weights or ignores measurements inconsistent with predictions.
• It employs a selective per-sensor rejection model with latent binary indicators to efficiently isolate and suppress outlier effects on the state estimate.
• A pseudo-measurement strategy transforms raw sensor data, allowing standard Gaussian filters to effectively track inliers while mitigating fat-tailed noise impacts.

Gaussian rejection filtering can be understood as a class of robust Gaussian-filtering procedures for nonlinear state-space models in which measurements that are inconsistent with the predicted state are down-weighted or selectively ignored during inference. In the literature considered here, two closely related constructions exemplify this idea. One is a pseudo-measurement approach that makes any Gaussian filter compatible with fat-tailed sensor models by replacing the physical measurement with a feature-transformed observation (Wüthrich et al., 2015). The other is a selective-outlier-rejecting formulation for independent sensors in which each measurement dimension receives its own latent outlier indicator and an effective measurement covariance is inferred by Variational Bayes within a general Gaussian-filter update (Chughtai et al., 2021).

1. Problem setting and motivation

The underlying estimation problem is the nonlinear state-space model

$$x_k = f(x_{k-1}) + q_{k-1}, \qquad q_{k-1} \sim \mathcal{N}(0,Q_{k-1}),$$

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$

with hidden state $x_k \in \mathbb{R}^n$, measurement $y_k \in \mathbb{R}^m$, and known covariances $Q_{k-1},R_k$ (Chughtai et al., 2021). The difficulty addressed by Gaussian rejection filtering arises when the nominal measurement model is violated by gross corruption, missing data, or other outliers.

The pseudo-measurement literature states that many sensors, such as range, sonar, radar, GPS and visual devices, produce measurements which are contaminated by outliers. It further observes that common fat-tailed models, including Student-$t$, Cauchy, and Gaussian mixtures with small outlier weight, better represent such measurements than a single Gaussian body (Wüthrich et al., 2015). At the same time, standard Gaussian filters are described as inherently incompatible with such fat-tailed sensor models: if the filter adopts a very broad effective covariance it can ignore all measurements, while if it forces a finite Gaussian approximation the posterior mean remains linear in the raw observation and large outliers can drive the estimate arbitrarily far (Wüthrich et al., 2015).

This motivates a rejection mechanism with two simultaneous properties. Inlier measurements should pull the estimate in the usual way, while measurements that look like outliers should be down-weighted or completely ignored. A common misconception is that merely substituting a fat-tailed likelihood into an EKF or UKF resolves the issue. The cited work argues that this is not generally the case within the Gaussian-filtering paradigm, and that an explicit modification of the update is required (Wüthrich et al., 2015).

2. Selective per-sensor rejection model

For measurements obtained from $m$ independent sensors or measurement dimensions, the selective-outlier-rejecting construction introduces a vector of binary indicators

$$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$

where $\mathcal{I}_k^i=1$ denotes no outlier in dimension $i$ and $$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$0 denotes an outlier (Chughtai et al., 2021). The prior factorizes across dimensions,

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$1

with $$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$2 the prior probability of no outlier at dimension $$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$3.

Conditioned on $$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$4, the measurement-noise precision becomes diagonal:

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$5

and the likelihood factorizes as

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$6

Equivalently, the overall measurement model is a mixture over all indicator configurations:

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$7

The significance of this construction is that outlier handling occurs at the level of individual dimensions rather than whole measurement vectors. This makes selective rejection possible: one corrupted sensor reading can be suppressed without discarding the remaining dimensions. A plausible implication is that this structure is especially suitable when many measurements are independent and only a subset is corrupted at a given time.

3. Variational-Bayes rejection update

The joint posterior

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$8

is intractable, so the approximation

$$y_k = h(x_k) + r_k, \qquad r_k \sim \mathcal{N}(0,R_k),$$9

is adopted and optimized by coordinate-ascent updates (Chughtai et al., 2021). For the state term,

$x_k \in \mathbb{R}^n$0

and, after inserting the Gaussian prior $x_k \in \mathbb{R}^n$1, the effective inverse measurement covariance becomes

$x_k \in \mathbb{R}^n$2

The update for $x_k \in \mathbb{R}^n$3 is then approximated by a Gaussian,

$x_k \in \mathbb{R}^n$4

using general Gaussian filtering.

For the indicator term, each $x_k \in \mathbb{R}^n$5 has two atoms and the posterior factorizes Bernoulli:

$x_k \in \mathbb{R}^n$6

with

$x_k \in \mathbb{R}^n$7

and closed-form update

$x_k \in \mathbb{R}^n$8

The rejection mechanism follows directly from this form. If the measurement residual variance $x_k \in \mathbb{R}^n$9 is large, then $y_k \in \mathbb{R}^m$0, $y_k \in \mathbb{R}^m$1, and that dimension is effectively turned off. If $y_k \in \mathbb{R}^m$2 is small, then $y_k \in \mathbb{R}^m$3 and the measurement is used normally. The method can use either a hard decision $y_k \in \mathbb{R}^m$4 to declare an outlier or the continuous weight $y_k \in \mathbb{R}^m$5 directly in $y_k \in \mathbb{R}^m$6 (Chughtai et al., 2021). This directly addresses another common misconception: selective rejection does not require a fixed external gate, because the posterior weight itself can serve as a soft rejection variable.

4. Embedding in general Gaussian filters

The selective-outlier-rejecting update is designed to be embedded in any Gaussian-filter framework, including EKF, UKF, and CKF, by replacing the nominal measurement noise $y_k \in \mathbb{R}^m$7 with the effective covariance $y_k \in \mathbb{R}^m$8 (Chughtai et al., 2021). The prediction step is

$y_k \in \mathbb{R}^m$9

$Q_{k-1},R_k$0

The measurement update is

$Q_{k-1},R_k$1

$Q_{k-1},R_k$2

$Q_{k-1},R_k$3

$Q_{k-1},R_k$4

$Q_{k-1},R_k$5

In practice, these integrals are evaluated with sigma-point methods or truncated Taylor expansions.

The operational algorithm initializes $Q_{k-1},R_k$6, chooses $Q_{k-1},R_k$7 and $Q_{k-1},R_k$8, performs a prediction step, initializes $Q_{k-1},R_k$9, and then alternates between computing $t$0, updating $t$1, recomputing $t$2, and rerunning a Gaussian-filter measurement update until the relative change in the posterior mean falls below a threshold (Chughtai et al., 2021).

The paper also gives an explicit complexity comparison:

Method	Complexity	KF analogue
SOR-UKF	$t$3	P-SPKF
mOD/mSEM/mROR-UKF	$t$4	SRD-SPKF
mSOR-UKF	$t$5	S-SPKF

The modified serial version, mSOR-UKF, exploits diagonal $t$6 and uses a serial SPKF with zero-mean scalar updates. This yields the stated $t$7 complexity and provides a direct complexity-versus-implementation trade-off within the same selective-rejection principle.

5. Pseudo-measurement robust Gaussian filtering

A second route to Gaussian rejection filtering uses a pseudo measurement rather than latent per-dimension indicators. The sensor model is written as

$t$8

where $t$9 is a Gaussian body and $m$0 is a fat-tailed outlier component (Wüthrich et al., 2015). The raw measurement is replaced by a feature-transformed pseudo measurement

$m$1

and a standard Gaussian filter is then applied to $m$2 instead of $m$3.

Under a linear Gaussian body, a Gaussian prior, and the approximation that the tail depends only weakly on $m$4 over the support of the prior, the posterior mean can be approximated by a ratio in which the inlier term and the tail term are explicitly separated. From this, a convenient three-dimensional feature is extracted:

$m$5

For $m$6, this reduces to $m$7 and recovers the standard Gaussian filter.

The update remains formally identical to an ordinary Gaussian-filter update. One computes the predicted moments, the body predictive statistics, the transformed covariances $m$8 and $m$9, and then uses

$$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$0

$$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$1

Only the measurement is replaced by $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$2 and the measurement model by $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$3 (Wüthrich et al., 2015).

This framework directly addresses the incompatibility between Gaussian filters and fat-tailed sensor models. The key claim in the paper is that $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$4 saturates for large $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$5, so outliers do not drive the estimate, yet preserves a linear response for inliers. The same source notes practical guidelines: $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$6 should be chosen near the expected outlier rate, the precise tail shape is not critical, and if outliers correlate strongly with $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$7, a richer feature or tail model may be required (Wüthrich et al., 2015).

6. Empirical behavior, applications, and significance

The selective-outlier-rejecting paper reports simulations on maneuvering-target 2D tracking with bearing/range sensors and additive mixture-Gaussian outliers or missing data. In those experiments, all selective filters—SOR-UKF, mSOR, mOD, mSEM, and mROR—achieve comparable RMSE trajectories, whereas non-selective baselines degrade rapidly as the outlier rate $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$8 increases (Chughtai et al., 2021). The same study states that mSOR-UKF shows the lowest empirical CPU time versus the number of measurements $$\mathcal{I}_k = [\mathcal{I}_k^1,\dots,\mathcal{I}_k^m]^\top, \qquad \mathcal{I}_k^i \in \{\epsilon,1\}, \qquad \epsilon \ll 1,$$9, in line with its $\mathcal{I}_k^i=1$0 complexity. For $\mathcal{I}_k^i=1$1 up to $\mathcal{I}_k^i=1$2 and $\mathcal{I}_k^i=1$3, mSOR’s RMSE remains approximately $\mathcal{I}_k^i=1$4–$\mathcal{I}_k^i=1$5 while CPU time grows linearly in $\mathcal{I}_k^i=1$6, whereas mOD, mSEM, and mROR grow superlinearly.

The same paper evaluates real-time indoor localization with Ultra-wide Band sensors. The experiment uses 11 anchors, with only 4 measurements available at once, and includes many missing-data outliers together with occasional NLoS biases. All methods have $\mathcal{I}_k^i=1$7 RMSE under $\mathcal{I}_k^i=1$8; mSOR-UKF achieves the best overall RMSE, approximately $\mathcal{I}_k^i=1$9–$i$0, and the lowest mean run time per step, approximately $i$1–$i$2 (Chughtai et al., 2021).

Method	Scenario 1/2/3 RMSE (m)	Scenario 1/2/3 RT (s)
mROR-UKF	0.16 / 0.16 / 9.26	0.24 / 0.21 / 0.25
mOD-UKF	0.18 / 0.21 / 0.39	0.18 / 0.17 / 0.17
mSEM-UKF	0.17 / 0.11 / 0.40	0.28 / 0.24 / 0.23
mSOR-UKF	0.15 / 0.10 / 0.36	0.08 / 0.07 / 0.06

The pseudo-measurement work reports a linear one-dimensional example with sensor model

$i$3

and a nonlinear five-dimensional radar-tracking example with range and bearing measurements contaminated by $i$4 high-variance Gaussian glint. In the first case, a thin-tailed Gaussian filter produces large estimation errors and a fat-tailed Gaussian filter ignores all measurements, whereas the robust Gaussian filter tracks correctly for inliers and rejects outliers. In the radar case, the thin Gaussian filter exhibits large spikes at outliers, the fat Gaussian filter is over-smooth and slow to respond, and the robust Gaussian filter is robust to glint outliers while accurately tracking maneuvers (Wüthrich et al., 2015).

Taken together, these results indicate two distinct but compatible interpretations of Gaussian rejection filtering. One interpretation performs explicit selective rejection through per-sensor latent variables and an effective covariance. The other reshapes the observation through a pseudo measurement so that the Gaussian filter itself becomes robust to tail events. A plausible implication is that the choice between them depends on the structure of the sensing problem: independent measurement dimensions favor selective rejection, while a known fat-tailed measurement model favors pseudo-measurement design.