Does iteration repeat the prediction step?

In the usual iterated measurement update, the predicted prior is held fixed while the measurement linearization is refined.

Is the method exact for nonlinear systems?

No. It remains a local approximation and generally does not recover the full nonlinear posterior.

Are iterated and invariant EKFs identical?

No. Their shared abbreviation can be confusing; inspect the full method name and equations.

AI Glossary / Probabilistic machine learning / Advanced

Iterated extended Kalman filter

Definition

The iterated extended Kalman filter refines an extended Kalman filter's measurement update by repeatedly linearizing the observation model around an improved state estimate. The iterations reuse the same measurement and predicted prior to solve a local nonlinear estimation problem more accurately.

What it means

A standard extended Kalman filter, or EKF, approximates a nonlinear measurement function by a straight-line or linear-map approximation at the predicted state. If the updated estimate moves far from that point, the original approximation may no longer describe the measurement relationship well.

The iterated version returns to the nonlinear function, builds a new local approximation and revises the estimate again. This inner loop occurs within a single measurement update. It is different from receiving new measurements at successive times.

The abbreviation IEKF is also used for the invariant extended Kalman filter. That is a different idea based on system structure and symmetry. An implementation or paper should spell out which meaning it uses; methods can combine iteration with invariant formulations.

How it works

For additive Gaussian measurement noise and a Gaussian predicted prior, the update can be expressed as a weighted least-squares problem. One term penalizes departure from the predicted state; another penalizes disagreement with the observed measurement.

The iterated filter applies successive local linearizations, closely related to Gauss - Newton optimization. Each iteration uses the original predicted prior and a correction accounting for the changed linearization point.

The loop ends when a stopping criterion is met or an iteration limit is reached. A covariance approximation is then formed consistently with the chosen final linearization. The procedure remains approximate: convergence to a useful local solution depends on the model, starting point and numerical method. More iterations do not guarantee a globally correct estimate.

Three practical examples

A tracking system refines an update when a sensor reports a nonlinear function of position, such as distance to a reference point.
A robotics estimator iterates a measurement update involving geometric residuals, using appropriate state representations for rotations.
An engineering team compares ordinary and iterated EKF updates on simulated sensor data with known hidden states.

Think of it this way

Think of using a short tangent segment to navigate a curved road. After moving, redraw the tangent near your new location before deciding the next correction.

The analogy concerns improving a local approximation. The algorithm is not collecting a fresh road survey at every inner step: it is reconsidering the same observation with a better local model. Recounting that observation as independent evidence would be a separate mistake.

Worked example

Refining one nonlinear observation

Use an original scalar example with predicted mean m = 1, predicted variance P = 1, observation y = 4 and measurement function h(x) = x squared. Let the measurement-noise variance R = 1.

Ignoring a constant factor, the update objective is (x − 1) squared + (4 − x squared) squared. Start at x0 = 1. The derivative H0 = 2, so K0 = 2/(4 + 1) = 0.4. The first EKF-style update gives x1 = 1 + 0.4(4 − 1) = 2.2.

Now linearize at 2.2. H1 = 4.4 and K1 = 4.4/(19.36 + 1). The corrected iterated update is 1 + K1 times [4 − 4.84 + 4.4(2.2 − 1)], approximately 1.9595.

The objective falls from 9 at the initial point to 2.1456 after the first update and approximately 0.9464 after the second. The original prior mean 1 and variance 1 are retained in both calculations. These two steps illustrate local improvement; they do not establish global convergence or provide an exact posterior distribution.

Illustrative case study

Fictional case: examining a nonlinear tracking update

A fictional lab tracks a moving cart using a sensor whose output depends nonlinearly on position. Some predicted positions are far enough from the measurement-consistent region that one linearization produces large corrections.

The lab compares a standard EKF with an iterated update on controlled recordings. It logs residuals, update sizes, iteration counts and errors against independent reference measurements. It also tests difficult initial positions rather than showing only successful runs.

The lab treats failed convergence and overconfident uncertainty estimates as findings to investigate. It does not assume that a lower local least-squares objective guarantees accurate tracking under every model mismatch.

Common misconceptions and mistakes

The same observation must not be assimilated repeatedly as if each iteration supplied independent new evidence. Doing so can shrink uncertainty for the wrong reason.

Iteration does not turn the nonlinear model into a globally linear one. Multiple plausible states can remain, and a single Gaussian approximation may poorly represent them. A measurement such as x squared can be compatible with both positive and negative states before prior information is considered.

A small change between iterations is a numerical stopping signal, not proof that the physical state is correct. Incorrect noise assumptions, Jacobians or sensor models can yield stable but misleading answers.

Business use

Consider this method when measurement nonlinearity causes meaningful errors in a standard EKF and additional update computation is affordable. Establish a baseline before adding iterations.

Specify the maximum iteration count, stopping tolerance, any damping strategy and what happens when the update fails. Validate derivatives and covariance calculations on small systems. Compare state errors and uncertainty coverage, not only residual reduction. For strongly non-Gaussian or multimodal problems, investigate whether a richer representation is necessary rather than expecting repeated local linearization to resolve every limitation.

Check your understanding

Should the second iteration in the worked example replace the predicted variance 1 with the first update's smaller posterior variance as if a new measurement arrived?

Answer: No. The inner iterations reuse the original predicted prior. Treating the same observation as new independent evidence would count it more than once.

Related terms

Extended Kalman filter
Jacobian
Gauss–Newton method
nonlinear least squares
state estimation
measurement update
invariant extended Kalman filter

Learn next

Derive the scalar iterated update from a linearized least-squares objective. Then compare its local covariance approximation with a numerically evaluated one-dimensional posterior that has more than one mode.

Least squares

Sources

Primary technical guide: Huai and Gao, A Quick Guide for the Iterated Extended Kalman Filter on Manifolds: https://arxiv.org/pdf/2307.09237

Primary textbook discussion: Särkkä and Svensson, Bayesian Filtering and Smoothing, section 7.4: https://users.aalto.fi/~ssarkka/pub/bfs_book_2023_online.pdf

The guide's update derivation and returned textbook discussion were reviewed; the textbook PDF did not fully open. The scalar calculation and cart case are original.

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