Kalman Filter Noise: Gaussianity and Estimation Error

Summary

The document asks whether practitioners commonly test the state and measurement noise assumptions in a linear Kalman filter, including whether the noise should be Gaussian. Its answer distinguishes the filter’s estimation objective from the stronger distributional assumptions used for full probabilistic inference.

For white noise, the linear Kalman filter can still minimize mean squared estimation error when the noise is non-Gaussian. Gaussianity matters because, in that case, the estimated mean and covariance fully characterize the posterior distribution. The answer therefore says Gaussianity tests are not strictly required for the filter to remain useful. It does not identify specific diagnostic tests, discuss how to verify whiteness, or address model misspecification and nonstationarity. The takeaway is limited: non-Gaussian noise alone does not invalidate the estimator’s mean-squared-error role, but it can affect what can be inferred from its reported distribution.

Key ideas

• A linear Kalman filter is framed as minimizing mean squared estimation error under white-noise assumptions.
• Non-Gaussian white noise does not by itself prevent the filter from producing useful estimates.
• With Gaussian noise, the estimated mean and covariance characterize the posterior distribution.
• The answer does not recommend tests for whiteness, Gaussianity, or other model assumptions.

Tags

# Kalman filter state/measurement noise


# Kalman filter state/measurement noise












In a linear Kalman filter, we assume that the state and measurement noise are white noise N(0,Q) and N(0,R) respectively. Is it common practice to test these hypothesis? And what are the most common statistical tests used in this case?

## Answer by rzch (score 1)

https://quant.stackexchange.com/a/42813

The linear Kalman filter still minimises the mean squared error of the estimate even if noise are non-Gaussian white, it's just that in the Gaussian case you will have the full posterior distribution from the mean and covariance estimates alone.

So in practice it wouldn't be absolutely necessary to test for Gaussian noise, and have the Kalman filter still 'work'.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.