Is this the extended Kalman filter?

No. The construction uses an invertible observation transformation rather than relying on local linearization of a general nonlinear model.

Does exact filtering imply exact parameter learning?

No. Learning model parameters can still involve numerical optimization and statistical uncertainty.

Why keep the Jacobian if it cancels in filtering?

It remains necessary for the observation density and likelihood.

AI Glossary / Probabilistic machine learning / Advanced

Normalizing-flow Kalman filter

Definition

A normalizing-flow Kalman filter combines a linear Gaussian state-space model with an invertible transformation of its noisy observations. It can represent nonlinear, non-Gaussian observations while retaining exact Gaussian filtering for the underlying state within this structured model. The method is also called a Normalizing Kalman Filter, or NKF.

What it means

An ordinary Kalman filter works with a linear relationship between the state and Gaussian observations. Here, that relationship holds in a transformed observation space. A normalizing flow maps those simpler observations into the more complex values that are actually recorded.
The transformation is part of the probabilistic model, not merely an arbitrary preprocessing step. Its inverse makes observations usable by the Gaussian filter, and its Jacobian supplies the density correction. Exact inference refers to the specified model with given parameters; it does not mean the model describes every nonlinear process exactly.

How it works

The latent state evolves through a linear transition plus Gaussian process noise. A linear emission of that state receives Gaussian observation noise, producing a pseudo-observation. An invertible nonlinear function then transforms the pseudo-observation into the observed value. Placing the observation noise before this transformation is important to the tractable construction.
For a fully observed measurement vector, invert the flow to obtain the pseudo-observation and perform the corresponding Kalman update. The transformation's Jacobian depends on the observation and flow parameters rather than the unknown state, so it cancels when normalizing the state posterior. It must still be included when evaluating the observation likelihood.
Forecasting propagates the state distribution, generates pseudo-observations and transforms them forward. Summaries on the observation scale must account for the transformation; simply transforming a Gaussian mean generally does not produce the transformed distribution's mean.

Three practical examples

Forecasting research: a team models related continuous measurements using simple latent dynamics and a flexible observation transformation. It compares forecast distributions with simpler baselines on held-out time periods.
Likelihood verification: an analyst checks that an invertible transformation's density includes the correct Jacobian. The filtering update alone cannot reveal every likelihood implementation error.
Uncertainty communication: researchers report forecast intervals or sampled trajectories on the original measurement scale, rather than presenting transformed-space Gaussian error bars as though the transformation were linear.

Think of it this way

Imagine observing a moving object through a precisely characterized distorting lens. Undoing the lens reveals coordinates in which the movement and observation errors follow a simpler model. The lens can make the visible distribution complicated, but the calculation remains manageable because the distortion is invertible and its effect on probability density is known.

Worked example

Consider an original scalar illustration with fixed parameters and an explicitly chosen transformation. Let the current prior state s be normal with mean 0 and variance 1. Form z = s + noise, where independent observation noise has variance 1. Observe y = sinh(z), an invertible function from the real line to itself.
Suppose y = sinh(2), approximately 3.62686. Inverting the transformation gives z = 2. The Gaussian innovation variance is 1 + 1 = 2, so the Kalman gain is 1/2. The filtered state mean is 0 + (1/2) times 2 = 1, and its variance is (1 - 1/2) times 1 = 1/2.
For the observation likelihood, the prior predictive distribution of z is normal with mean 0 and variance 2. The inverse transformation derivative is 1/sqrt(1+y squared), equal to 1/cosh(2) at this observation. Thus the density of y there is exp(-1) divided by [sqrt(4 pi) times cosh(2)], approximately 0.027584. This is a density value, not the probability of one exact observation.
Now let the next state equal the current state plus independent process noise of variance 1/2. Its predictive mean and variance are both 1. Adding observation noise of variance 1 makes the next z normal with mean 1 and variance 2. The mean of sinh(z) is exp(2/2) times sinh(1), approximately 3.19453. Transforming only the mean instead gives sinh(1), approximately 1.17520. The difference demonstrates why uncertainty must pass through the nonlinear transformation. No learned flow or complete forecasting experiment is claimed in this stipulated example.

Illustrative case study

Fictional company Alder Sensors studies forecasts for a panel of continuous measurements. Its researchers compare a linear Gaussian model with a model that adds an invertible observation transformation. They preserve the time order of training and evaluation data and assess likelihood as well as forecast errors.
The team checks inverse consistency and the likelihood correction on a small example before training a larger model. It also compares simulated observation-scale trajectories with the patterns seen in held-out records. This fictional workflow illustrates validation choices; it does not claim that the more flexible model wins or that a service has been deployed.

Common misconceptions and mistakes

Adding a normalizing flow anywhere in a state-space model does not automatically preserve Kalman tractability. The placement of noise and transformation matters. A nonlinear transition or additional noise after the flow generally requires separate analysis.
A Gaussian latent-state posterior does not imply Gaussian observations. Conversely, an invertible transformation alone does not guarantee that a chosen latent model fits the data. Missing components of a multivariate observation also require care when the flow mixes components; one cannot generally invert a partly observed vector as though it were complete. Parameter uncertainty is distinct from state uncertainty conditional on fitted parameters.

Business use

This approach can support probabilistic forecasting when an interpretable latent structure and flexible observation distribution are both useful. Teams must weigh the cost of learning and evaluating the flow against measured forecast quality. Validation should include distributional behavior, not only point accuracy, especially when downstream decisions depend on intervals or tail probabilities.

Check your understanding

Why can the example update use z = 2?

Answer: The observed y uniquely determines z through the inverse transformation.

Where was observation noise added?

Answer: Before applying sinh.

Why is the next observation mean not sinh(1)?

Answer: A nonlinear transformation does not generally commute with averaging over uncertainty.

Related terms

Kalman filter
Probabilistic forecasting
Extended Kalman filter
Normalizing flow
State-space model
Change of variables
Jacobian determinant

Learn next

Verify a scalar transformed Gaussian density, then compare filtering in observation and pseudo-observation coordinates. Study how observation-scale predictive summaries differ from latent Gaussian summaries.

Sources

De Bezenac and colleagues, Normalizing Kalman Filters for Multivariate Time Series Analysis, 2020, Section 2: https://proceedings.neurips.cc/paper_files/paper/2020/file/1f47cef5e38c952f94c5d61726027439-Paper.pdf . The scalar calculation and fictional company case are original teaching illustrations. Section 2 and Proposition 1 independently consulted for transformed observations and exact filtering.

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