Why is Information Filter called Information Filter?

Question

We all know Information Filter is a dual representation of Kalman Filter. The main difference between Information Filter and Kalman Filter is the way the Gaussian belief is represented. In Kalman Filter, the Gaussian belief is represented by their moments, whereas in Information Filter, the Gaussian belief is represented by the canonical form. The canonical form is comprised of an information matrix ($\Omega$) and an information vector ($\xi$).

As we all know, the name Kalman Filter is after Rudolf E. Kálmán, one of the primary developers of this theory.

Now I want to know where the name information filter came from? Why it is called information filter? What type of information is attached to this filter? Is there any significance behind the nomenclature?

I have the same question about the information matrix and information vector. Is there any significance behind the nomenclature?

I already read Probabilistic Robotics by Sebastian Thrun. Chapter 3 Gaussian filter, in subsection 3.4 The Information Filter. There are many equations and theories but that does not tell us about the nomenclature.

Answer 1

I am probably not qualified to answer your question, but I will attempt. At the very least I hope to start a discussion about the matter because I am unsure.

The (linear) Kalman filter uses the error Covariance as it's main contributor to the Kalman gain. We usually notice the Error covariance decreases over time which can be thought of as the decrease in uncertainty. In the information filter the inverse is true. Our "information matrix" increases with more measurements, because our information increases.

Additionally the measurement update of the information state vector y resembles the way we simply add information. Since

y(i+1) = y(i) + i(i+1)

here we are simply adding information.

This is how I reason and remember the Information filter. I'm not sure where the name came from.

Additionally: Arthur Mutambara mentions: "The information filter essentially tracks information about the states and not the states them selves". This may be more correct and concise.