[Deutscher]

Math newbie here.

Does 'filter' (as it is used here) have a precise mathematical definition? The only other place where I've seen this word used is the 'bloom filter', which according to Wikipedia is not an algorithm like the Kalman filter but a data structure.

[mbrudd]

In general, filters do what their names suggest -- they filter out noise or unwanted components of signals. The simple Kalman filter filters out white noise to estimate parameters that describe a system; other filters use things like the Fourier or wavelet transform to, say, select certain frequencies and remove others.

[alanbernstein]

IIRC, The kalman filter has a state-space formulation, in which it is clear that it is equivalent to an LTI filter (https://en.m.wikipedia.org/wiki/Linear_filter). However, I believe updating the statistics of the system in response to updated measurements breaks the "time invariant" part.

This is precisely the classic concept of an audio filter, which is, I believe, the inspiration for the term "filter" in other contexts. Bloom filters and selfie filters don't have much in common in a technical sense, but if you squint you can see the connection.

[billfruit]

Filter has very established definition in Signal Processing, and covers things which can do a lot of things, essentially something that can manipulate how energy or power is distributed over a set of frequencies.

A common analogy is yo imagine it as an entity that can turn a block of stone to a sculpture, by removing material.

[SimplyUnknown]

It is a filter in the sense that it can track the system state from one (or more) noisy measurements. The filter part thus reflects the input denoising.

[mpmpmpmp]

The wikipedia page for Kalman filter says it's also known as "linear quadratic estimation" so that might be a good place to start.

[sceadu]

In signals it's a term used to mean a processing step that removes unwanted parts of a signal, e.g. measurement noise