Just out of curiosity.
The two things sounds a little bit similar – 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer they have completely different setup.
Question May be nevertheless there is some relation ?
I googled a little, there are many papers with these two words inside,
but may be too many …
If someone can give self-contained comment it would be interesting, (well, any comment is welcome).
Let me comment on "uncertainty principle" and "Cramer-Rao bound".
Uncertainty principle in "quantum mechanics" – actually is a fact about matrices.
We know that if two matrices commute, then they have joint eigenbasis (assuming smth).
So we might ask if they do not commute – what happens ?
How to measure the deviation from the fact that they do not commute ?
The basic example is A, B: [A,B]=1. e.g. A = x, B = d/dx.
The answer goes as follows: take any vector v and decompose it in eigenbasis for A,
the "uncertainty principle" says that if vector is "localized" in eigenbasis for A,
then it should be unlocalized in eigenbasis for B.
In the basic example A= x, B = d/dx, the two eigenbasis related by the Fourier Transform
and so we get that if function is localized in "delta-function" basis – e.g. it is just localized in common sense, then its FT will be delocalized.
Cramer-Rao bound. (Statistics)
It basically says the following – one have a sample of some random variable
and we want estimate some parameter – then we cannot kill randomness completely,
e.g. there will always be certain inaccuracy in our estimation of parameter.
The Cramer-Rao inequality provides quantitative bound.
So you see setups are completely different – one is about matrices,
another is in probability theory.
Best Answer
More of a lurking variable than a connection, but:
Both Cramér-Rao and the Uncertainty Principle follow from the Cauchy-Schwarz inequality.