Noise Bottleneck of Nassim Taleb In several of Nassim Taleb's books he mentions a phenomenon referred to as a noise bottleneck where more sampling of something actually decreases your signal to noise relationship:
"Assume further that for what you are observing, at a yearly frequency, the ratio of signal to noise is about one to one (half noise, half signal)—this means that about half the changes are real improvements or degradations, the other half come from randomness. This ratio is what you get from yearly observations. But if you look at the very same data on a daily basis, the composition would change to 95 percent noise, 5 percent signal. And if you observe data on an hourly basis, as people immersed in the news and market price variations do, the split becomes 99.5 percent noise to 0.5 percent signal." - Antifragile pg. 126
All of the situations I can think of have the opposite effect, where longer sampling times (more data) average out the noise and improve your signal to noise ratio.  Can anyone give a mathematical example of this noise bottleneck?
 A: Your comment "longer sampling times (more data) average out the noise and improve your signal to noise ratio" in the third paragraph is consistent with Taleb's statement.  Suppose he had said something like:

Assume that for what you are observing, if you observe data on an hourly basis, as people immersed in the news and market price variations do, the split is 99.5 percent noise to 0.5 percent signal. But if you average the very same data on a daily basis, the composition would change to 95 percent noise, 5 percent signal. Now average it over a year and the ratio of signal to noise is about one to one (half noise, half signal)—this means that about half the changes are real improvements or degradations, the other half come from randomness.

I suspect you would find this broadly similar to your thought of longer sampling periods improving your signal to noise ratio, though perhaps not necessarily with precisely the numbers stated.  All I really did was reverse the order of Taleb's sentences without changing the meaning.
A: I assume that you have heard of the Cauchy (or Lorentz) distribution.
Despite its symmetry about a vertical axis, this is not a usual "well-behaved" distribution because it does not:

*

*Follow the law of large numbers, i.e. converge on a mean as the sample size increases


*Conform to the central limit theorem
So maybe this may have some relevance to the situations alluded to by Taleb, I don't know.
A: In reliability theory, what Taleb is talking about is called transient error. Transient error is systematic measurement error that is irrelevant to the signal, but only "cancels out" over longer periods of time. In one sense it is noise from the perspective of longer time periods, but from a short time perspective you could call it confounding.
For example, consider scoring someone on a measure of depression, but that person has jet lag. The jet lag is making that person irritable, increasing their depression scores.
Imagine that person is on the road to good mental health. You wont be able to pick up that signal until the effect of the jet lag wears off, maybe over a month later.
Scoring that person every hour will not capture the trend of improvement, but it may capture many irrelevant situational effects (transient error): they missed the bus :-(, they had a good date :-), they got a compliment :-), they got into an argument :-( etc.
