Let’s say I have a object where I’m uncertain about it’s temperature, but I can model it as a bell curve where the mean is 50C with a std dev of 10C. OK, fine, that’s simple enough, I can visualize that as like, a probability cloud.
But what if I’m uncertain about the uncertainty? Like, I could have a bell curve where the mean of the bell curve is a bell curve and the std dev is another bell curve. So like, now, what I have is a probability cloud of different possible bell curves that each also represent a probability cloud.
But obviously, this process can be applied recursively and by the 3rd level, I’ve already lost all of my mental tools. And like, what happens if the mean of a bell curve is a power law distribution?
Is this a defined field of mathematics? What keywords should I be searching for to understand how to think about problems like this?
Yes you can do this. It’s called a subordinated (normal) distribution. However, the std dev or variance could not have a normal distribution as that includes negative values. Two standard subordinated normals are a normal in which the mean is also normal and a normal in which the variance has a gamma distribution.
Note that the first is just a more spread out normal.
OldGuy’s subordinated distribution is a little over my head, but isn’t uncertainty the reason that we use confidence intervals? Rather than diving into an abyss of an uncertainty loop?
I believe that is exactly what is happening here: you make some measurements and then you can construct confidence intervals using Gosset’s t-distribution.
An object has only one temperature (unless we’re going to say that it’s not at equilibrium, which introduces a slew of additional issues to this problem), so to get your bell curve “model” you need to take a large number of measurements and see what the mean and standard deviation of those measurements are. That’s your sample mean and sample sd. They are properties of your measurements, not your object, and it’s just known that neither is a certain value.
If you increase the number of measurements your sample mean will get closer to the real (unknown) mean and the sd will shrink.
It doesn’t really make sense to say that the “mean” has a bell curve and sd has a bell curve in this situation, but let me, briefly, explain a little more about the t-distribution (Gosset’s or Student’s).
Imagine you are not measuring the temperature of a thing, but you are sampling a population. Say the height of 25-year olds in Japan. If you have a few hundred samples your sample mean and sample standard deviation will tell you a lot about the actual population mean and population sd. But if you only have a small sample both your sample mean and your sd will be quite uncertain. That means you can’t just normalize them, look at the bell curve and say “the real mean is likely to be within these bounds”.
If you repeat the small sample experiment over and over again you will actually get a bell curve, but there is no way to know, based on your sample, what the standard deviation and mean of that bell curve will be.
But if you imagine taking every such sample and normalizing it using the population mean, the sample size and the sample sd, you get the t-distribution. And what this means is that if you do only one experiment, you can use this same normalization to get a value that you can compare to the standard t-distribution and make statements about how probable it is to get this result given a hypothesized mean.
That was probably not very clear, so
tl;dr Read up on the t-distribution and its uses.
This is a principle in the frequentist paradigm, but in the Bayesian paradigm one would speak of the probability distribution for the temperature. Which probability interpretation you use depends on the question(s) you want to answer, and to some degree on the applicability of the methodology.
To the title topic, some points:
Errors on errors is a thing.
Whether errors on errors matter is a quantitative question. In practice, they often don’t, but sometimes they do. It depends on the questions you are trying to answer and on the influence the errors may have on that.
It takes very little complexity before a problem is best treated by simulation anyway, in which case it’s more straightforward to deal with errors on errors.
Properly assessing errors is typically the hardest part of any non-trivial measurement problem, and other approximations may be present that dominate over “errors on errors”. So the latter are often not practically relevant.
I am not aware of any dedicated term for “recursive uncertainty”. It’s all just uncertainty, even if some uncertainty is on uncertainty. At the end of the day it’s all getting rolled up.
In economics, the concept of ‘Knightian uncertainty’ seems relevant—it’s uncertainty that can’t be effectively quantified, so you’re in that sense uncertain about precisely how uncertain you are.