# Putting upper bounds from a null result

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• kelly0303
kelly0303
Hello! If I have a measurement of a (dimensionless) parameter, ##a##, say 998 +/- 3 and the theoretical prediction of that parameter (assuming known physics) is 1000 +/- 4. Let's say that I want to set limits on a new parameter (not included in the theoretical calculations), call it ##\alpha##, that would contribute additively to ##a##. By comparing the experiment with the theory, I get a = 2 +/- 5 (the errors are all gaussian and I added them in quadrature). I want to set, let's say, a 95% upper bound on the value of a. In general, I could just say a < 12. However, in my case I know that ##a## has to be positive. How would I define the upper bound in this case, knowing that part of the range allowed by a = 2 +/- 5 is in practice not allowed by the physical theory? Also, I am not totally sure if my range would be (1000-998) +/- 5 = 2 +/- 5 or (998-1000) +/- 5 = -2 +/- 5. How do I decide which one to use. The first one would be the conservative choice, but I am not sure if it is not too conservative). Thank you!

kelly0303 said:
the errors are all gaussian … I know that ##a## has to be positive.
I don’t see how this is possible.

Dale said:
I don’t see how this is possible.
I am not sure I understand

kelly0303 said:
I am not sure I understand
How can it be true both that the errors are Gaussian and also that they must be positive? The Gaussian distribution is not strictly positive.

Dale said:
How can it be true both that the errors are Gaussian and also that they must be positive? The Gaussian distribution is not strictly positive.
The errors on the theoretical calculations and on the measurement are Gaussian. The value of the new physics parameter must be positive. For example, the experimental value can be from a simple counting experiment. If the measured value is, say, 10000 and (I ignore the systematic uncertainties) the statistical error (assuming a Poisson process) is 100 (and it is very close to a Gaussian distribution) i.e. the measured value is 10000 +/- 100. The theoretical value can be, for example also 10000 +/- 100. Now I want to use this measurement to set upper bounds on a possible (positive value) new physics effect, given that its effect was not big enough to make the experiment deviate from the theoretical predictions (which don't include the new physics effect). I can share several papers where this is applied to some measurements if my explanation is not clear. Thank you!

Theoretically, the errors on the measurement of a positive parameter cannot be gaussian, because gaussians can be unboundedly negative. That said it's common to just ignore this and use a bunch of math that assumes gaussians are errors, and somehow by magic everything is basically correct anyway in a lot of examples.

Here you know it's not magically correct, which means if you want to say something mathematical, the details of exactly what you did are going to be important.

Office_Shredder said:
Theoretically, the errors on the measurement of a positive parameter cannot be gaussian, because gaussians can be unboundedly negative. That said it's common to just ignore this and use a bunch of math that assumes gaussians are errors, and somehow by magic everything is basically correct anyway in a lot of examples.

Here you know it's not magically correct, which means if you want to say something mathematical, the details of exactly what you did are going to be important.
But how do I do this in practice? Basically in my case I have a measurement and a theoretical calculation with gaussian errors. These values are from literature, so there is not much I can do about it. I want to test a new physics scenario which changes the expected results in a given way that depends on a parameter that needs to be positive. So the effect of this new physics can be written as ##k\alpha##, where ##k## can be calculated (we can ignore the uncertainty on k and assume it's just a constant) and ##\alpha > 0##. How do I use these (experimental and theoretical values and uncertainties, as well as k) to set an upper limit on ##\alpha## (assuming that the theory and experiment are consistent within uncertainties). Thank you!

It would help if you had more specifics about what you're doing. There also might be field standards techniques that you should be using.

Here's an example of a thing I might personally do. Let's suppose I have a 100 meter track and I use a radar to measure the speed of something traversing the track at constant speed. My radar gun says that it is traveling at ##20 \pm 3## meters per second. I want to think about this in terms of time traveled. Obviously if it's 20 on the nose it took 5 seconds to travel the whole length.

The stdev of 3 means there's a 95 percent chance that we're within 2 stdevs and hence the speed is in ##[14,26]##. This means the time it takes, with 95 percent probability, is in the interval ##[3.84,7.14]## seconds (100/26 and 100/14). Note this is not symmetric around 5. You might be able to do something similar where you construct a confidence interval for what the final parameter is

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