I How to implement proper error estimation using MC

[kelly0303]

Hello! I have a situation of the following form. I have a function $f = xy$. In my experiment $y$ is fixed, but it has some noise to it, such that at each measurement it is basically sampled from a Gaussian with the mean given by the fixed value and the standard deviation given by the known uncertainty on $y$, call it $dy$. At each instance of the measurement, $f$ is a number between 0 and 1, but when I actually record the measurement I get either 0 or 1, sampled from a binomial distribution with probability $f$ (it is a quantum projection measurement). What I need is, after a given number of measurements, to extract $x$ and I want to check how many measurements I need for a given uncertainty on $x$.

The way I am thinking of doing it is like this: Fix $x$ to a certain value (close to what I expect in practice). For each event, sample $y$ from its Gaussian distribution, calculate f, then get 0 or 1 based on Binomial sampling with probability f. Do this N times, which will give me $\sim fN$ non zero events. Now I do all these steps again a large number of times (e.g. 1000) and I get a Gaussian distribution over the values of f, with a mean and standard deviation. Now, in order to estimate $x$ I can divide the central value of f by the central value of y. But I am not sure how to estimate the uncertainty on x. Should I just divide the standard deviation of f by y? Or do I need to account for the spread in y, too? Given that I already used the spread in y in the first step, that feels like double counting it, so I am not sure what is the right way. Thank you!

 

[Office_Shredder]

Are y and the standard deviation of dy exactly known to you?

 

[kelly0303]

Are y and the standard deviation of dy exactly known to you?

Yes

 

[Office_Shredder]

You can kind of just compute this exactly. For any possible choice of x, you know what fraction of the time your sampling will return a 0 instead of a 1. Then you can compute things like what value of x makes it so you would only see at least as extreme a result as you got 5% of the time (in either direction, either x being a large number and being surprised by the number of 0s, or x is a small number and you are surprised by the number of 1s). I would argue this doesn't tell you the uncertainty you have for x in the same way that no p value actually measures your uncertainty in the parameter that is being estimated, but it will tell you what values of x would likely be able to give the result that you actually measured.

Depending on how much data you have to crunch through you might want to make some estimate here, for example modeling the percentage of f value that are 1 using a gaussian assuming you know what x is to help figure out how unlikely the given result was.