Poisson error question

[Malamala]

Hello! I have a fit to a histogram $y(x)$. Now I want to predict the number of counts at some other point, not in the original data, using this fitted function and assign an error to it. Let's say that the the point where I want to compute this value, $x_0$ gives $y(x_0) = 100$. As the function $y(x)$ was obtained from a fit, it has some error associated to the error propagation of the errors on the parameters of $y$. Let's assume that that error is 2. Now, as this functions measure counts, the error associated to it can be considered a Poisson error i.e. assuming that I wouldn't have that error of 2, I would give my result as $100 \pm \sqrt{10} = 100 \pm 10$. Now given that I have that 2, can I just add the two sources of error in quadrature and give my result as $100 \pm \sqrt{4+100} = 100 \pm 10.2$? Is this correct?

 

[BvU]

Hi,

With numbers like that you can ignore the error in the fitted parameters: you shouldn't quote 100 $\pm$ 10.2 because your estimate has no decimals $\rightarrow$ your error estimate doesn't either.

If the fit is a lot less accurate, say with an error of 10, then yes, I would add in quadrature if I only had to report $y(x_0)$.
If I had to erport a series of $y(x_i)$, I would keep the fit error separated (it's a systematic error).

 

[Malamala]

Hi,

With numbers like that you can ignore the error in the fitted parameters: you shouldn't quote 100 $\pm$ 10.2 because your estimate has no decimals $\rightarrow$ your error estimate doesn't either.

If the fit is a lot less accurate, say with an error of 10, then yes, I would add in quadrature if I only had to report $y(x_0)$.
If I had to erport a series of $y(x_i)$, I would keep the fit error separated (it's a systematic error).

Thank you! So ideally I should quote my result (for a fit error of 10) as $100 \pm (10)_{stat} \pm (10)_{sys}$. Is this right? One more question about my initial example. Assuming that using the parameters obtained by that fit i.e. $y(x)$, I try to fit to some other data and I see a bump in a given bin, and I want to see how statistically significant it is. So in order to do that I would calculate $|y_{bump} - y(x_{bump})|/\sigma_y$. Should I use in this case $10.2$ as the value for $\sigma_y$? Or can I still safely use just 10?

 

[BvU]

Ah, the ice gets thinner !

With your fit you calculate $y(x_\text{bump}) = 100\pm 10_\text{sys}$ but your measurement give you $y_\text{bump}\pm\sqrt{y_\text{bump}}$ and therefore the net height of the bump has error $\sqrt{100 + y_\text{bump}}$ .

In other words: I would ignore the statistical error in $y(x_0)$ ... (*)

This goes to show you really want a low background, and if that's done, you also want a very good fit of that background !


(*) don't feel 100 ($\pm$10 % ) certain here, could use some help from e.g. @mfb

 

[mfb]

Your prediction will be the fit result with uncertainties from the fit only. The actual results will have some spread from their Poisson distribution, but typically this is not considered an uncertainty of your prediction, it is an uncertainty from the experimental realization testing this prediction. Take 10 times as much data and your relative uncertainty will be lower.

 

[Stephen Tashi]

Hello! I have a fit to a histogram $y(x)$. Now I want to predict the number of counts at some other point, not in the original data, using this fitted function and assign an error to it.

Is the prediction $y(x)$actually an integer number of counts at the value $x$ (whatever $x$ represents)? If the prediction predicts the parameter $\lambda$ of Poission distribution, it can be used to predict the mean number of counts at the value $x$. This mean number of counts need not be an integer.