Maximum Likelihood Fitting normalization (Bevington)

[stoked]

Hi,
I'm posting this in this particular forum because, though this's a statistics question, my application is in high energy.

My question is regarding a problem in Bevington's book (Data Reduction and Error Analysis..., Page 193, Ex. 10.1), but I'll give a general description here...

Say you want to fit to a scattering distribution function: f = (a + b cos² theta) using the likelihood method. cos$$\theta$$ ranging from -1 to 1, you would get a normalization integral, norm = 2(a + b/3). However, what goes into the actual code is the normalized fuction (a + b cos² theta) / 2(a + b/3), so that the fitter is only sensitive to (a/b) and not a and b separately. So if you were asked to estimate a and b (as does Bevington's problem)...what do you do?

I am aware of the "extended" likelihood method. Is that applicable here? If so, how?

Thanks a bunch.

 

[Valerie Park]

Yes, the extended likelihood method is applicable here. The extended maximum likelihood method allows you to estimate the parameters (a and b) separately while incorporating the normalization factor in the fitting procedure. To do this, you will need to define the likelihood function as: L(a,b) = product(f(theta_i | a,b)/norm)Where f(theta_i | a,b) is the scattering distribution function, and norm is the normalization factor. The extended maximum likelihood method then involves taking the partial derivatives of L(a,b) with respect to a and b, setting them equal to zero, and solving for a and b. This will give you the maximum likelihood estimates for a and b.

 

[sir-pinski]

Hi there,

Thank you for your question. The Maximum Likelihood Fitting normalization is a commonly used method in statistics to estimate the parameters of a distribution function. In this case, you are fitting a scattering distribution function using the likelihood method. The normalization integral, as you mentioned, is necessary to ensure that the fitted function is normalized to 1. However, in order to estimate the individual parameters a and b, you need to consider the normalized function (a + b cos2 theta) / 2(a + b/3). This is because the fitter is only sensitive to the ratio of a and b, not the individual values themselves.

One approach to estimating a and b separately is to use the "extended" likelihood method, as you mentioned. This method takes into account the normalization factor in the likelihood function, allowing you to estimate a and b separately. Another approach is to use a non-linear least squares fitting method, such as the Levenberg-Marquardt algorithm, which can handle non-linear functions and give better estimates for a and b compared to the Maximum Likelihood Fitting normalization method.

I hope this helps answer your question. Best of luck with your fitting process!