LogLikelihood - Poisson distribution

[alobamons]

Hello I try to fit some parameters of the particle (e.g. energy, direction) be means of log-likelihood minimization. Input data to likelihood function are pulses amplitudes, while Poisson distribution is used. However, the problem is that Poisson distribution is as follows

i.e. for higher pulse amplitute there is a lower Poisson probability and thus higher likelihood value. However, pulses with higher amplitudes are very important in the event and the probability for them should be higher, not? Please, do you know how to correct this effect? Or what would you suggest to do? Thank you very much in advance.

 

[BigBeachBanana]

Hello I try to fit some parameters of the particle (e.g. energy, direction) be means of log-likelihood minimization. Input data to likelihood function are pulses amplitudes, while Poisson distribution is used. However, the problem is that Poisson distribution is as follows
View attachment 33794
i.e. for higher pulse amplitute there is a lower Poisson probability and thus higher likelihood value. However, pulses with higher amplitudes are very important in the event and the probability for them should be higher, not? Please, do you know how to correct this effect? Or what would you suggest to do? Thank you very much in advance.

Confused about your terminology, what are you referring to as "Poisson probability" vs. "likelihood value"?
Second, how are you computing the log-likelihood?

 

[alobamons]

Sorry for bad terminology.
I mean:
Curly L as "likelihood value", which is numerically minimized. P^hit is a Poisson probability.

 

[BigBeachBanana]

Sorry for bad terminology.
I mean: View attachment 33797
Curly L as "likelihood value", which is numerically minimized. P^hit is a Poisson probability.

When you simplify that loglikelihood function of Poisson distribution, it turns out to be just the ordinary average of your data.


Is that what you did?

 

[alobamons]

No, I try to fit the energy E and direction of particle d and these two observables are parameters in the likelihood minimization.
Poisson probability is given as follows:
where N_i is measured number of photoelectrons (p.e.) and lambda_i is expected number of p.e. (taken from pregenerated Monte Carlo tables) and this lambda depends on the parameters E and d. Then, numerical minimization is performed. According to minimal value of likelihood parameters are determined.

 

[BigBeachBanana]

No, I try to fit the energy E and direction of particle d and these two observables are parameters in the likelihood minimization.
Poisson probability is given as follows: View attachment 33807
where N_i is measured number of photoelectrons (p.e.) and lambda_i is expected number of p.e. (taken from pregenerated Monte Carlo tables) and this lambda depends on the parameters E and d. Then, numerical minimization is performed. According to minimal value of likelihood parameters are determined.

Yes, I'm aware of the MLE setup. See this link. and you'll know what I mean. (If you don't then we can talk about it).
If you've computed the MLE correctly, it should be the same as the sample average of your # of p.e from the simulation.

 

[alobamons]

Thank you for the link However, I do not try to fit lambda. I have different lambda for each impulse i. My lambda depends on the parameter of energy and that energy is fit.

 

[BigBeachBanana]

Thank you for the link However, I do not try to fit lambda. I have different lambda for each impulse i. My lambda depends on the parameter of energy and that energy is fit.

If you're not looking to fit lambda then I'm not sure what you're asking/doing with MLE. Because MLE is used to fit the parameter of a distribution i.e. lambda.

 

[alobamons]

In my case lambda is a function of energy and I am looking for that energy - the energy for which the likelihood value is minimal.

 

[BigBeachBanana]

In my case lambda is a function of energy and I am looking for that energy - the energy for which the likelihood value is minimal.

And your energy follows a Poisson distribution? Keep in mind that Poisson is a discrete distribution, and not continous.

 

[alobamons]

And your energy follows a Poisson distribution? Keep in mind that Poisson is a discrete distribution, and not continous.

no, energy is generated according to some theoretical model.

 

[BigBeachBanana]

no, energy is generated according to some theoretical model.

I think you're misusing the MLE for your purposes. The MLE is used to estimate the parameter of a distribution using the observations.

Since your number of p.e is discrete you can assume the # of p.e follows a Poisson distribution and use the observations from the Monte Carlo simulation to estimate [imath]\lambda[/imath] via MLE. So you should only have 1 value of lambda. I don't understand how you came up with 1 lambda for each impulse.

But what you're trying to do is estimate the parameter of the distribution for energy, that is continuous and does not follow the Poisson distribution. You can't use what you showed me in post #3.

Perhaps post a small sample of your calculation of the MLE.