LogLikelihood - Poisson distribution

In summary, the conversation discusses using log-likelihood minimization to fit parameters of a particle, with input data being pulses amplitudes and a Poisson distribution being used. However, there is a problem where higher pulse amplitudes have a lower Poisson probability and thus a higher likelihood value. The person is seeking advice on how to correct this effect or what to do in this situation.
  • #1
Zuzana
12
1
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
1661160531826.png

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.
 
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  • #2
What are you comparing?
 
  • #3
mathman said:
What are you comparing?
number of photoelectrons. measured vs expected
 

FAQ: LogLikelihood - Poisson distribution

What is the LogLikelihood for a Poisson distribution?

The LogLikelihood for a Poisson distribution is a statistical measure used to estimate the probability of observing a set of data based on a given set of parameters. It is calculated by taking the natural logarithm of the likelihood function, which is the product of the probability of each data point occurring. In simpler terms, it is a measure of how well the Poisson distribution fits the observed data.

How is the LogLikelihood used in the context of a Poisson distribution?

The LogLikelihood is used to determine the best-fit parameters for a Poisson distribution. By comparing the LogLikelihood values for different sets of parameters, scientists can identify the set of parameters that best describes the observed data. This is often done using a process called maximum likelihood estimation.

What are the assumptions made when using LogLikelihood for a Poisson distribution?

When using LogLikelihood for a Poisson distribution, it is assumed that the data follows a Poisson distribution and that the data points are independent and identically distributed. Additionally, it is assumed that the data points are countable and that the mean and variance of the distribution are equal.

How is the LogLikelihood related to other measures of fit for a Poisson distribution?

The LogLikelihood is closely related to the chi-square statistic, which is another measure of fit for a Poisson distribution. In fact, for large sample sizes, the LogLikelihood is approximately equal to the negative of the chi-square statistic. This means that a smaller LogLikelihood value corresponds to a better fit for the data.

Can the LogLikelihood be used for other types of distributions?

Yes, the LogLikelihood can be used for other types of distributions, such as the normal distribution or the binomial distribution. However, the specific formula for calculating the LogLikelihood will vary depending on the distribution being used. It is important to use the appropriate formula for the specific distribution being analyzed.

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