april 2011 question 10

On the last part of the question, with the poisson distribution, can you please explain me why doesn't he calculate the likelihood as the products of all the distributions? Why does he take only 1 distribution?

Thanks a lot

 

Hi

It does compute using all distributions here. To simplify things, it starts off with one component first and then it invokes the simple algebra that log-likelihood = Sum of each component log-likelihood (see line 4 in the solution)

You can start taking the whole function and you will see you reach the same place as in line 4.

Trust this helps.

 

At some point in the solution, they end up doing a summation but am not sure about their method. However, if you do it the usual way and take the product, it all comes down to the same solution. Try it.

 

thanks a lot for the reply !
Are there any cases that we take only 1 function? If it is asked to find the mle of 1 variable? Or we always calculate it the normal way?

 

Hi

I think you (johnpe21) have missed the point.

In the solution, the examiner is showing that you can simplify your calculations by first computing the log likelihood for the i-th sample point. As all sample points comes from same distribution, the final result will be just the sum of the log likelihood of the i-th sample point (the sum is taken over all values of i from 1 to n). This approach will not work if the sample points come from different distributions like a mixture or some sample points are censored observations.

You will usually not be asked to derive MLE of an unknown parameter for a given data with 1 sample point ie when n =1.

PS: Given this is now so close to the exam day, I would suggest you forget this approach to the solution and only think of doing it in the usual way i.e. writing the likelihood for the whole sample. You can never go wrong if you attempt this way.

 

clear now thanks a lot !