Negative Log Likelihood is a negative number?

Hello, I'd like to ask what does it mean when Negative Log Likelihood is negative number. I have witnessed negative value of parameter Negative Log likelihood return by gmdistribution.fit and by fitdist, both from statistical toolbox. Negative Log likelihood can not be basically positive number... The fact is that likelihood can be in range 0 to 1. The Log likelihood values are then in range -Inf to 0. Negative log likelihood is finally number in range 0 to + Inf. What happens when the number is positive?

Thank you!!!

Answer 1

[Wayne King]

I think your problem is that this statement is not true: "The fact is that likelihood can be in range 0 to 1. "

There's nothing wrong with a positive log likelihood. The likelihood is the product of the density evaluated at the observed values. Probability densities can take "large" positive values depending on how concentrated they are. Don't confuse PDFs with PMFs (probability mass functions). Remember, PDFs integrate to 1, they are not bound by values in the interval [0,1]

For example, consider the N(0,0.1^2) PDF

   x = -1:0.01:1;
   y = normpdf(x,0,0.1);
   plot(x,y)

Note you can clearly find values of the PDF in the above that result in positive logs.