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To my understanding, likelihood has no meaning per se. Only by comparing likelihoods do they become intepretable. Likelihood is unbounded and probability is bounded.

In maximum likelihood estimation, we have likelihood as the function of some parameter value give the data. Here the parameter is not a random variable. It is what we are optimizing. Howerver, maximum likelihood estimation procedure generates a likelihood function that range from 0 to 1. The log-likelihood is therefore from -infinity to 0. Here, the likelihood is bounded (0,1)?

I'm always confused about 'likelihood'. Is the 'likelihood' in maximum likelihood estimation a 'true likelihood'? Can someone help to clarify the concept? Very much appreciated!

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    $\begingroup$ "a likelihood function that range from 0 to 1." Likelihood is similar to probability density function, so it > 0, but can be >1. $\endgroup$
    – user158565
    Commented Oct 12, 2018 at 18:12
  • $\begingroup$ It basically means your regression line has the maximum likelihood of being the best fitting line to your data. $\endgroup$
    – Kreitz Gigs
    Commented Oct 12, 2018 at 18:13
  • $\begingroup$ Yes, likelihood function in maximum likelihood estimation is a true likelihood. For "What is likelihood?", see stats.stackexchange.com/questions/112451/… $\endgroup$
    – kjetil b halvorsen ♦
    Commented Oct 12, 2018 at 19:24

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Likelihood function is defined in terms of probability mass function for discrete variables (bounded in (0, 1)) or probability density function for continuous variables (>0). Likelihood function itself is a function of varying parameters given fixed data, so it does not integrate to unity. Intuitively, in maximum likelihood you seek for parameters such that were most likely to lead to generating the data you observed, hence you maximize the function describing the probability (or probability density) of observing your data.

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