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I'm sorry for asking probably elementary question, but I cannot understand how estimating probability distribution parameters using maximum likelihood estimation method differs from calculating these parameters from observed data manually. For MLE we need to know the type of probability distribution anyway so why don't we just use the known formulas for calculating the corresponding parameters from observed data? I believe that MLE is somehow more general method but I cannot see what is the real advantage of MLE compared to getting these parameters "manually".

Thanks for explanation.

Tomas

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These "known formulas" that you are thinking about are precisely the ones that maximize the likelihood of the distribution (e.g. taking the mean of a sample gives you the maximum likelihood estimate of the $\mu$ parameter of a normal distribution fit to that sample).

In many cases however, there aren't any closed formulas for the "best" parameters of distributions, in which case you have to follow an iterative optimization approach (e.g. generalized linear models), if that's what you meant to ask.

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I think you are asking about how estimates for these parameters from your data sample (obtained by calculating moments) would be different from maximum likelihood estimation. If so, check out: https://stats.stackexchange.com/questions/252936/what-is-the-method-of-moments-and-how-is-it-different-from-mle http://gradquant.ucr.edu/wp-content/uploads/2013/11/Maximum-Likelihood-and-Method-of-Moments-Estimation.pdf

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