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I'm having a hard time understanding the use of MLE. I understand what it does, it is intended to give the best parameters for a particular model to represent a population.

My question is, is MLE necessary for getting the best model parameters? Isn't if possible to do this calculation manually?

For example, if I want to calculate the best parameters of a Gaussian Distribution for the following population...

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I would need the mean and standard deviation, I can calculate those myself, they come out to be 49.8 and 11.37 respectively. So my question is, what benefit does MLE give over the standard way of computing these parameters?

I apologize if this question is naive, I have just been introduced to MLE and my background is not in statistics so I am trying to wrap my head around this.

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If you fit to the mean and variance (or other moments), that's called Method of Moments fitting.

Maximum Likelihood estimation is using the full dataset to do the estimation, whereas method of moments just uses the moments. If you do an MLE fitting, you will often not fit to the moments exactly, but the fit of the model to the observations is closer.

There are advantages and disadvantages to both methods. Often it is hard to maximise the likelihood, i.e. when the likelihood is nonlinear in the parameters. Sometimes it is very easy to do a Method of Moments fit, e.g. normal distribution and many others. However when a distribution does not have an easy linear relationship between the mean and variance (in terms of parameters), an MoM is just as computationally demanding as an MLE, requiring optimisation algorithms and the like.

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