How can I see that the maximum likelihood approach finds the parameter values of the probability distribution that maximize the probability of the observed sample? Maximum likelihood is not the maximum of probability in general because for, say, two continuous random variables $X_1, X_2$ we have $P(X_1=x_1,X_2=x_2)=0$.
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1Does this answer your question? [What is the difference between "likelihood" and "probability"?](https://stats.stackexchange.com/questions/2641/what-is-the-difference-between-likelihood-and-probability) – Arya McCarthy Apr 08 '21 at 14:21
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Or perhaps this: [Difference between density and probability](https://stats.stackexchange.com/questions/156883/difference-between-density-and-probability) – Arya McCarthy Apr 08 '21 at 14:21
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Thank you! But what is that theoretical link that makes maximum likelihood maximum probability? This likelihood function appears as axiom, but I guess it should come from some other argument that makes it maximum probability. – Alex Apr 08 '21 at 14:27