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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1$\begingroup$ Does this answer your question? What is the difference between "likelihood" and "probability"? $\endgroup$– Arya McCarthyApr 8, 2021 at 14:21
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$\begingroup$ Or perhaps this: Difference between density and probability $\endgroup$– Arya McCarthyApr 8, 2021 at 14:21
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$\begingroup$ 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. $\endgroup$– AlexApr 8, 2021 at 14:27
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