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Maximum Likelihood Estimation

Suppose we have a probability distribution with parameter \boldsymbol\theta. We also have some samples drawn from the distribution \mathbf X. We would like to estimate \boldsymbol\theta based on \mathbf X to maximize the following probability.

{\hat {\boldsymbol\theta }}={\underset {\boldsymbol\theta \in \Theta }{\operatorname {arg\;max} }}\ p(\mathbf X\mid \boldsymbol\theta )

Intuitively, we would like to find the parameters, which would most likely to generate these samples.