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For example, MLE is defined as estimating the distrubution that gives the maximum probability of the observations [itex]x_i[/itex] given the distribution parameters, [itex]p(x|\theta)[/itex] . However my instructor stressed that since [itex]\theta[/itex] is not a random variable but a parameter of the distribution, it is meaningless to take a conditional on it, and uses the notation [itex]p(x;\theta)[/itex].

I understand that the word "given" can be misleading since in natural language it can very well mean "given the specific value of the parameter" but in probability it refers to conditional probability.

My question is , if the use of the | notation is valid, or the view of [itex]\theta[/itex] as a random variable is illegal.

Isn't MLE, however, in comparison to the MAP, viewed as a specific case of MAP where the probability of [itex] \theta [\itex] is uniform? So is the use of the | valid ?