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In explaining MLE, some texts (such as this) formulate the likelihood function as: $\prod_{i=1}^n f(x_i; \theta)$
while some texts (such as this) formulale the likelihood function as: $\prod_{i=1}^n f(x_i| \theta)$
The basic difference is that, in the latter, $f$ is given as a conditional probability. Do they mean the same thing? What are their differences?
This is merely a matter of convention for denoting the dependence of the density on the unknown parameter. This dependence becomes a probabilistic dependence on the random variable $\theta$ only when $\theta$ itself is a random variable, namely in the Bayesian setting.
