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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?

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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.

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  • $\begingroup$ How do we actually know whether $\theta$ can or not be considered a random variable? To be considered a random variable, there must be an associated probability distribution. Can you give some examples of when this is the case and when it is not the case? Why only in the Bayesian setting the parameters can be considered a random variable? $\endgroup$ – nbro Nov 12 '19 at 16:34

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