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The joint pmf/pdf can be written as a product because the sample observations are assumed to be independent. Independence allows the joint distribution to factor into the product of the corresponding univariate pmf/pdf terms.

Conditions: The observations $x_1, \dots, x_n$ are independent.; The joint distribution is written as $f(x_1, \dots, x_n \mid \theta)$.; Each observation shares the same marginal pmf/pdf $f(x_i \mid \theta)$.

The joint pmf/pdf can be written as a product because the sample observations are assumed to be independent. Independence allows the joint distribution to factor into the product of the corresponding univariate pmf/pdf terms.

Conditions: The observations $x_1, \dots, x_n$ are independent.; The joint distribution is written as $f(x_1, \dots, x_n \mid \theta)$.; Each observation shares the same marginal pmf/pdf $f(x_i \mid \theta)$.

The lecturer states that once the data have been collected, they should be treated as constants because there is no more variability. This fixed-data perspective is what allows the likelihood to be discussed as a function of the parameter alone.

Conditions: The data have already been observed.; The discussion is in the inference stage rather than the sampling stage.

The lecturer states that once the data have been collected, they should be treated as constants because there is no more variability. This fixed-data perspective is what allows the likelihood to be discussed as a function of the parameter alone.

Conditions: The data have already been observed.; The discussion is in the inference stage rather than the sampling stage.