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The likelihood function is a function of the unknown parameter $\theta$ after the observed data $x_1, \dots, x_n$ are fixed. It assigns to each candidate parameter value a measure of how likely the observed data are under that parameter.

Conditions: The data $x_1, \dots, x_n$ are treated as fixed observed constants.; The argument being varied is the parameter $\theta$.

The likelihood function is a function of the unknown parameter $\theta$ after the observed data $x_1, \dots, x_n$ are fixed. It assigns to each candidate parameter value a measure of how likely the observed data are under that parameter.

Conditions: The data $x_1, \dots, x_n$ are treated as fixed observed constants.; The argument being varied is the parameter $\theta$.

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 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.

In likelihood inference, the observed data $x_1, \dots, x_n$ are held fixed, and the parameter $\theta$ is varied to compare different candidate values.

Conditions: The data have already been observed.; One is comparing different candidate values of $\theta$.

In likelihood inference, the observed data $x_1, \dots, x_n$ are held fixed, and the parameter $\theta$ is varied to compare different candidate values.

Conditions: The data have already been observed.; One is comparing different candidate values of $\theta$.