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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 $\theta$.

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 $\theta$.