Context \to The likelihood function is the same, but with the y_1, \ldots, y_n considered fixed and with \theta considered the independent variable. You usually use this function when you know the sample values y_1, \ldots, y_n (because you’ve observed them by collecting data), but don’t know the parameter \theta

What does this y terms mean? Is it just the labels or combination of features and label (y_i = \text{flatten} ( \vec X_i, y_i ))

They say “you know the sample values”, so it must mean they are labels of some sort or perhaps input data. If they are labels, then you’d need to compute the corresponding output values (the \hat{y_i}) based on your parameter \theta and apply the loss function.

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