Hi @StarryNight, I am maybe wrong, but it looks like from the notation that you are fitting a power spectrum/periodogram (S) as a function of frequency (f), with a power law model. If the power spectrum has come from the application of a Discrete Fourier Transform, then the likelihood is not a Normal. It should be chi-squared.
If I am wrong, then feel free to ignore me 
Happy to provide further information if I am right though. You can send me a direct message.