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No they don't. Only because some people ignore the function which is the second parameter is where it fails.

Isn't the likelihood function of given fit with any parameter theta with this silly function almost always 0, making it wrong to use either AIC or BIC?

Ok, assume Gaussian noise with a fixed variance, hence 0 additional parameters.

AWGN would not help with fit probabilities, you get an additional constant term in log likelihood L. You still get to at least evaluate the log likelihood function or show that AWGN dominates the other term.

With the proposed method, you can fit arbitrarily closely to the data, so you can get your likelihood as good as you like, still using a single parameter. So you get good k(=1) and good likelihood, thus good AIC. The likelihood does not need to dominate the other term, it just has to be as good as the likelihood of the model you are comparing to.

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