Fourier transform is an estimator of frequency. Because its an estimator, there is always some uncertainty associated with my coefficients as described by the Fourier/Gabor limit. I'm wondering how to generate a probability density of frequency from the transformed data. My intuition is that I like what they do in physics with wave function probability amplitudes: modulus square the coefficients and normalize.
Edit: Fourier transform is NOT an estimator of frequency when integrating across time from -infinity to infinity. When you do not do this, you do not get distinct poles, but gaussian-like curves. First question, does it make sense or can you think of a time-limited fourier-transform in terms of probability? Given my time-limited fourier transform, what is the probability that frequency f is precsent?