Working in the Fourier domain will definitely be helpful when I start to move away white Gaussian noise, as the Covariance Matrix can get pretty complex if there is any underlying frequency structure. If my hunch is right, this also requires me to include the frequency response of a boxcar time-domain filter on each of the sinusoids. But i am not completely sure about this, so I need to do some more scribbling.
But on the subject of priors, at least in the time domain
- Probably the cleanest way to model a single sinusoid with a phase offset is to have the amplitude be positive and the phase to range across the full 2pi. You could also use a sine and a cosine with positive amplitudes and derive the phase shift from those values. It’s unclear which one would be more computationally with NUTS/ADVI. Probably the latter but recommendations welcome.
- The prior on the sinusoidal frequency is tricky as the relevant band may span many orders of magnitude. In my original example, I was lazy and used a uniform distribution despite knowing it would weight higher frequencies more.
- The most interesting & troublesome one is the number of sinusoids. I didn’t see anything in the documentation on having reversible-jump/model-scale evaluation built into the sampler. How to address this prior is currently a bit beyond me at the moment.