Supplementary material: Probabilistic amplitude and frequency demodulation

t=1 exp (k1 cos(θt − θt−1) + k2 cos(θt − θt−2)) , (2) p(yt|at, θt) = Norm(yt; at cos(θt + ω̄t), σ 2 y). (3) The joint distribution over the amplitudes is a multivariate truncated Gaussian. Therefore, when the von Mises process becomes an independent, uniform distribution (k1 = 0 and k2 = 0), the model reduces to that of Sell and Slaney with a particular spectral weighting function determined by the dynamical parameter, λ1:τ . 3 Reformulating the model as a constrained Gaussian AR(1) process The model can be massaged into a form which simplifies the application of EP. First, the distribution over the amplitudes is written as a multi-dimensional Gaussian AR(1) process with an associated positivity constraint. Defining at = [a1,t, a2,t, . . . , aτ,t] = [a1,t, a1,t−1, . . . , a1,t−τ+1], we have p(a1:T |Λa,Σa) ∝ T