  Nuclear magnetic resonance (NMR) spectroscopy exploits the magnetic
properties of atomic nuclei to discover the structure, reaction state and
chemical environment of molecules. We propose a probabilistic generative model
and inference procedures for NMR spectroscopy. Specifically, we use a weighted
sum of trigonometric functions undergoing exponential decay to model free
induction decay (FID) signals. We discuss the challenges in estimating the
components of this general model -- amplitudes, phase shifts, frequencies,
decay rates, and noise variances -- and offer practical solutions. We compare
with conventional Fourier transform spectroscopy for estimating the relative
concentrations of chemicals in a mixture, using synthetic and experimentally
acquired FID signals. We find the proposed model is particularly robust to low
signal to noise ratios (SNR), and overlapping peaks in the Fourier transform of
the FID, enabling accurate predictions (e.g., 1% sensitivity at low SNR) which
are not possible with conventional spectroscopy (5% sensitivity).
