Information Theory Applied to Parameter Inference of Pake Doublet Spectra

Abstract

Magnetic resonance absorption lineshapes can have subtle dependencies on the model parameters that specify the lineshape. To quantify how the model parameters influence the lineshape, it is useful to study simple model systems for which analytical expressions are available. We propose that information theory is a useful tool to quantify how well model parameters may be inferred from a noisy signal. Information theory also allows us to assess the importance of missing parameters from an incomplete model. We do this by monitoring the magnitude of a partition function determined from a suitably defined probability mass function as the model parameters are varied. The optimum parameter set makes the partition function a maximum, which establishes a computable criterion for determining the best model parameter set. Given the availability of a partition function, one may define thermodynamic functions such as the entropy. The optimum parameter set in this interpretation corresponds to the state of maximum entropy. In this work, we observe that at sufficiently low signal to noise ratio, the entropy landscape has no clear maximum, while a related quantity, the Fisher information, always has a clear minimum at the optimum parameter set. The qualitative information we are able to gather from the entropy landscapes is also difficult to assess when the parameters are far from their optimum values, at least for the model system studied here.