Bayesian and information theory analysis of MAS sideband patterns in spin 1/2 systems

Bayesian statistics and information theory are used to analyze the reliability of extracting chemical shift parameters from spinning sideband patterns of spin 1/2 systems. Efficient code has been written to calculate the two-dimensional posterior probability as a function of the chemical shift anisotropy, δ, and the asymmetry parameter, η, given the sideband intensities and the signal-to-noise ratio. This method has the advantage of assuming only that the noise in the sideband intensities is distributed as a Gaussian. It assumes nothing about the distribution of the values of parameters δ and η, which are shown in some cases to be highly non-Gaussian. The utility of Bayesian analysis is demonstrated on 1D slow-spinning MAS spectra and on sideband patterns extracted from 2D PASS spectra. Previous investigations have shown that there is an optimal range of spinning frequencies for determining δ. In this study, information theory is used to determine the signal-to-noise ratio dependence of the entropy in δ, η, and total entropy in spinning sideband spectra. The entropy is a measure of the information content of a probability distribution. When the entropy is zero, there is perfect information on a system, while if it is infinite, there is no information on the system. It is found that for all values of η and for signal-to-noise ratios in the range 50-1000, an entropy minimum in νδ/νr occurs for values 1.5 ⩽ νδ/νr ⩽ 3. In the same range of signal-to-noise ratios, the entropy in η is a monotonically decreasing function of νδ/νr. The global information content of a spinning sideband pattern (i.e., the total entropy) is dependent on the signal-to-noise ratio and has an optimal value at νδ/νr ≈ 2 at a signal-to-noise ratio of 50 and increasing to ≈2.5 for signal-to-noise ratios of 1000. Finally, the increase of information in a sideband pattern as a function of the number of sidebands used in the analysis is examined. Most of the information about δ and η is contained in the five central sidebands; i.e., sidebands −2 to 2.