A Metropolis Monte Carlo Implementation of Bayesian Time-Domain Parameter Estimation: Application to Coupling Constant Estimation from Antiphase Multiplets

✍ Scribed by Michael Andrec; James H. Prestegard

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 391 KB
• Volume: 130
• Category: Article
• ISSN: 1090-7807
• DOI: 10.1006/jmre.1997.1304

✦ Synopsis

The Bayesian perspective on statistics asserts that it makes sense synthetic data with substantial added random noise, we demto speak of a probability of an unknown parameter having a particonstrate that the error estimates obtained using the Bayesian ular value. Given a model for an observed, noise-corrupted signal, approach are consistent with a Monte Carlo error analysis in we may use Bayesian methods to estimate not only the most probathe case of two well-resolved Lorentzians, and that coupling ble value for each parameter but also their distributions. We preconstants can be reliably obtained from overlapping antisent an implementation of the Bayesian parameter estimation forphase doublets in cases where the splitting is substantially malism developed by G. L. Bretthorst (1990, J. Magn. Reson. 88, smaller than one-half of the linewidth. An illustration of 533) using the Metropolis Monte Carlo sampling algorithm to application to the extraction of coupling constants from experform the parameter and error estimation. This allows us to perimental data containing an antiphase doublet with passive make very few assumptions about the shape of the posterior distrisplittings is given.

bution, and allows the easy introduction of prior knowledge about constraints among the model parameters. We present evidence that the error estimates obtained in this manner are realistic, and THEORY that the Monte Carlo approach can be used to accurately estimate coupling constants from antiphase doublets in synthetic and exper-

The Bayesian perspective on statistics asserts that a probaimental data. ᭧ 1998 Academic Press bility represents a degree of belief rather than a frequency of occurrence (4). In other words, it is possible to speak of the probability of a particular vector of parameter values A common task in NMR spectroscopy is the accurate among all possible parameter vectors in a statistical model.

Thus, the process of parameter estimation and error estima-estimation of spectral parameters (such as splittings, linewidths, or intensities) and their uncertainties from time-do-tion is intimately connected from a Bayesian point of view.

For example, one could choose the best parameter estimate main data (FIDs). In addition, it is often possible to specify constraints among the parameters. For example, the presence as the parameter vector U which maximizes the probability density P(UÉD) given a data vector D. Similarly, the uncer-of a known multiplet structure could lead to a significant reduction in the number of adjustable parameters, since one tainty in U could be expressed by a credible interval, i.e., a hyperrectangle in U space that encloses a given fraction could specify constraints among the frequencies, phases, and intensities. Thus, the ideal quantitative NMR data analysis of the probability density P(UÉD). Similarly, correlations between the model parameters can be determined from the tool not only would estimate parameter values and uncertainties but also would allow the flexible specification of rela-covariance structure of P(UÉD). The formal justification of such methods and a discussion of their relationship to classi-tionships among the model parameters.

We present here a new implementation of Bayesian pa-cal point and interval estimation is beyond the scope of this paper, but has been amply discussed in the statistical litera-rameter estimation based on the work of Bretthorst (1). Although it is certainly not the ideal quantitative NMR data ture (e.g., Refs. ( 5,).

Previous use of Bayesian methods in the statistical analy-analysis tool, it does possess many of the desirable attributes mentioned above. In addition, we hope to demonstrate that sis of NMR data was pioneered by Bretthorst, who developed the theoretical methodology (1, 7-11) and with co-workers it is robust and flexible and that it can give reasonable estimates of the reliability of the extracted parameters. We have evaluated its performance using synthetic NMR data containing well-resolved signals (12,. Applications of chosen to use the Metropolis-Hastings Monte Carlo algorithm (2, 3) to generate points in the parameter space ac-Bayesian methods by other workers have included the estimation of coupling constants from poorly resolved in-phase cording to the Bayesian posterior probability density. Using 217