There is a class of statistical problems that arises in several contexts, the Lattice QCD problem of particle physics being one that has attracted the most attention. In essence, the problem boils down to the estimation of an infinite number of parameters from a finite number of equations, each equation being an infinite sum of exponential functions. By introducing a latent parameter into the QCD system, we are able to identify a pattern which tantamounts to reducing the system to a telescopic series. A statistical model is then endowed on the series, and inference about the unknown parameters done via a Bayesian approach. A computationally intensive Markov Chain Monte Carlo (MCMC) algorithm is invoked to implement the approach. The algorithm shares some parallels with that used in the particle Kalman filter. The approach is validated against simulated as well as data generated by a physics code pertaining to the quark masses of protons. The value of our approach is that we are now able to answer questions that could not be readily answered using some standard approaches in particle physics.

The structure of the Lattice QCD equations is not unique to physics. Such architectures also appear in mathematical biology, nuclear magnetic imaging, network analysis, ultracentrifuge, and a host of other relaxation and time decay phenomena. Thus, the methodology of this paper should have an appeal that transcends the Lattice QCD scenario which motivated us.

The purpose of this paper is twofold. One is to draw attention to a class of problems in statistical estimation that has a broad appeal in science and engineering. The second is to outline some essentials of particle physics that give birth to the kind of problems considered here. It is because of the latter that the first few sections of this paper are devoted to an overview of particle physics, with the hope that more statisticians will be inspired to work in one of the most fundamental areas of scientific inquiry.

## References

*Statistical Theory of Reliability and Life Testing*. Holt, Rinehart and Winston, New York. MR438625 0379.62080Barlow, R. E. and Proschan, F. (1975).

*Statistical Theory of Reliability and Life Testing*. Holt, Rinehart and Winston, New York. MR438625 0379.62080

*Concepts in Magnetic Resonance Part A*

**27A**55–63.Bretthorst, G. L., Hutton, W. C., Garbow, J. R. and Ackerman, J. J. H. (2005). Exponential parameter estimation (in NMR) using Bayesian probability theory.

*Concepts in Magnetic Resonance Part A*

**27A**55–63.

*Phys. Rev. D*. Available at http://arxiv.org/pdf/hep-lat/0405001.Chen, Y., Draper, T., Dong, S. J., Horvath, I., Lee, F. X., Liu, K. F., Mathur, N., Srinivasan, C., Tamhankar, S. and Zhang, J. B. (2004). The sequential empirical Bayes method: An adaptive constrained-curve fitting algorithm for lattice QCD.”

*Phys. Rev. D*. Available at http://arxiv.org/pdf/hep-lat/0405001.

*Biochemistry*

**10**3233–3241.Dyson, R. D. and Isenberg, I. (1971). Analysis of exponential curves by a method of moments, with special attention to sedimentation equilibrium and fluorescence decay.

*Biochemistry*

**10**3233–3241.

*Phys. Rev. D*

**65**094512.Fiebig, H. R. (2002). Spectral density analysis of time correlation functions in lattice QCD using the maximum entropy method.

*Phys. Rev. D*

**65**094512.

*EURASIP J. Appl. Signal Process.*

**8**1159–1173. MR2168617 1110.62338 10.1155/ASP.2005.1159Giurcăneanu, C. D., Tăbuş, I. and Astola, J. (2005). Clustering time series gene expression data based on sum-of-exponentials fitting.

*EURASIP J. Appl. Signal Process.*

**8**1159–1173. MR2168617 1110.62338 10.1155/ASP.2005.1159

*Physics in Medicine and Biology*

**12**379–388.Glass, H. I. and de Garreta, A. C. (1967). Quantitative analysis of exponential curve fitting for biological applications.

*Physics in Medicine and Biology*

**12**379–388.

*IEE Proceedings F*

**140**107–113.Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation.

*IEE Proceedings F*

**140**107–113.

*Introduction to Elementary Particles*. Wiley, New York. 1162.00012Griffiths, D. (1987).

*Introduction to Elementary Particles*. Wiley, New York. 1162.00012

*Nuclear Physics B Proceedings Supplements*

**106**12–20.Lepage, G. P., Clark, B., Davies, T. H., Hornbostel, K., Mackenzie, P. B., Morningstar, C. and Trottier, H. (2002). Constrained curve fitting.

*Nuclear Physics B Proceedings Supplements*

**106**12–20.

*Electron. Trans. Numer. Anal.*

**34**163–169. MR2597808 1188.65073Paluszny, M., Martín-Landrove, M., Figueroa, G. and Torres, W. (2008/09). Boosting the inverse interpolation problem by a sum of decaying exponentials using an algebraic approach.

*Electron. Trans. Numer. Anal.*

**34**163–169. MR2597808 1188.65073

*IEEE Transactions on Power Systems*

**14**995–1002.Sanchez-Gasca, J. J. and Chow, J. H. (1999). Performance comparison of three identification methods for the analysis of electromagnetic oscillations.

*IEEE Transactions on Power Systems*

**14**995–1002.

*Internat. J. Modern Phys. A*

**20**5753–5777. MR2189131 10.1142/S0217751X05029022Wilczek, F. A. (2005). Asymptotic freedom: From paradox to paradigm.

*Internat. J. Modern Phys. A*

**20**5753–5777. MR2189131 10.1142/S0217751X05029022