To this date the best known application of the method is towards distributional comparison. Here I provide a list of non-standard applications outside of this (very wide) area.

Table of contents (last update 22 July 2017)

Statistical mechanics

  • Eichelsbacher, P., & Löwe, M. (2009). Stein's method for dependent random variables occuring in statistical mechanics (No. arXiv: 0908.1909). MFO.
  • Chatterjee, S., & Shao, Q. M. (2011). Nonnormal approximation by Stein's method of exchangeable pairs with applications to the Curie-Weiss Model. The Annals of applied probability, 464-483.
  • Röllin, A., & Ross, N. (2015). Local limit theorems via Landau–Kolmogorov inequalities. Bernoulli, 21(2), 851-880.
  • Eichelsbacher, P., & Martschink, B. (2015). On rates of convergence in the Curie–Weiss–Potts model with an external field. In Annales de l'Institut Henri Poincaré, Probabilités et Statistiques (Vol. 51, No. 1, pp. 252-282). Institut Henri Poincaré.
  • Goldstein, L., & Wiroonsri, N. (2016). Stein's method for positively associated random variables with applications to Ising, percolation and voter models. arXiv:1603.05322

Quantum mechanics

  • McKeague, I. W., & Levin, B. (2014). Convergence of empirical distributions in an interpretation of quantum mechanics. arXiv preprint arXiv:1412.1563.
  • McKeague, I. W., Peköz, E. A., & Swan, Y. (2016). Stein's method, many interacting worlds and quantum mechanics. arXiv preprint arXiv:1606.06618.

Information theory

  • Sason, I. (2012). An information-theoretic perspective of the Poisson approximation via the Chen-Stein method. arXiv preprint arXiv:1206.6811.
  • Ley, C., & Swan, Y. (2013). Stein’s density approach and information inequalities. Electron. Comm. Probab, 18(7), 1-14.
  • Ley, C., & Swan, Y. (2013). Local Pinsker inequalities via Stein's discrete density approach. Information Theory, IEEE Transactions on, 59(9), 5584-5591.
  • Nourdin, I., Peccati, G., & Swan, Y. (2014). Entropy and the fourth moment phenomenon. Journal of Functional Analysis, 266(5), 3170-3207.
  • Nourdin, I., Peccati, G., & Swan, Y. (2014, June). Integration by parts and representation of information functionals. In Information Theory (ISIT), 2014 IEEE International Symposium on (pp. 2217-2221). IEEE.

Log-Sobolev (LS, LSI,...)

  • Ledoux, M., Nourdin, I., & Peccati, G. (2015). Stein’s method, logarithmic Sobolev and transport inequalities. Geometric and Functional Analysis, 25(1), 256-306.
  • Ledoux, M., Nourdin, I., & Peccati, G. (2016). A Stein deficit for the logarithmic Sobolev inequality. arXiv preprint arXiv:1602.08235.

Sparse estimation and random matrices

  • Mackey, L., Jordan, M. I., Chen, R. Y., Farrell, B., & Tropp, J. A. (2014). Matrix concentration inequalities via the method of exchangeable pairs. The Annals of Probability, 42(3), 906-945.
  • Goldstein, L., Nourdin, I., & Peccati, G. (2014). Gaussian phase transitions and conic intrinsic volumes: Steining the Steiner formula. arXiv preprint arXiv:1411.6265.

High dimensional statistics

  • Chernozhukov, V., Chetverikov, D., & Kato, K. (2014). Central limit theorems and bootstrap in high dimensions. arXiv preprint arXiv:1412.3661.

Machine learning

  • Janzamin, M., Sedghi, H., & Anandkumar, A. (2014). Score Function Features for Discriminative Learning: Matrix and Tensor Framework. arXiv preprint arXiv:1412.2863.
  • Liu, Q. (2017). Stein Variational Gradient Descent as Gradient Flow. arXiv preprint arXiv:1704.07520.

Monte-Carlo estimation

  • Chris Oates, Mark Girolami, and Nicolas Chopin (2016) "Control Functionals for Monte Carlo Integration" to appear Journal of Royal Statistical Society - Series B. arXiv preprint arXiv:1410.2392v4
  • Chris J. Oates, Jon Cockayne, François-Xavier Briol, Mark Girolami (March 2016). Convergence Rates for a Class of Estimators Based on Stein's Identity. arXiv preprint arXiv:1603.03220

Bayesian statistics

  • Ley, C., Reinert, G., & Swan, Y. (2015). Distances between nested densities and a measure of the impact of the prior in Bayesian statistics. arXiv preprint arXiv:1510.05826.

Signal processing

  • Weinberg, G. V. (2005). Stein’s method and its application in radar signal processing. DSTO Formal Reports (DSTO-TR-1735). Electronic Warfare & Radar Division, Dept. Defence (Australian Goverment). Available at dspace. dsto. defence. gov. au/dspace/handle/1947/4064.
  • Weinberg, G. V. (2016). Error bounds on the Rayleigh approximation of the K-distribution. IET Signal Processing.

Number Theory

  • Arras, B., Mijoule, G., Poly, G., & Swan, Y. (2016). Distances between probability distributions via characteristic functions and biasing. arXiv preprint arXiv:1605.06819.

Mean-field analysis

  • Ying, L. (2017). Stein's Method for Mean Field Approximations in Light and Heavy Traffic Regimes. Proceedings of the ACM on Measurement and Analysis of Computing Systems, 1(1), 12.

Goodness of fit testing

  • Liu, Q., Lee, J., & Jordan, M. (2016, June). A kernelized Stein discrepancy for goodness-of-fit tests. In International Conference on Machine Learning (pp. 276-284).
  • Chwialkowski, K., Strathmann, H., & Gretton, A. (2016, June). A kernel test of goodness of fit. In International Conference on Machine Learning (pp. 2606-2615).
  • Gaunt, R. E., Pickett, A. M., & Reinert, G. (2017). Chi-square approximation by Stein’s method with application to Pearson’s statistic. The Annals of Applied Probability, 27(2), 720-756.
  • Jitkrittum, W., Xu, W., Szabo, Z., Fukumizu, K., & Gretton, A. (2017). A Linear-Time Kernel Goodness-of-Fit Test. arXiv preprint arXiv:1705.07673.