{"id":155,"date":"2023-06-08T12:59:33","date_gmt":"2023-06-08T19:59:33","guid":{"rendered":"https:\/\/live-usc-dornsife.pantheonsite.io\/larry-goldstein\/?page_id=155"},"modified":"2023-06-20T15:57:39","modified_gmt":"2023-06-20T22:57:39","slug":"math-605-steins-method","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-605-steins-method\/","title":{"rendered":"Math 605: Stein’s Method"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
\n\n \n \n
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Since its introduction and application to the Central Limit Theorem in 1972, Stein’s method has offered a novel way of evaluating the quality of distributional approximations through its use of characterizing equations. The method can often produce not only asymptotic information on the error made when approximating a complicated distribution by a simpler one, but also rates of convergence, and in some cases, finite sample bounds with computable constants. In addition, Stein’s method can often be applied in the presence of complicated dependence, generalizing, in the case of Normal approximation, the classical Berry-Esseen theorem in a number of directions. The characterizing equation approach to distributional approximation is not specific to the normal, and Stein’s method has successfully been applied to dozens of distributions, both classical ones, and those less known.
\nFollowing on its success in distributional approximation, the techniques developed for the method have found useful connections to Gaussian inequalities and Mallivin Calculus, Concentration of measure inequalities, and high dimensional statistics.<\/p>\n

The course will cover the fundamentals of Stein’s method, starting with distribuitonal approximation for the Normal and Poisson to illustrate the construction of the Stein equation and the derivation of the properties of its solution. A number of coupling methods for use in the Stein equation will be presented. In addition to the basic case of independence, a sampling of potential applications involving the Poisson and Normal include sequence matching, the birthday problem and random graphs, hierarchical sequences, cone measure projections, the combinatorial central limit theorem, simple random sampling, geometric coverage processes, character ratios, the anti-voter model, random graphs and the lightbulb process; non-normal examples include the Curie-Weiss model from physics and the time complexity of the Quick-Select algorithm.<\/p>\n

The course will also include the derivation of concentration inequalities using Stein type couplings, with applications to permutations and random regular graphs, and applications to high dimensional statistics, drawn from the study of the recovery threshold in high dimensional regression and its relation to Gaussian inequalities, and the relaxation of Gaussian conditions in high dimensional single index models, such as in one bit compressed sensing and shrinkage estimation.<\/p>\n

Input from course members will help determine the selection of topics from the wide range of choices available. Students will be evaluated on the basis of course participation and a final project presentation.<\/p>\n

Schedule: Mondays and Wednesdays, 8:40-10AM, to be held in KAP 414<\/p>\n

Recommended Text:<\/p>\n

Normal Approximation by Stein’s Method
\nChen, L., Goldstein, L., and Shao, Q.M.
\nSpringer Verlag, 2010 [Springer Link<\/a>]<\/p>\n

Supplemental Reference:<\/p>\n

Fundamental’s of Stein’s Method
\nRoss, N.\u00a0http:\/\/arxiv.org\/abs\/1109.1880<\/a><\/p>\n

Sampling of additional books and articles of\u00a0interest:<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

\n\n \n \n
\n\n \n

Poisson Approximation<\/strong><\/p>\n

Poisson Approximation
\nBarbour, A.D., Holst, L., and Janson, S.
\nOxford Science Publications, 1992
\nMR1163825<\/strong>\u00a0(93g:60043)<\/strong><\/a><\/p>\n

Two Moments Suffice for Poisson Approximations: The Chen-Stein Method
\n<\/strong>\u00a0Arratia, R.,\u00a0Goldstein, L., and\u00a0Gordon, L.
\nThe Annals of Probability, Vol. 17, No. 1. (Jan., 1989), pp. 9-25
\nMR0972770<\/strong>\u00a0(90b:60021)<\/strong><\/a><\/p>\n

\u00a0<\/strong>Poisson Approximation and the Chen-Stein Method
\nArratia; R.,\u00a0Goldstein, L. and\u00a0Gordon, L.
\nStatistical Science, Vol. 5, No. 4. (Nov., 1990), pp. 403-424.
\n<\/strong>MR1092983<\/strong>\u00a0(92e:62036)<\/strong><\/a><\/p>\n

Total Variation Distance for Poisson Subset Numbers
\nGoldstein, L, and Reinert, G.
\nAnnals of Combinatorics\u00a0(2006), vol 10, \u00a0pp. 333–341
\n[pdf<\/a>][Springer<\/a>]<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

\n\n \n \n
\n\n \n

Normal Approximation<\/strong><\/p>\n

A Probabilistic Proof of the Lindeberg-Feller Central Limit Theorem
\nGoldstein, L.
\nAmerican Mathematical Monthly\u00a0(2009), vol 116, pp. 45–60 [pdf<\/a>]<\/p>\n

Bounds on the Constant in the Mean Central Limit Theorem
\nGoldstein, L.
\nAnnals of Probability<\/a>\u00a0(2010),\u00a0vol 38, pp. 1672-1689.
\n[pdf<\/a>][arXiv:0906.5145<\/a>]<\/p>\n

<\/small>Spin glasses and Stein’s method
\nChatterjee,\u00a0S.
\nProbab. Theory Related Fields\u00a0 148\u00a0 (2010),\u00a0 no. 3-4, 567\u2013600
\nMR2678899<\/a>\u00a0\u00a0<\/small><\/span><\/span><\/p>\n

On coupling constructions and rates in the CLT for dependent summandswith applications to the anti-voter model and weighted U-statisticss
\nRinott Y. and Rotar V.
\nAnnals of Applied Probability 7, pp 1080-1105<\/a><\/p>\n

Multivariate Normal Approximation by Stein’s Method and Size Bias Couplings
\nGoldtein, L. and Rinott Y
\nhttps:\/\/arxiv.org\/abs\/math\/0510586<\/a>\u00a0<\/big><\/p>\n

Multivariate Normal Approximation by Stein’s Method: The Concentration Inequality Approach
\nChen, L.H.Y., and\u00a0Fang, X.
\nhttp:\/\/arxiv.org\/abs\/1111.4073<\/a><\/p>\n

Fluctuations of Eigenvalues and Second Order Poincar\u00e9 Inequalities
\nChatterjee, S.
\nhttps:\/\/arxiv.org\/abs\/0705.1224<\/a><\/p>\n

Stein’s method on Wiener chaos
\nNourdin I, and\u00a0Peccati, G.
\nhttp:\/\/arxiv.org\/abs\/0712.2940<\/a><\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

\n\n \n \n
\n\n \n

Other distributional Approximations<\/strong><\/big><\/p>\n

<\/big>\u00a0Non asymptotic distributional bounds for the Dickman approximation of the running time of the\u00a0 \u00a0 \u00a0 Quickselect algorithm
\nGoldstein, L.
\nElectronic Journal of Probability, (2018) vol. 23, pp. 1–13\u00a0DOI: 10.1214\/18-EJP227<\/a>
\n[https:\/\/arxiv.org\/abs\/1703.00505<\/a>]<\/p>\n

Dickman approximation in simulation, summations and perpetuities
\nBhattacharjee, C. and Goldstein, L.
\nBernoulli,\u00a0(2019) vol 25, No. 4A, pp. 2758\u20132792<\/em>
\n[https:\/\/arxiv.org\/abs\/1706.08192<\/a>]<\/p>\n

Non normal approximation by Stein’s method of exchangeable pairs with application to the Curie-Weiss model
\nChatterjee, S., Shao, Q.M.
\nAnn. Appl. Probab. 21 (2011), no. 2, 464\u2013483
\nMR2807964<\/a><\/p>\n

Degree asymptotics with rates for preferential attachment random graphs
\nPek\u00f6z, E.,\u00a0R\u00f6llin, A., and Ross, Nathan R.
\nhttp:\/\/arxiv.org\/abs\/1108.5236<\/a><\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

\n\n \n \n
\n\n \n

Concentration Inequalities<\/strong><\/p>\n

Stein’s method for concentration inequalities
\nChatterjee, S.
\n[https:\/\/arxiv.org\/abs\/math\/0604352]<\/a><\/p>\n

Applications of size biased couplings for concentration of measures
\nGhosh, S., and Goldstein, L.
\nElectronic Communications in Probability (2011), vol 16, pp. 70-83.
\n[pdf<\/a>][ECP<\/a>]<\/p>\n

Bounded size biased couplings, log concave distributions and concentration of measure for occupancy models
\nBartroff, J., Goldstein, L. and I\u015flak, \u00dc
\nBernoulli, (2018) vol\u00a0 24, No. 4B, pp. 3283-3317.
\n[http:\/\/arxiv.org\/abs\/1402.6769<\/a>]<\/p>\n

Concentration inequalities via zero bias couplings
\nGoldstein, L. and I\u015flak,\u00a0\u00dc.
\nStatistics and Probability Letters, (2014), vol 86, pp. 17-23
\n[http:\/\/arxiv.org\/abs\/1304.5001<\/a>][Elsevier<\/a>] [10.1016\/j.spl.2013.12.001]<\/p>\n

Size biased couplings and the spectral gap for random regular graphs
\nCook, N., Goldstein, L. and\u00a0Johnson, T.
\nAnnals of Probability, (2018), vol 46, No.1, 72-125
\n[http:\/\/arxiv.org\/abs\/1510.06013<\/a>]<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

\n\n \n \n
\n\n \n

Statistics and Machine Learning Applications
\n<\/strong><\/p>\n

\u00a0<\/strong>Non-Gaussian Observations in Nonlinear Compressed Sensing via Stein Discrepancies
\nGoldstein, L. and Wei, X
\nInformation and Inference: A Journal of the IMA, (2019) vol 8.1, pp.\u00a0125-159.\u00a0iay006<\/a>.
\n[https:\/\/arxiv.org\/abs\/1609.08512<\/a>]<\/p>\n

Gaussian Phase Transitions and Conic Intrinsic Volumes:\u00a0 Steining the Steiner Formula
\nGoldstein, L., Nourdin, I. and Peccati, G.
\nAnnals of Applied Probability (2017), vol 27, pp. 1-47
\n[http:\/\/arxiv.org\/abs\/1411.6265]<\/a><\/p>\n

A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation
\nQiang L., Q, Lee, J., Jordan, M.
\n[https:\/\/arxiv.org\/abs\/1602.03253<\/a>]<\/p>\n

Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm
\nQiang L. and Wang, D.
\n[https:\/\/arxiv.org\/abs\/1608.04471<\/a>]<\/p>\n

Relaxing the Gaussian assumption in Shrinkage and SURE in high dimension
\n<\/strong>Fathi, M., Goldstein, L., Reinert, G.\u00a0and Saumard, A.
\n[https:\/\/arxiv.org\/abs\/2004.01378<\/a>]<\/p>\n

 <\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n\n\n\n\n \n \n\n\n\n\n\n\n\n

\n\n \n \n
\n\n \n

Additional material of potential interest<\/strong><\/p>\n

Exchangeable pairs from switchings
\nJohnson, T.
\nhttp:\/\/arxiv.org\/abs\/1112.0704<\/a><\/p>\n

On a connection between Stein characterizations and Fisher information
\nLey, C., and\u00a0Swan, Y.
\nhttp:\/\/arxiv.org\/abs\/1111.2368<\/a><\/p>\n

Estimation of the mean of a multivariate normal distribution
\nStein, C.
\nMR0630098 (83a:62080)<\/a>
\n<\/big><\/p>\n

A General Size biased distribution
\nhttps:\/\/arxiv.org\/abs\/1903.01983<\/a><\/p>\n

Size bias for one and all
\nArratia, R., Goldstein, L. and Kochman, F.
\nProbability Surveys, (2019) vol 16, pp. 1-61
\n[http:\/\/arxiv.org\/abs\/<\/a>1308.2729<\/a>]<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":370,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-155","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nMath 605: Stein's Method - Larry Goldstein<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-605-steins-method\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Math 605: Stein's Method - Larry Goldstein\" \/>\n<meta property=\"og:url\" content=\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-605-steins-method\/\" \/>\n<meta property=\"og:site_name\" content=\"Larry Goldstein\" \/>\n<meta property=\"article:modified_time\" content=\"2023-06-20T22:57:39+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-605-steins-method\/\",\"url\":\"https:\/\/dornsife.usc.edu\/larry-goldstein\/math-605-steins-method\/\",\"name\":\"Math 605: Stein's Method - 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