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題名 馬可夫鏈蒙地卡羅收斂的研究與貝氏漸進的表現
A study of mcmc convergence and performance evaluation of bayesian asymptotics
作者 許正宏
Hsu, Cheng Hung
貢獻者 翁久幸
Weng, Chiu Hsing
許正宏
Hsu, Cheng Hung
關鍵詞 Edgeworth 展開式
馬可夫鏈蒙地卡羅
個別後驗分配
Stein’s 等式
Edgeworth expansion
Markov chain Monte Carlo
marginal posterior distribution
Stein`s identity
日期 2010
上傳時間 5-十月-2011 14:31:53 (UTC+8)
摘要 本論文主要討論貝氏漸近的比較，推導出參數的聯合後驗分配與利用圖形來診斷馬可夫鏈蒙地卡羅的收斂。Johnson (1970)利用泰勒展開式得到個別後驗分配的展開式，此展開式是根據概似函數與先驗分配。 Weng (2010b) 和 Weng and Hsu (2011) 利用 Stein’s 等式且由概似函數與先驗分配估計後驗動差；將這些後驗動差代入Edgeworth 展開式得到近似後驗分配，此近似分配的誤差可精確到大O的負3/2次方與Johnson’s 相同。另外Weng and Hsu (2011)發現Weng (2010b) 和Johnson (1970)的近似展開式各別項誤差到大O的負1次方不一致，由模擬結果得到Weng’s 在此項表現比Johnson’s 好。另外由Weng (2010b)得到一維參數 的Edgeworth 近似後驗分配延伸到二維參數的聯合後驗分配；並應用二維參數的聯合後驗分配於多階段資料。本論文我們提出利用圖形來診斷馬可夫鏈蒙地卡羅收斂的方法，並且應用一般化線性模型與混合常態模型做為模擬。


關鍵字: Edgeworth 展開式；馬可夫鏈蒙地卡羅；個別後驗分配；Stein’s 等式
Johnson (1970) obtained expansions for marginal posterior
distributions through Taylor expansions. The expansion in
Johnson (1970) is expressed in terms of the likelihood and the prior. Weng (2010b) and Weng and Hsu (2011) showed that by using
Stein`s identity we can approximate the posterior moments in terms
of the likelihood and the prior; then substituting these
approximations into an Edgeworth series one can obtain an expansion
which is correct to O(t{-3/2}), similar to Johnson`s.
Weng and Hsu (2011) found that the O(t{-1}) terms in
Weng (2010b) and Johnson (1970) do not agree and further
compared these two expansions by simulation study. The simulations
confirmed this finding and revealed that our O(t{-1}) term gives
better performance than Johnson`s. In addition to the comparison of
Bayesian asymptotics, we try to extend Weng (2010a)`s Edgeworth
series for the distribution of a single parameter to the joint
distribution of all parameters. Since the calculation is quite
complicated, we only derive expansions for the two-parameter case
and apply it to the experiment of multi-stage data. Markov Chain
Monte Carlo (MCMC) is a popular method for making Bayesian
inference. However, convergence of the chain is always an issue.
Most of convergence diagnosis in the literature is based solely on
the simulation output. In this dissertation, we proposed a graphical
method for convergence diagnosis of the MCMC sequence. We used some
generalized linear models and mixture normal models for simulation
study. In summary, the goals of this dissertation are threefold: to
compare some results in Bayesian asymptotics, to study the expansion
for the joint distribution of the parameters and its applications,
and to propose a method for convergence diagnosis of the MCMC sequence.

Key words: Edgeworth expansion; Markov Chain Monte Carlo;
marginal posterior distribution; Stein`s identity.
參考文獻 Albert, J. H. and Chib, S. (1993), “Bayesian analysis of binary and polychotomous responsedata”, Journal of the American Statistical Association, 88, 669–679.
Brooks, S. P. and Gelman, A. (1998), “General methods for monitoring convergence ofiterative simulations”, Journal of Computational and Graphical Statistics, 7, 434–455.
Dellaportas, P. and Smith, A. F. M. (1993), “Bayesian inference for generalized linear andproportional hazards models via gibbs sampling”, Appl. Statist., 42, 443–460.
Erdelyi, A. (1956), Asympotic Expansions, New York: Dover Publications.
Finney, D. J. (1947), “The estimation from individual records of the relationship betweendose and quantal response”, Biometrika, 34, 320–334.
Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995), Bayesian data analysis,New York : Chapman and Hall.4.
Gelman, A. and Rubin, D. B. (1992), “Inference form iterative simulation using multiplesequences”, Statistical Science, 7, 457–511.
Geman, S. and Geman, D. (1984), “Stochastic relaxation, gibbs distributions and thebayesian restoration of images”, IEEE Trans. Pattn. Anal. Mach. Intel, 6, 721–741.
Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1995), Markov Chain Monte Carloin Practice, London: Chapman and Hall.
Gilks, W. R. and Wild, P. (1992), “Adaptive rejection sampling for gibbs sampling”, Appl.Statist., 41, 337–348.
Glickman, Mark E. (1993), “Paired comparison models with time-varying parameters”,Ph.D. thesis, Department of Statistics, Harvard University.
Hinde, J. (1982), Compound Poisson regression models, New York: Springer.
Hurn, M. W., Barker, N. W., and Magath, T. D. (1945), “The determination of prothrombintime following the administration of dicumarol with specific reference to thromboplastin”,Med., 30, 432–447.
Johnson, R. (1967), “An asymptotic expansion for posterior distributions”, Ann. Math.Statist., 38, 1899–1906.
Johnson, R. (1970), “Asymptotic expansions associated with posterior distributions”, Ann. Math. Statist., 41, 851–864.
Kutner, M. H., Nachtsheim, C. J., and Neter, J. (2004), Applied linear regression models,Boston: McGraw.
Lunn, D.J., Thomas, A., Best, N., and Spiegelhalter, D. (2000), “Winbugs – a Bayesian modelling framework: concepts structure and extensibility”, Statistics and Computing,10, 325–337.
Martin, Andrew D., Quinn, Kevin M., and Park, Jong Hee (2010), “MCMCpack:
Markov chain Monte Carlo (MCMC) Package”, R package version 2.10.0, URL
http://CRAN.R-project.org/package=MCMCpack.
McCullagh, P. and Nelder, J. A. (1989), Generalized linear models, New York : Chapman
and Hall.
Mendenhall, W. M., Parsons, J. T., Stringer, S. P., Cassissi, N. J., and Million, R. R.(1989), “T2 oral tongue carcinoma treated with radiotherapy: analysis of local control and complications”, Radiotherapy and Oncology, 16, 275–282.
Minka, T. P. (2001), “A family of algorithms for approximate bayesian inference”, Department of Electrical Engineering and Computer Science, 1–75.
Myers, R. H. (1990), Classical and Modern Regression with Applications, Boston: PWSKent.
R Development Core Team (2010), “R: A language and environment for statistical computing”, R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0, URL http://www.R-project.org.
Tanner, M. A. (1993), Tools for Statistical Inference, New York: Springer.
Tierney, L. and Kadane, J. B. (1986), “Accurate approximations for posterior moments and marginal densities”, Journal of the American Statistical Association, 81, 82–86.
Weng, R. C. (2003), “On Stein’s identity for posterior normality”, Statistica Sinica, 13, 495–506.
Weng, R. C. (2010a), “A Bayesian Edgeworth expansion by Stein’s Identity”, Bayesian Analysis, 5, 741–764.
Weng, R. C. (2010b), “A note on posterior distributions”, in JSM Proceedings, Section on
Bayesian Statistical Science, 2599–2608, Alexandria, VA: American Statistical Association.
Weng, R. C. and Hsu, C.-H. (2011), “A study of expansions of posterior distributions”, Communications in Statistics - Theory and Methods.
Weng, R. C. and Tsai, W.-C. (2008), “Asymptotic posterior normality for multiparameter problems”, Journal of Statistical Planning and Inference, 138, 4068–4080.
Woodroofe, M. (1989), “Very weak expansions for sequentially designed experiments: linear models”, Ann. Statist., 17, 1087–1102.
描述 博士
國立政治大學
統計研究所
93354504
99
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0093354504
資料類型 thesis
dc.contributor.advisor 翁久幸zh_TW
dc.contributor.advisor Weng, Chiu Hsingen_US
dc.contributor.author (作者) 許正宏zh_TW
dc.contributor.author (作者) Hsu, Cheng Hungen_US
dc.creator (作者) 許正宏zh_TW
dc.creator (作者) Hsu, Cheng Hungen_US
dc.date (日期) 2010en_US
dc.date.accessioned 5-十月-2011 14:31:53 (UTC+8)-
dc.date.available 5-十月-2011 14:31:53 (UTC+8)-
dc.date.issued (上傳時間) 5-十月-2011 14:31:53 (UTC+8)-
dc.identifier (其他 識別碼) G0093354504en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/51199-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計研究所zh_TW
dc.description (描述) 93354504zh_TW
dc.description (描述) 99zh_TW
dc.description.abstract (摘要) 本論文主要討論貝氏漸近的比較，推導出參數的聯合後驗分配與利用圖形來診斷馬可夫鏈蒙地卡羅的收斂。Johnson (1970)利用泰勒展開式得到個別後驗分配的展開式，此展開式是根據概似函數與先驗分配。 Weng (2010b) 和 Weng and Hsu (2011) 利用 Stein’s 等式且由概似函數與先驗分配估計後驗動差；將這些後驗動差代入Edgeworth 展開式得到近似後驗分配，此近似分配的誤差可精確到大O的負3/2次方與Johnson’s 相同。另外Weng and Hsu (2011)發現Weng (2010b) 和Johnson (1970)的近似展開式各別項誤差到大O的負1次方不一致，由模擬結果得到Weng’s 在此項表現比Johnson’s 好。另外由Weng (2010b)得到一維參數 的Edgeworth 近似後驗分配延伸到二維參數的聯合後驗分配；並應用二維參數的聯合後驗分配於多階段資料。本論文我們提出利用圖形來診斷馬可夫鏈蒙地卡羅收斂的方法，並且應用一般化線性模型與混合常態模型做為模擬。


關鍵字: Edgeworth 展開式；馬可夫鏈蒙地卡羅；個別後驗分配；Stein’s 等式		zh_TW
dc.description.abstract (摘要) Johnson (1970) obtained expansions for marginal posterior
distributions through Taylor expansions. The expansion in
Johnson (1970) is expressed in terms of the likelihood and the prior. Weng (2010b) and Weng and Hsu (2011) showed that by using
Stein`s identity we can approximate the posterior moments in terms
of the likelihood and the prior; then substituting these
approximations into an Edgeworth series one can obtain an expansion
which is correct to O(t{-3/2}), similar to Johnson`s.
Weng and Hsu (2011) found that the O(t{-1}) terms in
Weng (2010b) and Johnson (1970) do not agree and further
compared these two expansions by simulation study. The simulations
confirmed this finding and revealed that our O(t{-1}) term gives
better performance than Johnson`s. In addition to the comparison of
Bayesian asymptotics, we try to extend Weng (2010a)`s Edgeworth
series for the distribution of a single parameter to the joint
distribution of all parameters. Since the calculation is quite
complicated, we only derive expansions for the two-parameter case
and apply it to the experiment of multi-stage data. Markov Chain
Monte Carlo (MCMC) is a popular method for making Bayesian
inference. However, convergence of the chain is always an issue.
Most of convergence diagnosis in the literature is based solely on
the simulation output. In this dissertation, we proposed a graphical
method for convergence diagnosis of the MCMC sequence. We used some
generalized linear models and mixture normal models for simulation
study. In summary, the goals of this dissertation are threefold: to
compare some results in Bayesian asymptotics, to study the expansion
for the joint distribution of the parameters and its applications,
and to propose a method for convergence diagnosis of the MCMC sequence.

Key words: Edgeworth expansion; Markov Chain Monte Carlo;
marginal posterior distribution; Stein`s identity.		en_US
dc.description.tableofcontents 1 Introduction 1
2 Preliminaries 3
2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
2.2 Stein’s Identity and Bayesian Edgeworth Expansion . . . . . . . . . . . . .4
2.3 The Laplace Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
2.4 The Gibbs Sampling and MCMC Convergence Diagnostic . . . . . . . . . . 8
2.5 The Generalized Linear Model . . . . . . . . . . . . . . . . . . . . . . . . .9
3 Theoretical Results 10
3.1 Validation of Simulation Results . . . . . . . . . . . . . . . . . . . . . . . .10
3.2 Joint Posterior Distributions . . . . . . . . . . . . . . . . . . . . . . . . . .13
4 Experimental Results 20
4.1 Comparison of Second Order Approximations . . . . . . . . . . . . . . . . .20
4.1.1 Comparison with Johnson (1970) . . . . . . . . . . . . . . . . . . . . 20
4.1.2 Comparison with Tierney and Kadane (1986) . . . . . . . . . . . . . 25
4.2 Multi-stage Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
4.2.1 Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28
4.2.2 Two-parameter Logit Model . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Logit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
4.4 Poisson Regression Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
4.5 Gamma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33
4.6 Mixture Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
5 Concluding Remarks..................37		zh_TW
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0093354504en_US
dc.subject (關鍵詞) Edgeworth 展開式zh_TW
dc.subject (關鍵詞) 馬可夫鏈蒙地卡羅zh_TW
dc.subject (關鍵詞) 個別後驗分配zh_TW
dc.subject (關鍵詞) Stein’s 等式zh_TW
dc.subject (關鍵詞) Edgeworth expansionen_US
dc.subject (關鍵詞) Markov chain Monte Carloen_US
dc.subject (關鍵詞) marginal posterior distributionen_US
dc.subject (關鍵詞) Stein`s identityen_US
dc.title (題名) 馬可夫鏈蒙地卡羅收斂的研究與貝氏漸進的表現zh_TW
dc.title (題名) A study of mcmc convergence and performance evaluation of bayesian asymptoticsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) Albert, J. H. and Chib, S. (1993), “Bayesian analysis of binary and polychotomous responsedata”, Journal of the American Statistical Association, 88, 669–679.zh_TW
dc.relation.reference (參考文獻) Brooks, S. P. and Gelman, A. (1998), “General methods for monitoring convergence ofiterative simulations”, Journal of Computational and Graphical Statistics, 7, 434–455.zh_TW
dc.relation.reference (參考文獻) Dellaportas, P. and Smith, A. F. M. (1993), “Bayesian inference for generalized linear andproportional hazards models via gibbs sampling”, Appl. Statist., 42, 443–460.zh_TW
dc.relation.reference (參考文獻) Erdelyi, A. (1956), Asympotic Expansions, New York: Dover Publications.zh_TW
dc.relation.reference (參考文獻) Finney, D. J. (1947), “The estimation from individual records of the relationship betweendose and quantal response”, Biometrika, 34, 320–334.zh_TW
dc.relation.reference (參考文獻) Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995), Bayesian data analysis,New York : Chapman and Hall.4.zh_TW
dc.relation.reference (參考文獻) Gelman, A. and Rubin, D. B. (1992), “Inference form iterative simulation using multiplesequences”, Statistical Science, 7, 457–511.zh_TW
dc.relation.reference (參考文獻) Geman, S. and Geman, D. (1984), “Stochastic relaxation, gibbs distributions and thebayesian restoration of images”, IEEE Trans. Pattn. Anal. Mach. Intel, 6, 721–741.zh_TW
dc.relation.reference (參考文獻) Gilks, W. R., Richardson, S., and Spiegelhalter, D. J. (1995), Markov Chain Monte Carloin Practice, London: Chapman and Hall.zh_TW
dc.relation.reference (參考文獻) Gilks, W. R. and Wild, P. (1992), “Adaptive rejection sampling for gibbs sampling”, Appl.Statist., 41, 337–348.zh_TW
dc.relation.reference (參考文獻) Glickman, Mark E. (1993), “Paired comparison models with time-varying parameters”,Ph.D. thesis, Department of Statistics, Harvard University.zh_TW
dc.relation.reference (參考文獻) Hinde, J. (1982), Compound Poisson regression models, New York: Springer.zh_TW
dc.relation.reference (參考文獻) Hurn, M. W., Barker, N. W., and Magath, T. D. (1945), “The determination of prothrombintime following the administration of dicumarol with specific reference to thromboplastin”,Med., 30, 432–447.zh_TW
dc.relation.reference (參考文獻) Johnson, R. (1967), “An asymptotic expansion for posterior distributions”, Ann. Math.Statist., 38, 1899–1906.zh_TW
dc.relation.reference (參考文獻) Johnson, R. (1970), “Asymptotic expansions associated with posterior distributions”, Ann. Math. Statist., 41, 851–864.zh_TW
dc.relation.reference (參考文獻) Kutner, M. H., Nachtsheim, C. J., and Neter, J. (2004), Applied linear regression models,Boston: McGraw.zh_TW
dc.relation.reference (參考文獻) Lunn, D.J., Thomas, A., Best, N., and Spiegelhalter, D. (2000), “Winbugs – a Bayesian modelling framework: concepts structure and extensibility”, Statistics and Computing,10, 325–337.zh_TW
dc.relation.reference (參考文獻) Martin, Andrew D., Quinn, Kevin M., and Park, Jong Hee (2010), “MCMCpack:zh_TW
dc.relation.reference (參考文獻) Markov chain Monte Carlo (MCMC) Package”, R package version 2.10.0, URLzh_TW
dc.relation.reference (參考文獻) http://CRAN.R-project.org/package=MCMCpack.zh_TW
dc.relation.reference (參考文獻) McCullagh, P. and Nelder, J. A. (1989), Generalized linear models, New York : Chapmanzh_TW
dc.relation.reference (參考文獻) and Hall.zh_TW
dc.relation.reference (參考文獻) Mendenhall, W. M., Parsons, J. T., Stringer, S. P., Cassissi, N. J., and Million, R. R.(1989), “T2 oral tongue carcinoma treated with radiotherapy: analysis of local control and complications”, Radiotherapy and Oncology, 16, 275–282.zh_TW
dc.relation.reference (參考文獻) Minka, T. P. (2001), “A family of algorithms for approximate bayesian inference”, Department of Electrical Engineering and Computer Science, 1–75.zh_TW
dc.relation.reference (參考文獻) Myers, R. H. (1990), Classical and Modern Regression with Applications, Boston: PWSKent.zh_TW
dc.relation.reference (參考文獻) R Development Core Team (2010), “R: A language and environment for statistical computing”, R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0, URL http://www.R-project.org.zh_TW
dc.relation.reference (參考文獻) Tanner, M. A. (1993), Tools for Statistical Inference, New York: Springer.zh_TW
dc.relation.reference (參考文獻) Tierney, L. and Kadane, J. B. (1986), “Accurate approximations for posterior moments and marginal densities”, Journal of the American Statistical Association, 81, 82–86.zh_TW
dc.relation.reference (參考文獻) Weng, R. C. (2003), “On Stein’s identity for posterior normality”, Statistica Sinica, 13, 495–506.zh_TW
dc.relation.reference (參考文獻) Weng, R. C. (2010a), “A Bayesian Edgeworth expansion by Stein’s Identity”, Bayesian Analysis, 5, 741–764.zh_TW
dc.relation.reference (參考文獻) Weng, R. C. (2010b), “A note on posterior distributions”, in JSM Proceedings, Section onzh_TW
dc.relation.reference (參考文獻) Bayesian Statistical Science, 2599–2608, Alexandria, VA: American Statistical Association.zh_TW
dc.relation.reference (參考文獻) Weng, R. C. and Hsu, C.-H. (2011), “A study of expansions of posterior distributions”, Communications in Statistics - Theory and Methods.zh_TW
dc.relation.reference (參考文獻) Weng, R. C. and Tsai, W.-C. (2008), “Asymptotic posterior normality for multiparameter problems”, Journal of Statistical Planning and Inference, 138, 4068–4080.zh_TW
dc.relation.reference (參考文獻) Woodroofe, M. (1989), “Very weak expansions for sequentially designed experiments: linear models”, Ann. Statist., 17, 1087–1102.zh_TW