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.

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