馬可夫鏈蒙地卡羅法（Markov chain Monte Carlo，MCMC）在貝氏統計中是一種隨機化方法，在處理複雜函數時表現較好，但需要較長時間收斂。相對而言，貝氏統計中的拉普拉斯近似法較為快速，雖然對於後驗分配的估計較不精準，但在處理大規模資料上，使用這些方法仍是可考慮的選項。
關於顧客購買時間間隔的資料，過去的研究或使用最大概似估計，或使用層級貝氏模型並以MCMC方法進行估計。前者雖計算速度較快，但有過度擬合（overfitting）的情形。後者雖準確度較高，但以MCMC方法運算費時，在大規模資料較不適用。本論文考慮層級貝氏模型以避免過度擬合，再以最大後驗估計取代MCMC之估計，來處理較大比數的資料並提升預測率。
總結而言，本研究結合層級貝氏模型與最大後驗估計，應用於顧客購買時間間隔的問題。本研究的結果顯示，最大後驗估計相較於最大概似估計，雖計算時間稍長，但在AUC有得到提升；而最大後驗估計在計算量上遠低於MCMC，故較適合應用於大規模資料。

Markov chain Monte Carlo (MCMC) is a stochastic method in Bayesian statistics that performs well when dealing with complex functions but requires a longer convergence time. In contrast, several approximate methods in Bayesian statistics are faster but may have lower accuracy. However, these methods can still be considered when handling large-scale data. In previous studies on customers’ interpurchase times , researchers have either used maximum likelihood estimation or employed hierarchical Bayesian models with MCMC for estimation. The former method is faster in computation but prone to overfitting. The latter method offers higher accuracy but is time-consuming when using MCMC, making it less suitable for large-scale data. This paper considers hierarchical Bayesian models to avoid overfitting and replaces MCMC estimation with maximum a posteriori estimation to handle larger datasets and improve prediction accuracy. In summary, this study combines hierarchical Bayesian models with maximum a posteriori estimation for the problem of customers’ interpurchase times. The results show that, compared to maximum likelihood estimation, maximum a posteriori estimation has slightly longer computation time but yields improvements in AUC estimation. On the other hand, maximum a posteriori estimation has significantly lower computational requirements than MCMC, making it more suitable for large-scale data applications.

郭瑞祥、蔣明晃、陳薏棻、楊凱全，”應用層級被式理論於跨商品類別之顧客 購 買期間預測模型”，管理學報，2009 蔣宛蓉，”廣義伽馬分配於顧客購買時間模型之應用”，2020
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