高維度且高度相關的迴歸資料是一個具有挑戰性的問題，因為解釋變數 之間若存在強烈共線性，可能導致迴歸係數估計不穩定、變數選取結果不可 靠，以及重要訊號被過度收縮。雖然TPBN-MBSP 模型提供了一個有效的 Bayesian global-local shrinkage 架構，可用於高維度稀疏迴歸分析，但其原始 先驗結構並未明確納入解釋變數之間的相關性資訊。為了改善此限制，本論 文提出一個修正後的TPBN-MBSP方法，透過引入考量相關性結構的先驗共 變異矩陣來描述迴歸係數之間的相依關係。 本文所提出的方法先由解釋變數的樣本相關係數矩陣建立 predictor graph，並利用對應的 graph Laplacian matrix 將相依資訊納入迴歸係數的 先驗分布中。具體而言，本文將先驗共變異矩陣修改為 Σβ =DψRβDψ +ϕL， 其中 Dψ 保留 local shrinkage 結構，Rβ 用以捕捉迴歸係數之間的相關性， 而 L 則反映高度相關解釋變數之間的圖形結構。此外，本文也使用onion method 在後驗計算過程中產生有效且正定的相關矩陣。 模擬研究結果顯示，在不同相關性設定下，本文所提出的方法特別適用 於解釋變數具有高度相關或近乎完全相關的情境。相較於原始TPBN-MBSP 模型，本文方法能夠降低相關雜訊變數所造成的假訊號，並在嚴重共線性的 情況下提升估計穩定性。進一步透過鑽石價格資料與房價資料進行實際資料 分析，結果顯示本文方法在維持具競爭力的預測表現下，能夠提供更具彈性 且考量相關性結構的變數選取架構。整體而言，本文所提出的方法可被視為 TPBN-MBSP 在處理高維度且高度相關迴歸問題時的一個結構化延伸。 關鍵字: 變數選擇; 收縮先驗; 回歸分析; 相關性結構;High-dimensional regression with highly correlated predictors is a challeng ing problem because strong multicollinearity may lead to unstable coefficient estimation， unreliable variable selection， and excessive shrinkage of important signals. Although the TPBN-MBSP model provides an effective Bayesian global-local shrinkage framework for sparse high-dimensional regression， its original prior structure does not explicitly incorporate the correlation infor mation among predictors. To address this limitation， this thesis proposes a modified TPBN-MBSP method by introducing a correlation-aware prior co variance structure for the regression coefficients. The proposed method constructs a predictor graph from the empirical corre lation matrix and uses the corresponding graph Laplacian matrix to incorporate dependence information into the coefficient prior. Specifically， the prior covari ance matrix is modified as Σβ =DψRβDψ +ϕL， where Dψ retains the local shrinkage structure， Rβ captures coefficient-level cor relation， and L reflects the graph structure among highly correlated predictors. The onion method is also used to generate valid positive definite correlation matrices during posterior computation. Simulation studies under different correlation settings show that the pro posed method is especially useful when predictors are strongly or nearly per fectly correlated. Compared with the original TPBN-MBSP model， the pro posed method can reduce false signals among correlated noise variables and improve estimation stability under severe multicollinearity. Real data analyses using diamond price data and house price data further demonstrate that the proposed method maintains competitive predictive performance while provid ing a more flexible and correlation-aware variable selection framework. Overall， the proposed method can be viewed as a structured extension of TPBN-MBSP for high-dimensional regression problems involving highly correlated predictors. Keywords: Variable selection; Shrinkage prior; Regression analysis; Correla tion structure
