贝叶斯多元线性回归

统计学中，贝叶斯多元线性回归是一种多元线性回归（预测结果为相关随机变量向量，而非标量随机变量的线性回归）的贝叶斯推断方法。这种方法的更一般论述见最小均方误差。 考虑一回归问题，其中需要预测的自变量不是实标量而是相关实数组成的m维向量。与标准回归设置一样，有n个观测值，其中每个观测i包含k−1个解释变量，归为k维向量 x i {\displaystyle \mathbf {x} _{i}} （添加值为1的虚拟变量，以允许截距系数）。对每个观测i，可以视作m个相关回归问题： y i , 1 = x i T β 1 + ϵ i , 1 ⋮ y i , m = x i T β m + ϵ i , m {\displaystyle {\begin{aligned}y_{i,1}&=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}_{1}+\epsilon _{i,1}\\&\;\;\vdots \\y_{i,m}&=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}_{m}+\epsilon _{i,m}\end{aligned}}} 其中误差集 { ϵ i , 1 , … , ϵ i , m } {\displaystyle \{\epsilon _{i,1},\ldots ,\epsilon _{i,m}\}} 都是相关的。等价地，也可以视作单一回归问题，其中结果是行向量 y i T {\displaystyle \mathbf {y} _{i}^{\mathsf {T}}} ，回归系数向量排在一起： y i T = x i T B + ϵ i T . {\displaystyle \mathbf {y} _{i}^{\mathsf {T}}=\mathbf {x} _{i}^{\mathsf {T}}\mathbf {B} +{\boldsymbol {\epsilon }}_{i}^{\mathsf {T}}.} 系数矩阵B是 k × m {\displaystyle k\times m} 矩阵，其中每个回归问题的系数向量 β 1 , … , β m {\displaystyle {\boldsymbol {\beta }}_{1},\ldots ,{\boldsymbol {\beta }}_{m}} 垂直排列在一起： B = [ ( β 1 ) ⋯ ( β m ) ] = [ ( β 1 , 1 ⋮ β k , 1 ) ⋯ ( β 1 , m ⋮ β k , m ) ] . {\displaystyle \mathbf {B} ={\begin{bmatrix}{\begin{pmatrix}\\{\boldsymbol {\beta }}_{1}\\\\\end{pmatrix}}\cdots {\begin{pmatrix}\\{\boldsymbol {\beta }}_{m}\\\\\end{pmatrix}}\end{bmatrix}}={\begin{bmatrix}{\begin{pmatrix}\beta _{1,1}\\\vdots \\\beta _{k,1}\end{pmatrix}}\cdots {\begin{pmatrix}\beta _{1,m}\\\vdots \\\beta _{k,m}\end{pmatrix}}\end{bmatrix}}.} 每个观测i的噪声向量 ϵ i {\displaystyle {\boldsymbol {\epsilon }}_{i}} 服从联合正态分布，因此给定观测的结果是相关的： ϵ i ∼ N ( 0 , Σ ϵ ) . {\displaystyle {\boldsymbol {\epsilon }}_{i}\sim N(0,{\boldsymbol {\Sigma }}_{\epsilon }).} 可以将整个回归问题写成矩阵形式： Y = X B + E , {\displaystyle \mathbf {Y} =\mathbf {X} \mathbf {B} +\mathbf {E} ,} 其中Y、E是 n × m {\displaystyle n\times m} 矩阵。设计矩阵X是 n × k {\displaystyle n\times k} 矩阵，观测如标准线性回归垂直排列： X = [ x 1 T x 2 T ⋮ x n T ] = [ x 1 , 1 ⋯ x 1 , k x 2 , 1 ⋯ x 2 , k ⋮ ⋱ ⋮ x n , 1 ⋯ x n , k ] . {\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {x} _{1}^{\mathsf {T}}\\\mathbf {x} _{2}^{\mathsf {T}}\\\vdots \\\mathbf {x} _{n}^{\mathsf {T}}\end{bmatrix}}={\begin{bmatrix}x_{1,1}&\cdots &x_{1,k}\\x_{2,1}&\cdots &x_{2,k}\\\vdots &\ddots &\vdots \\x_{n,1}&\cdots &x_{n,k}\end{bmatrix}}.} 经典频率学派线性最小二乘（英语：Linear least squares）解利用摩尔－彭若斯广义逆，简单地估计回归系数矩阵 B ^ {\displaystyle {\hat {\mathbf {B} }}} ： B ^ = ( X T X ) − 1 X T Y . {\displaystyle {\hat {\mathbf {B} }}=(\mathbf {X} ^{\mathsf {T}}\mathbf {X} )^{-1}\mathbf {X} ^{\mathsf {T}}\mathbf {Y} .} 求贝叶斯解，要先指定条件似然，再找到适当的共轭先验。与线性贝叶斯回归不同，可以指定一个自然的条件共轭先验（与规模相关）。 把条件似然写成 ρ ( E | Σ ϵ ) ∝ | Σ ϵ | − n / 2 exp ⁡ ( − 1 2 tr ⁡ ( E T E Σ ϵ − 1 ) ) , {\displaystyle \rho (\mathbf {E} |{\boldsymbol {\Sigma }}_{\epsilon })\propto |{\boldsymbol {\Sigma }}_{\epsilon }|^{-n/2}\exp \left(-{\tfrac {1}{2}}\operatorname {tr} \left(\mathbf {E} ^{\mathsf {T}}\mathbf {E} {\boldsymbol {\Sigma }}_{\epsilon }^{-1}\right)\right),} 误差 E {\displaystyle \mathbf {E} } 表为 Y , X , B {\displaystyle \mathbf {Y} ,\mathbf {X} ,\mathbf {B} } ，则有 ρ ( Y | X , B , Σ ϵ ) ∝ | Σ ϵ | − n / 2 exp ⁡ ( − 1 2 tr ⁡ ( ( Y − X B ) T ( Y − X B ) Σ ϵ − 1 ) ) , {\displaystyle \rho (\mathbf {Y} |\mathbf {X} ,\mathbf {B} ,{\boldsymbol {\Sigma }}_{\epsilon })\propto |{\boldsymbol {\Sigma }}_{\epsilon }|^{-n/2}\exp(-{\tfrac {1}{2}}\operatorname {tr} ((\mathbf {Y} -\mathbf {X} \mathbf {B} )^{\mathsf {T}}(\mathbf {Y} -\mathbf {X} \mathbf {B} ){\boldsymbol {\Sigma }}_{\epsilon }^{-1})),} 寻找一个自然共轭先验——联合密度 ρ ( B , Σ ϵ ) {\displaystyle \rho (\mathbf {B} ,\Sigma _{\epsilon })} ，其泛函形式与似然相同。由于似然在 B {\displaystyle \mathbf {B} } 中是二次的，因此我们重写似然使其在 ( B − B ^ ) {\displaystyle (\mathbf {B} -{\hat {\mathbf {B} }})} （与经典样本估计的差）是正态的。 用与贝叶斯线性回归（英语：Bayesian linear regression）相同的技术，可用矩阵形式的平方和分解指数项。不过此处还要用到矩阵微分（克罗内克积和向量化变换）。 首先，应用平方和得到新的似然表达式： ρ ( Y | X , B , Σ ϵ ) ∝ | Σ ϵ | − ( n − k ) / 2 exp ⁡ ( − tr ⁡ ( 1 2 S T S Σ ϵ − 1 ) ) | Σ ϵ | − k / 2 exp ⁡ ( − 1 2 tr ⁡ ( ( B − B ^ ) T X T X ( B − B ^ ) Σ ϵ − 1 ) ) , {\displaystyle \rho (\mathbf {Y} |\mathbf {X} ,\mathbf {B} ,{\boldsymbol {\Sigma }}_{\epsilon })\propto |{\boldsymbol {\Sigma }}_{\epsilon }|^{-(n-k)/2}\exp(-\operatorname {tr} ({\tfrac {1}{2}}\mathbf {S} ^{\mathsf {T}}\mathbf {S} {\boldsymbol {\Sigma }}_{\epsilon }^{-1}))|{\boldsymbol {\Sigma }}_{\epsilon }|^{-k/2}\exp(-{\tfrac {1}{2}}\operatorname {tr} ((\mathbf {B} -{\hat {\mathbf {B} }})^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} (\mathbf {B} -{\hat {\mathbf {B} }}){\boldsymbol {\Sigma }}_{\epsilon }^{-1})),} S = Y − X B ^ {\displaystyle \mathbf {S} =\mathbf {Y} -\mathbf {X} {\hat {\mathbf {B} }}} 我们想开发一种先验的条件形式： ρ ( B , Σ ϵ ) = ρ ( Σ ϵ ) ρ ( B | Σ ϵ ) , {\displaystyle \rho (\mathbf {B} ,{\boldsymbol {\Sigma }}_{\epsilon })=\rho ({\boldsymbol {\Sigma }}_{\epsilon })\rho (\mathbf {B} |{\boldsymbol {\Sigma }}_{\epsilon }),} 其中 ρ ( Σ ϵ ) {\displaystyle \rho ({\boldsymbol {\Sigma }}_{\epsilon })} 服从逆威沙特分布， ρ ( B | Σ ϵ ) {\displaystyle \rho (\mathbf {B} |{\boldsymbol {\Sigma }}_{\epsilon })} 是矩阵 B {\displaystyle \mathbf {B} } 中某种形式的正态分布。这是通过向量化变换实现的，它将似然从矩阵 B , B ^ {\displaystyle \mathbf {B} ,{\hat {\mathbf {B} }}} 的函数变换为向量 β = vec ⁡ ( B ) , β ^ = vec ⁡ ( B ^ ) {\displaystyle {\boldsymbol {\beta }}=\operatorname {vec} (\mathbf {B} ),{\hat {\boldsymbol {\beta }}}=\operatorname {vec} ({\hat {\mathbf {B} }})} 的函数。 tr ⁡ ( ( B − B ^ ) T X T X ( B − B ^ ) Σ ϵ − 1 ) = vec ⁡ ( B − B ^ ) T vec ⁡ ( X T X ( B − B ^ ) Σ ϵ − 1 ) {\displaystyle \operatorname {tr} ((\mathbf {B} -{\hat {\mathbf {B} }})^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {X} (\mathbf {B} -{\hat {\mathbf {B} }}){\boldsymbol {\Sigma }}_{\epsilon }^{-1})=\operatorname {vec} (\mathbf {B} -{\hat {\mathbf {B} }})^{\mathsf {T}}\operatorname {vec} (\mathbf {X} ^{\mathsf {T}}\mathbf {X} (\mathbf {B} -{\hat {\mathbf {B} }}){\boldsymbol {\Sigma }}_{\epsilon }^{-1})} 令 vec ⁡ ( X T X ( B − B ^ ) Σ ϵ − 1 ) = ( Σ ϵ − 1 ⊗ X T X ) vec ⁡ ( B − B ^ ) , {\displaystyle \operatorname {vec} (\mathbf {X} ^{\mathsf {T}}\mathbf {X} (\mathbf {B} -{\hat {\mathbf {B} }}){\boldsymbol {\Sigma }}_{\epsilon }^{-1})=({\boldsymbol {\Sigma }}_{\epsilon }^{-1}\otimes \mathbf {X} ^{\mathsf {T}}\mathbf {X} )\operatorname {vec} (\mathbf {B} -{\hat {\mathbf {B} }}),} 其中 A ⊗ B {\displaystyle \mathbf {A} \otimes \mathbf {B} } 表示矩阵A、B的克罗内克积，其是外积的推广。 则 vec ⁡ ( B − B ^ ) T ( Σ ϵ − 1 ⊗ X T X ) vec ⁡ ( B − B ^ ) = ( β − β ^ ) T ( Σ ϵ − 1 ⊗ X T X ) ( β − β ^ ) {\displaystyle {\begin{aligned}&\operatorname {vec} (\mathbf {B} -{\hat {\mathbf {B} }})^{\mathsf {T}}({\boldsymbol {\Sigma }}_{\epsilon }^{-1}\otimes \mathbf {X} ^{\mathsf {T}}\mathbf {X} )\operatorname {vec} (\mathbf {B} -{\hat {\mathbf {B} }})\\&=({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})^{\mathsf {T}}({\boldsymbol {\Sigma }}_{\epsilon }^{-1}\otimes \mathbf {X} ^{\mathsf {T}}\mathbf {X} )({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})\end{aligned}}} 产生的似然在 ( β − β ^ ) {\displaystyle ({\boldsymbol {\beta }}-{\hat {\boldsymbol {\beta }}})} 中正态。 有了更易理解的似然，就可以找到自然的（条件）共轭先验了。 由向量化的 β {\displaystyle {\boldsymbol {\beta }}} 得到的自然共轭先验形式为 ρ ( β , Σ ϵ ) = ρ ( Σ ϵ ) ρ ( β | Σ ϵ ) , {\displaystyle \rho ({\boldsymbol {\beta }},{\boldsymbol {\Sigma }}_{\epsilon })=\rho ({\boldsymbol {\Sigma }}_{\epsilon })\rho ({\boldsymbol {\beta }}|{\boldsymbol {\Sigma }}_{\epsilon }),} 其中 ρ ( Σ ϵ ) ∼ W − 1 ( V 0 , ν 0 ) {\displaystyle \rho ({\boldsymbol {\Sigma }}_{\epsilon })\sim {\mathcal {W}}^{-1}(\mathbf {V} _{0},{\boldsymbol {\nu }}_{0})} ρ ( β | Σ ϵ ) ∼ N ( β 0 , Σ ϵ ⊗ Λ 0 − 1 ) . {\displaystyle \rho ({\boldsymbol {\beta }}|{\boldsymbol {\Sigma }}_{\epsilon })\sim N({\boldsymbol {\beta }}_{0},{\boldsymbol {\Sigma }}_{\epsilon }\otimes {\boldsymbol {\Lambda }}_{0}^{-1}).} 利用上述先验与似然，可得到后验 ρ ( β , Σ ϵ | Y , X ) ∝ | Σ ϵ | − ( ν 0 + m + 1 ) / 2 exp ⁡ ( − 1 2 tr ⁡ ( V 0 Σ ϵ − 1 ) ) × | Σ ϵ | − k / 2 exp ⁡ ( − 1 2 tr ⁡ ( ( B − B 0 ) T Λ 0 ( B − B 0 ) Σ ϵ − 1 ) ) × | Σ ϵ | − n / 2 exp ⁡ ( − 1 2 tr ⁡ ( ( Y − X B ) T ( Y − X B ) Σ ϵ − 1 ) ) , {\displaystyle {\begin{aligned}\rho ({\boldsymbol {\beta }},{\boldsymbol {\Sigma }}_{\epsilon }|\mathbf {Y} ,\mathbf {X} )\propto {}&|{\boldsymbol {\Sigma }}_{\epsilon }|^{-({\boldsymbol {\nu }}_{0}+m+1)/2}\exp {(-{\tfrac {1}{2}}\operatorname {tr} (\mathbf {V} _{0}{\boldsymbol {\Sigma }}_{\epsilon }^{-1}))}\\&\times |{\boldsymbol {\Sigma }}_{\epsilon }|^{-k/2}\exp {(-{\tfrac {1}{2}}\operatorname {tr} ((\mathbf {B} -\mathbf {B} _{0})^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}(\mathbf {B} -\mathbf {B} _{0}){\boldsymbol {\Sigma }}_{\epsilon }^{-1}))}\\&\times |{\boldsymbol {\Sigma }}_{\epsilon }|^{-n/2}\exp {(-{\tfrac {1}{2}}\operatorname {tr} ((\mathbf {Y} -\mathbf {XB} )^{\mathsf {T}}(\mathbf {Y} -\mathbf {XB} ){\boldsymbol {\Sigma }}_{\epsilon }^{-1}))},\end{aligned}}} 其中 vec ⁡ ( B 0 ) = β 0 {\displaystyle \operatorname {vec} (\mathbf {B} _{0})={\boldsymbol {\beta }}_{0}} 。 记 Λ 0 = U T U {\displaystyle {\boldsymbol {\Lambda }}_{0}=\mathbf {U} ^{\mathsf {T}}\mathbf {U} } ，涉及 B {\displaystyle \mathbf {B} } 的项可以分类为 ( B − B 0 ) T Λ 0 ( B − B 0 ) + ( Y − X B ) T ( Y − X B ) = ( [ Y U B 0 ] − [ X U ] B ) T ( [ Y U B 0 ] − [ X U ] B ) = ( [ Y U B 0 ] − [ X U ] B n ) T ( [ Y U B 0 ] − [ X U ] B n ) + ( B − B n ) T ( X T X + Λ 0 ) ( B − B n ) = ( Y − X B n ) T ( Y − X B n ) + ( B 0 − B n ) T Λ 0 ( B 0 − B n ) + ( B − B n ) T ( X T X + Λ 0 ) ( B − B n ) , {\displaystyle {\begin{aligned}&\left(\mathbf {B} -\mathbf {B} _{0}\right)^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}\left(\mathbf {B} -\mathbf {B} _{0}\right)+\left(\mathbf {Y} -\mathbf {XB} \right)^{\mathsf {T}}\left(\mathbf {Y} -\mathbf {XB} \right)\\={}&\left({\begin{bmatrix}\mathbf {Y} \\\mathbf {U} \mathbf {B} _{0}\end{bmatrix}}-{\begin{bmatrix}\mathbf {X} \\\mathbf {U} \end{bmatrix}}\mathbf {B} \right)^{\mathsf {T}}\left({\begin{bmatrix}\mathbf {Y} \\\mathbf {U} \mathbf {B} _{0}\end{bmatrix}}-{\begin{bmatrix}\mathbf {X} \\\mathbf {U} \end{bmatrix}}\mathbf {B} \right)\\={}&\left({\begin{bmatrix}\mathbf {Y} \\\mathbf {U} \mathbf {B} _{0}\end{bmatrix}}-{\begin{bmatrix}\mathbf {X} \\\mathbf {U} \end{bmatrix}}\mathbf {B} _{n}\right)^{\mathsf {T}}\left({\begin{bmatrix}\mathbf {Y} \\\mathbf {U} \mathbf {B} _{0}\end{bmatrix}}-{\begin{bmatrix}\mathbf {X} \\\mathbf {U} \end{bmatrix}}\mathbf {B} _{n}\right)+\left(\mathbf {B} -\mathbf {B} _{n}\right)^{\mathsf {T}}\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}\right)\left(\mathbf {B} -\mathbf {B} _{n}\right)\\={}&\left(\mathbf {Y} -\mathbf {X} \mathbf {B} _{n}\right)^{\mathsf {T}}\left(\mathbf {Y} -\mathbf {X} \mathbf {B} _{n}\right)+\left(\mathbf {B} _{0}-\mathbf {B} _{n}\right)^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}\left(\mathbf {B} _{0}-\mathbf {B} _{n}\right)+\left(\mathbf {B} -\mathbf {B} _{n}\right)^{\mathsf {T}}\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}\right)\left(\mathbf {B} -\mathbf {B} _{n}\right),\end{aligned}}} 其中 B n = ( X T X + Λ 0 ) − 1 ( X T X B ^ + Λ 0 B 0 ) = ( X T X + Λ 0 ) − 1 ( X T Y + Λ 0 B 0 ) . {\displaystyle \mathbf {B} _{n}=\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}\right)^{-1}\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} {\hat {\mathbf {B} }}+{\boldsymbol {\Lambda }}_{0}\mathbf {B} _{0}\right)=\left(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}\right)^{-1}\left(\mathbf {X} ^{\mathsf {T}}\mathbf {Y} +{\boldsymbol {\Lambda }}_{0}\mathbf {B} _{0}\right).} 现在可以用更有用的形式来写后验： ρ ( β , Σ ϵ | Y , X ) ∝ | Σ ϵ | − ( ν 0 + m + n + 1 ) / 2 exp ⁡ ( − 1 2 tr ⁡ ( ( V 0 + ( Y − X B n ) T ( Y − X B n ) + ( B n − B 0 ) T Λ 0 ( B n − B 0 ) ) Σ ϵ − 1 ) ) × | Σ ϵ | − k / 2 exp ⁡ ( − 1 2 tr ⁡ ( ( B − B n ) T ( X T X + Λ 0 ) ( B − B n ) Σ ϵ − 1 ) ) . {\displaystyle {\begin{aligned}\rho ({\boldsymbol {\beta }},{\boldsymbol {\Sigma }}_{\epsilon }|\mathbf {Y} ,\mathbf {X} )\propto {}&|{\boldsymbol {\Sigma }}_{\epsilon }|^{-({\boldsymbol {\nu }}_{0}+m+n+1)/2}\exp {(-{\tfrac {1}{2}}\operatorname {tr} ((\mathbf {V} _{0}+(\mathbf {Y} -\mathbf {XB_{n}} )^{\mathsf {T}}(\mathbf {Y} -\mathbf {XB_{n}} )+(\mathbf {B} _{n}-\mathbf {B} _{0})^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}(\mathbf {B} _{n}-\mathbf {B} _{0})){\boldsymbol {\Sigma }}_{\epsilon }^{-1}))}\\&\times |{\boldsymbol {\Sigma }}_{\epsilon }|^{-k/2}\exp {(-{\tfrac {1}{2}}\operatorname {tr} ((\mathbf {B} -\mathbf {B} _{n})^{\mathsf {T}}(\mathbf {X} ^{T}\mathbf {X} +{\boldsymbol {\Lambda }}_{0})(\mathbf {B} -\mathbf {B} _{n}){\boldsymbol {\Sigma }}_{\epsilon }^{-1}))}.\end{aligned}}} 其形式为逆威沙特分布乘以矩阵正态分布： ρ ( Σ ϵ | Y , X ) ∼ W − 1 ( V n , ν n ) {\displaystyle \rho ({\boldsymbol {\Sigma }}_{\epsilon }|\mathbf {Y} ,\mathbf {X} )\sim {\mathcal {W}}^{-1}(\mathbf {V} _{n},{\boldsymbol {\nu }}_{n})} ρ ( B | Y , X , Σ ϵ ) ∼ M N k , m ( B n , Λ n − 1 , Σ ϵ ) . {\displaystyle \rho (\mathbf {B} |\mathbf {Y} ,\mathbf {X} ,{\boldsymbol {\Sigma }}_{\epsilon })\sim {\mathcal {MN}}_{k,m}(\mathbf {B} _{n},{\boldsymbol {\Lambda }}_{n}^{-1},{\boldsymbol {\Sigma }}_{\epsilon }).} 此后验的参数由下式给出 V n = V 0 + ( Y − X B n ) T ( Y − X B n ) + ( B n − B 0 ) T Λ 0 ( B n − B 0 ) {\displaystyle \mathbf {V} _{n}=\mathbf {V} _{0}+(\mathbf {Y} -\mathbf {XB_{n}} )^{\mathsf {T}}(\mathbf {Y} -\mathbf {XB_{n}} )+(\mathbf {B} _{n}-\mathbf {B} _{0})^{\mathsf {T}}{\boldsymbol {\Lambda }}_{0}(\mathbf {B} _{n}-\mathbf {B} _{0})} ν n = ν 0 + n {\displaystyle {\boldsymbol {\nu }}_{n}={\boldsymbol {\nu }}_{0}+n} B n = ( X T X + Λ 0 ) − 1 ( X T Y + Λ 0 B 0 ) {\displaystyle \mathbf {B} _{n}=(\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0})^{-1}(\mathbf {X} ^{\mathsf {T}}\mathbf {Y} +{\boldsymbol {\Lambda }}_{0}\mathbf {B} _{0})} Λ n = X T X + Λ 0 {\displaystyle {\boldsymbol {\Lambda }}_{n}=\mathbf {X} ^{\mathsf {T}}\mathbf {X} +{\boldsymbol {\Lambda }}_{0}} 贝叶斯线性回归（英语：Bayesian linear regression） 矩阵正态分布 Peter E. Rossi, Greg M. Allenby, Rob McCulloch. Bayesian Statistics and Marketing. John Wiley & Sons, 2012, p. 32. Box, G. E. P.; Tiao, G. C. 8. Bayesian Inference in Statistical Analysis. Wiley. 1973. ISBN 0-471-57428-7.  Geisser, S. Bayesian Estimation in Multivariate Analysis. The Annals of Mathematical Statistics. 1965, 36 (1): 150–159. JSTOR 2238083.  Tiao, G. C.; Zellner, A. On the Bayesian Estimation of Multivariate Regression. Journal of the Royal Statistical Society. Series B (Methodological). 1964, 26 (2): 277–285. JSTOR 2984424.