Appendix: Orthogonal Design Case
Lemma A.1.
Let,
$$ {\upbeta}_{j} \mid y \sim\frac{\alpha_{j}w_{1,j}}{W_{j}} \mathcal{N}\left( \hat{\upbeta}_{1,j}, {\sigma_{1}^{2}}\right) +\frac{(1-\alpha_{j})w_{0,j}}{W_{j}} \mathcal{N}\left( \hat{\upbeta}_{0,j}, {\sigma_{0}^{2}}\right). $$
(1)
Then, \(\mathcal {N}\left (\hat {\upbeta }_{1,j}, {\sigma _{1}^{2}}\right )\) dominates the posterior if
$$ \begin{array}{@{}rcl@{}} \hat{{{\upbeta}_{j}^{2}}} > \frac{\sigma^{2}}{n}\frac{(n{\tau_{1}^{2}}+1)(n{\tau_{0}^{2}}+1)}{n{\tau_{1}^{2}}-n{\tau_{0}^{2}}} \left[ 2\ln\left( \frac{1-\epsilon_{1}}{\epsilon_{1}}\right) +\ln\left( \frac{n{\tau_{1}^{2}}+1}{n{\tau_{0}^{2}}+1}\right)\right] \end{array} $$
(2)
and \(\mathcal {N}\left (\hat {\upbeta }_{0,j}, {\sigma _{0}^{2}}\right )\) dominates the posterior if,
$$ \begin{array}{@{}rcl@{}} \hat{{{\upbeta}_{j}^{2}}} < \frac{\sigma^{2}}{n}\frac{(n{\tau_{1}^{2}}+1)(n{\tau_{0}^{2}}+1)}{n{\tau_{1}^{2}}-n{\tau_{0}^{2}}} \left[ 2\ln\left( \frac{\epsilon_{2}}{1-\epsilon_{2}}\right) +\ln\left( \frac{n{\tau_{1}^{2}}+1}{n{\tau_{0}^{2}}+1}\right)\right]. \end{array} $$
(3)
Proof 4.
Exploiting the monotonicity property of the posterior odds, we can say that \(\mathcal {N}\left (\hat {\upbeta }_{1,j}, {\sigma _{1}^{2}}\right )\) dominates the posterior if \(\frac {\epsilon _{1}}{(1-\epsilon _{1})}\cdot \frac {w_{1,j}}{w_{0,j}}>1\). That is, if
$$ \begin{array}{@{}rcl@{}} \exp\left( -\frac{n\hat{{{\upbeta}_{j}^{2}}}}{2(n\sigma^{2}{\tau_{1}^{2}}+\sigma^{2})} + \frac{n\hat{{{\upbeta}_{j}^{2}}}}{2(n\sigma^{2}{\tau_{0}^{2}}+\sigma^{2})}\right) \!\!\!\!\!\!\!\!&&>\!\frac{(1 - \epsilon_{1})\sqrt{n{\tau_{1}^{2}}+1}}{\epsilon_{1}\sqrt{n{\tau_{0}^{2}}+1}} \end{array} $$
(4)
$$ \begin{array}{@{}rcl@{}} -\frac{n\hat{{{\upbeta}_{j}^{2}}}}{2(n\sigma^{2}{\tau_{1}^{2}}+\sigma^{2})} +\frac{n\hat{{{\upbeta}_{j}^{2}}}}{2(n\sigma^{2}{\tau_{0}^{2}}+\sigma^{2})} \!\!\!\!\!\!\!\!&&>\!\ln\left( \frac{(1 - \epsilon_{1})\sqrt{n{\tau_{1}^{2}}+1}}{\epsilon_{1}\sqrt{n{\tau_{0}^{2}}+1}}\right) \end{array} $$
(5)
$$ \begin{array}{@{}rcl@{}} \frac{n\hat{{{\upbeta}_{j}^{2}}}}{2\sigma^{2}} \left[-\frac{1}{(n{\tau_{1}^{2}}+1)}+\frac{1}{(n{\tau_{0}^{2}}+1)}\right] \!\!\!\!\!\!\!\!&&>\!\ln\left( \frac{(1 - \epsilon_{1})\sqrt{n{\tau_{1}^{2}}+1}}{\epsilon_{1}\sqrt{n{\tau_{0}^{2}}+1}}\right) \end{array} $$
(6)
$$ \begin{array}{@{}rcl@{}} \frac{n\hat{{{\upbeta}_{j}^{2}}}}{2\sigma^{2}} \frac{n{\tau_{1}^{2}}-n{\tau_{0}^{2}}}{(n{\tau_{1}^{2}}+1)(n{\tau_{0}^{2}}+1)} \!\!\!\!\!\!\!\!&&>\!\ln\left( \frac{(1 - \epsilon_{1})\sqrt{n{\tau_{1}^{2}}+1}}{\epsilon_{1}\sqrt{n{\tau_{0}^{2}}+1}}\right).~~~~~~~~~ \end{array} $$
(7)
Then after rearranging the terms on both sides, we get:
$$ \begin{array}{@{}rcl@{}} \hat{{{\upbeta}_{j}^{2}}} > \frac{\sigma^{2}}{n}\frac{(n{\tau_{1}^{2}}+1)(n{\tau_{0}^{2}}+1)}{n{\tau_{1}^{2}}-n{\tau_{0}^{2}}} \left[ 2\ln\left( \frac{1-\epsilon_{1}}{\epsilon_{1}}\right) +\ln\left( \frac{n{\tau_{1}^{2}}+1}{n{\tau_{0}^{2}}+1}\right)\right]. \end{array} $$
(8)
Similarly we can prove the other inequality. □
Lemma A.2.
Let \(g:(0,1)\to \mathbb {R}\) be defined as
$$ g(\alpha) := \frac{a\alpha + b}{c\alpha + d} $$
(9)
for some constants a, b, c, and \(d\in \mathbb {R}\) such that cα + d > 0 ∀α ∈ (0,1). Then g is monotonically increasing when ad − bc > 0 and monotonically decreasing when ad − bc < 0.
Proof 5.
The proof is straightforward and therefore we omit it. □
Note that, the posterior mean of βj can be written as:
$$ \begin{array}{@{}rcl@{}} E({\upbeta}_{j}\mid y) = \frac{(w_{1,j}\hat{\upbeta}_{1,j} - w_{0,j}\hat{\upbeta}_{1,j})\alpha_{j} + w_{0,j}\hat{\upbeta}_{0,j}} {(w_{1,j}-w_{0,j})\alpha_{j} + w_{0,j}}. \end{array} $$
(10)
Therefore, we have
$$ \begin{array}{@{}rcl@{}} \frac{d}{d\alpha_{j}}E({\upbeta}_{j}\mid y) \!\!\!\!\!\!\!\!\!&&= \frac{w_{0, j}w_{1, j} (\hat{\upbeta}_{1,j}-\hat{\upbeta}_{0,j})}{[(w_{1,j}-w_{0,j})\alpha_{j} + w_{0,j}]^{2}} \end{array} $$
(11)
$$ \begin{array}{@{}rcl@{}} &&= \frac{w_{0, j}w_{1, j}} {[(w_{1,j}-w_{0,j})\alpha_{j} + w_{0,j}]^{2}} \left( \frac{n{\tau_{1}^{2}}\hat{{\upbeta}_{j}}}{n{\tau_{1}^{2}} +1} -\frac{n{\tau_{0}^{2}}\hat{{\upbeta}_{j}}}{n{\tau_{0}^{2}} +1}\right) \end{array} $$
(12)
$$ \begin{array}{@{}rcl@{}} &&= \frac{w_{0, j}w_{1, j}\hat{\upbeta}_{j}} {[(w_{1,j}-w_{0,j})\alpha_{j} + w_{0,j}]^{2}} \left( \frac{n{\tau_{1}^{2}}}{n{\tau_{1}^{2}} +1} - \frac{n{\tau_{0}^{2}}}{n{\tau_{0}^{2}} +1}\right) \end{array} $$
(13)
$$ \begin{array}{@{}rcl@{}} &&= \frac{w_{0, j}w_{1, j}} {[(w_{1,j}-w_{0,j})\alpha_{j} + w_{0,j}]^{2}} \frac{n{\tau_{1}^{2}} - n{\tau_{0}^{2}}}{(n{\tau_{1}^{2}} +1)(n{\tau_{0}^{2}} +1)}\hat{\upbeta}_{j}. ~~~~~~~~~~~~~~\end{array} $$
(14)
Since τ1 > τ0, the posterior mean of βj is monotonically increasing with respect to αj for \(\hat {\upbeta }_{j} > 0\) and monotonically decreasing with respect to αj for \(\hat {\upbeta }_{j} < 0\).
Lemma A.3.
Let
$$ X\sim w_{1} f_{1} + w_{2} f_{2} $$
(15)
where fi denotes a normal density with mean μi and variance \({\sigma ^{2}_{i}}\) for i = 1,2. Then,
$$ Var(X) = {\sum}_{i=1}^{2} w_{i}({\sigma^{2}_{i}} + {\mu_{i}^{2}}) - \left( {\sum}_{i=1}^{2} w_{i}\mu_{i}\right)^{2}. $$
(16)
Proof 6.
First, note that
$$ \begin{array}{@{}rcl@{}} E(X^{2}){} &&=\int x^{2}[w_{1}f_{1}(x) + w_{2} f_{2}(x)]dx \end{array} $$
(17)
$$ \begin{array}{@{}rcl@{}} && = w_{1}\int x^{2}f_{1}(x)dx + w_{2}\int x^{2}f_{2}(x)dx \end{array} $$
(18)
$$ \begin{array}{@{}rcl@{}} && = w_{1}({\sigma^{2}_{1}} + {\mu_{1}^{2}}) + w_{2}({\sigma^{2}_{2}} + {\mu_{2}^{2}}). \end{array} $$
(19)
Consequently, Then, the variance of X is given by:
$$ \begin{array}{@{}rcl@{}} \text{Var}(X) &&= E(X^{2}) - \left[E(X)\right]^{2} \end{array} $$
(20)
$$ \begin{array}{@{}rcl@{}} && = {\sum}_{i=1}^{2}w_{i}({\sigma^{2}_{i}} + {\mu_{i}^{2}}) - \left( {\sum}_{i=1}^{2} w_{i}\mu_{i}\right)^{2}. \end{array} $$
(21)
□
Now, we know that,
$$ \begin{array}{@{}rcl@{}} {\upbeta}_{j} \mid y \sim\frac{\alpha_{j}w_{1,j}}{W_{j}} \mathcal{N}\left( \hat{\upbeta}_{1,j}, {\sigma_{1}^{2}}\right) +\frac{(1-\alpha_{j})w_{0,j}}{W_{j}} \mathcal{N}\left( \hat{\upbeta}_{0,j}, {\sigma_{0}^{2}}\right). \end{array} $$
(22)
Then from above lemma, we can show that the variance of βj∣y is given by:
$$ \begin{array}{@{}rcl@{}} \text{Var}({\upbeta}_{j}\mid y) &=& \frac{\alpha_{j}w_{1,j}{\sigma^{2}_{1}} + (1-\alpha_{j})w_{0,j}{\sigma^{2}_{0}}}{W_{j}}\\ &&+ \frac{\alpha(1-\alpha)w_{1,j}w_{0,j}(\hat{\upbeta}_{1,j} - \hat{\upbeta}_{0,j})^{2}} {{W_{j}^{2}}}. \end{array} $$
(23)
Appendix: Regression Coefficients
$$ \begin{array}{@{}rcl@{}} P(\upbeta\mid y) \!\!\!\!\!\!\!\!\!&&\overset{\upbeta}{\propto}\int \int {\sum}_{\gamma} P(\upbeta, \gamma, \sigma^{2}, q \mid y) dq d\sigma^{2} \end{array} $$
(1)
$$ \begin{array}{@{}rcl@{}} &&\overset{\upbeta}{\propto}\int P(y\mid \upbeta, \sigma^{2}) {\sum}_{\gamma}P(\upbeta \mid \gamma, \sigma^{2}) \left[\int P(\gamma\mid q) P(q) dq\right]\\&& P(\sigma^{2}) d \sigma^{2} \end{array} $$
(2)
$$ \begin{array}{@{}rcl@{}} &&\overset{\upbeta}{\propto}\int P(y\mid \upbeta, \sigma^{2}) {\prod}_{j}\left[\alpha_{j}f_{1}({\upbeta}_{j})+ (1-\alpha_{j})f_{0}({\upbeta}_{j})\right] P(\sigma^{2}) d \sigma^{2}.~~~~~~~~~~~ \end{array} $$
(3)
To simplify the product term in the above expression, we first provide the following identity:
Lemma B.1.
Let,
$$ f_{\gamma_{j}}({\upbeta}_{j}) := \frac{1}{\sqrt{2\pi\sigma^{2}\tau_{1}^{2\gamma_{j}}\tau_{0}^{2(1-\gamma_{j})}}} \exp\left( -\frac{{{\upbeta}_{j}^{2}}}{2\sigma^{2}\tau_{1}^{2\gamma_{j}}\tau_{0}^{2(1-\gamma_{j})}}\right). $$
(4)
Then,
$$ \begin{array}{@{}rcl@{}} &&{\prod}_{j}\left[\alpha_{j}f_{1}({\upbeta}_{j})+ (1-\alpha_{j})f_{0}({\upbeta}_{j})\right]\\ &&= \frac{1}{\sqrt{(2\pi\sigma^{2}{\tau_{1}^{2}})^{p}}}\exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}{\tau_{1}^{2}}}\right) \left( {\prod}_{j}\alpha_{j}\right)\left( {\sum}_{k=0}^{p}g_{k}(\upbeta, \sigma^{2})\right) , \end{array} $$
(5)
where
$$ \begin{array}{@{}rcl@{}} &&g_{0}(\upbeta, \sigma^{2}) = 1, \end{array} $$
(6)
$$ \begin{array}{@{}rcl@{}} &&g_{p}(\upbeta, \sigma^{2}) = \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}\exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}} \left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right) {\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}}, \end{array} $$
(7)
$$ \begin{array}{@{}rcl@{}} &&\text{and for }\ 1\le k \le p-1,\\ &&g_{k}(\upbeta, \sigma^{2}) = \frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}}\exp\\ &&\left( -\frac{{\upbeta}_{j_{1}}^{2}+{\upbeta}_{j_{2}}^{2}+\cdots+{\upbeta}_{j_{k}}^{2}}{2\sigma^{2}} \left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right). \end{array} $$
(8)
Proof 7.
From the left hand side, we have
$$ \begin{array}{@{}rcl@{}} &&{\prod}_{j}\left[\alpha_{j}f_{1}({\upbeta}_{j})+ (1-\alpha_{j})f_{0}({\upbeta}_{j})\right]\\ &&={\prod}_{j}\alpha_{j}f_{1}({\upbeta}_{j}){\prod}_{j}\left[1+ \frac{1-\alpha_{j}}{\alpha_{j}} \frac{f_{0}({\upbeta}_{j})}{f_{1}({\upbeta}_{j})}\right] \end{array} $$
(9)
$$ \begin{array}{@{}rcl@{}} &&={\prod}_{j}\frac{\alpha_{j}}{\sqrt{2\pi\sigma^{2}{\tau_{1}^{2}}}} \exp\left( -\frac{{{\upbeta}_{j}^{2}}}{2\sigma^{2}{\tau_{1}^{2}}}\right){\prod}_{j}\\ &&\left[1+ \frac{\tau_{1}(1-\alpha_{j})}{\tau_{0}\alpha_{j}} \exp\left( -\frac{{{\upbeta}_{j}^{2}}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)\right] \end{array} $$
(10)
$$ \begin{array}{@{}rcl@{}} &&=\frac{1}{\sqrt{(2\pi\sigma^{2}{\tau_{1}^{2}})^{p}}}\exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}{\tau_{1}^{2}}}\right) {\prod}_{j}\alpha_{j}{\prod}_{j}\\ && \left[1+ \frac{\tau_{1}(1-\alpha_{j})}{\tau_{0}\alpha_{j}} \exp\left( -\frac{{{\upbeta}_{j}^{2}}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)\right]. \end{array} $$
(11)
Now to evaluate the product term \({\prod }_{j} \left [1+ \frac {\tau _{1}(1-\alpha _{j})}{\tau _{0}\alpha _{j}} \exp \left (-\frac {{{\upbeta }_{j}^{2}}}{2\sigma ^{2}}\left (\frac {1}{{\tau _{0}^{2}}} -\frac {1}{{\tau _{1}^{2}}}\right )\right )\right ]\), we use the following identity
$$ \begin{array}{@{}rcl@{}} &&{\prod}_{j}(1+a_{j}) \\ &&= 1 + {\sum}_{j}a_{j} + {\sum}_{j_{1}<j_{2}}a_{j_{1}}a_{j_{2}} + {\sum}_{j_{1}<j_{2}<j_{3}}a_{j_{1}}a_{j_{2}}a_{j_{3}}\\ &&+ {\cdots} + {\sum}_{j_{1}<\cdots<j_{(p-1)}}(a_{j_{1}} {\cdots} a_{j_{(p-1)}})+ {\prod}_{j}a_{j}. \end{array} $$
(12)
Then, combining eq. 11 and eq. 12, we have
$$ \begin{array}{@{}rcl@{}} &&{\prod}_{j} \left[1+ \frac{\tau_{1}(1-\alpha_{j})}{\tau_{0}\alpha_{j}} \exp\left( -\frac{{{\upbeta}_{j}^{2}}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)\right]\\ &&=1 + \frac{\tau_{1}}{\tau_{0}}{\sum}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \exp\left( -\frac{{{\upbeta}_{j}^{2}}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)\\ && + \frac{{\tau_{1}^{2}}}{{\tau_{0}^{2}}} {\sum}_{j_{1}<j_{2}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \exp\left( -\frac{{\upbeta}_{j_{1}}^{2}+{\upbeta}_{j_{2}}^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)\\ && + \frac{{\tau_{1}^{3}}}{{\tau_{0}^{3}}} {\sum}_{j_{1}<j_{2}<j_{3}}\frac{1 - \alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1 - \alpha_{j_{2}}}{\alpha_{j_{2}}} \frac{1 - \alpha_{j_{3}}}{\alpha_{j_{3}}} \exp\left( -\frac{{\upbeta}_{j_{1}}^{2}+{\upbeta}_{j_{2}}^{2}+{\upbeta}_{j_{3}}^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} - \frac{1}{{\tau_{1}^{2}}}\right)\right)\\ && + \cdots\\ && + \frac{\tau_{1}^{(p-1)}}{\tau_{0}^{(p-1)}} {\sum}_{j_{1}<\cdots<j_{p-1}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{(p-1)}}}{\alpha_{j_{(p-1)}}}\\ &&\exp\left( -\frac{{\upbeta}_{j_{1}}^{2}+{\upbeta}_{j_{2}}^{2}+\cdots+{\upbeta}_{j_{(p-1)}}^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)\\ && + \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}\exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right) {\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}}. \end{array} $$
(13)
Therefore,
$$ \begin{array}{@{}rcl@{}} &&{\prod}_{j}\left[\alpha_{j}f_{1}({\upbeta}_{j})+ (1-\alpha_{j})f_{0}({\upbeta}_{j})\right]\\ &&= \frac{1}{\sqrt{(2\pi\sigma^{2}{\tau_{1}^{2}})^{p}}}\exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}{\tau_{1}^{2}}}\right) \left( {\prod}_{j}\alpha_{j}\right)\left( {\sum}_{k=0}^{p}g_{k}(\upbeta, \sigma^{2})\right)~~~~~~~~~~~~ \end{array} $$
(14)
where gk(β,σ2) is as specified before. □
Now, using our result from lemma B.1 in eq. 3, we get
$$ \begin{array}{@{}rcl@{}} &&P(\upbeta\mid y)\\ &&\overset{\upbeta}{\propto}\int P(y\mid \upbeta, \sigma^{2}) \frac{1}{\sqrt{(2\pi\sigma^{2}{\tau_{1}^{2}})^{p}}} \exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}{\tau_{1}^{2}}}\right) {\prod}_{j}\alpha_{j}\left( {\sum}_{k=0}^{p}g_{k}(\upbeta, \sigma^{2})\right)\\ &&P(\sigma^{2})d\sigma^{2} \end{array} $$
(15)
$$ \begin{array}{@{}rcl@{}} &&\overset{\upbeta}{\propto}{\sum}_{k=0}^{p}\int\frac{1}{\sigma^{2(n/2+p/2)}} \exp\left( -\frac{1}{\sigma^{2}} \left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}}\right)\right)\\ &&g_{k}(\upbeta, \sigma^{2})\frac{1}{\sigma^{2(a+1)}} \exp\left( -\frac{b}{\sigma^{2}}\right)d\sigma^{2} \end{array} $$
(16)
$$ \begin{array}{@{}rcl@{}} &&\overset{\upbeta}{\propto} {\sum}_{k=0}^{p}\int\frac{1}{\sigma^{2(n/2+p/2 + a +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)\\ &&g_{k}(\upbeta, \sigma^{2})d\sigma^{2}. \end{array} $$
(17)
Before evaluating the integrals, we need to show some identities. For that, we first need to define some expressions. Let, \(D_{\tau _{1}} = \tau _{1}^{-2}\mathbf {I}_{p}\), then we define
$$ \begin{array}{@{}rcl@{}} L_{\tau_{1}} &=& (\mathbf{x}^{T}\mathbf{x} + D_{\tau_{1}})^{-1}, \mu_{\tau_{1}} = L_{\tau_{1}}\mathbf{x}^{T}y,\\ r_{\tau_{1}} &=& \frac{y^{T}y - y^{T}\mathbf{x}L_{\tau_{1}}\mathbf{x}^{T}y}{2} +b, \text{ and } {\Sigma}^{-1}_{\tau_{1}} = \frac{n+2a}{2r_{\tau_{1}}}L_{\tau_{1}}^{-1}. \end{array} $$
(18)
We also use similar expressions using \(D_{\tau _{0}}\) and \(D_{j_{1},j_{2},\cdots ,j_{k}}\) where
$$ D_{\tau_{0}} = \tau_{0}^{-2}\mathbf{I}_{p} \text{ and } D_{j_{1},j_{2},\cdots,j_{k}} = \text{diag}((1-\mathbb{I}_{j_{1},j_{2},\cdots,j_{k}}(j))\tau_{1}^{-2} +\mathbb{I}_{j_{1},j_{2},\cdots,j_{k}}(j)\tau_{0}^{-2}). $$
(19)
Lemma B.2.
Let a∗/2 = n/2 + a. Then,
$$ \begin{array}{@{}rcl@{}} &&\int\frac{1}{\sigma^{2(n/2+p/2 + a +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)g_{k}(\upbeta, \sigma^{2})d\sigma^{2}\\ &&= {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2}h_{k}(\upbeta) \end{array} $$
(20)
where,
$$ \begin{array}{@{}rcl@{}} h_{0}(\upbeta) \!\!\!\!\!\!\!\!&&= \frac{\sqrt{|{\Sigma}_{\tau_{1}}|}} {r_{\tau_{1}}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}(\mu_{\tau_{1}}, {\Sigma}_{\tau_{1}}) \end{array} $$
(21)
$$ \begin{array}{@{}rcl@{}} h_{p}(\upbeta) \!\!\!\!\!\!\!\!&&= \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \frac{\sqrt{|{\Sigma}_{\tau_{0}}|}} {r_{\tau_{0}}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}(\mu_{\tau_{0}}, {\Sigma}_{\tau_{0}}) \end{array} $$
(22)
and for 1 ≤ k ≤ p − 1
$$ \begin{array}{@{}rcl@{}} h_{k}(\upbeta) &=& \frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}} \frac{\sqrt{|{\Sigma}_{j_{1},j_{2},\cdots,j_{k}}|}} {r_{j_{1},j_{2},\cdots,j_{k}}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}\\ &&\left( \mu_{j_{1},j_{2},\cdots,j_{k}}, {\Sigma}_{j_{1},j_{2},\cdots,j_{k}}\right). \end{array} $$
(23)
Proof 8.
We compute the integrals using the properties of inverse gamma distribution followed by some adjustments to obtain the expression of multivariate t distribution.
For k = 0, we have
$$ \begin{array}{@{}rcl@{}} &&{}\int\frac{1}{\sigma^{2(n/2+p/2 + a +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)g_{0}(\upbeta, \sigma^{2})d\sigma^{2}~~\\ &&{} = \int\frac{1}{\sigma^{2(a^{*}/2 + p/2 +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)d\sigma^{2} \end{array} $$
(24)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{\Gamma(a^{*}/2+p/2)}{\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} +b\right)^{a^{*}/2 + p/2}} \end{array} $$
(25)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{\Gamma(a^{*}/2+p/2)}{\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{{\upbeta}^{t} D_{\tau_{1}}\upbeta}{2} +b\right)^{a^{*}/2 + p/2}} \end{array} $$
(26)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{\Gamma(a^{*}/2+p/2)}{\left( \frac{(\upbeta-\mu_{\tau_{1}})^{T}L_{\tau_{1}}^{-1} (\upbeta-\mu_{\tau_{1}})}{2} +r_{\tau_{1}}\right)^{a^{*}/2 + p/2}} \end{array} $$
(27)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{\Gamma(a^{*}/2+p/2)}{r_{\tau_{1}}^{a^{*}/2 + p/2} \left( \frac{1}{a^{*}}\frac{(\upbeta-\mu_{\tau_{1}})^{T}L_{\tau_{1}}^{-1} (\upbeta-\mu_{\tau_{1}})}{2r_{\tau_{1}}/a^{*}} +1\right)^{a^{*}/2 + p/2}} \end{array} $$
(28)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{\Gamma(a^{*}/2+p/2)}{r_{\tau_{1}}^{a^{*}/2 + p/2} \left( 1+\frac{1}{a^{*}}(\upbeta-\mu_{\tau_{1}})^{T} {\Sigma}^{-1}_{\tau_{1}}(\upbeta-\mu_{\tau_{1}})\right)^{a^{*}/2 + p/2}} \end{array} $$
(29)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{\Gamma(a^{*}/2)(a^{*}\pi)^{p/2}\sqrt{|{\Sigma}_{\tau_{1}}|}} {r_{\tau_{1}}^{a^{*}/2 + p/2}} \\&&{}\frac{\Gamma(a^{*}/2+p/2)}{\Gamma(a^{*}/2)(a^{*}\pi)^{p/2}\sqrt{|{\Sigma}_{\tau_{1}}|} \left( 1+\frac{1}{a^{*}}(\upbeta-\mu_{\tau_{1}})^{T} {\Sigma}^{-1}_{\tau_{1}}(\upbeta-\mu_{\tau_{1}})\right)^{a^{*}/2 + p/2}} \end{array} $$
(30)
$$ \begin{array}{@{}rcl@{}} &&{}= {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2}\frac{\sqrt{|{\Sigma}_{\tau_{1}}|}} {r_{\tau_{1}}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}(\mu_{\tau_{1}}, {\Sigma}_{\tau_{1}}) \end{array} $$
(31)
$$ \begin{array}{@{}rcl@{}} &&{} = {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2} h_{0}(\upbeta). \end{array} $$
(32)
For k = p, we have
$$ \begin{array}{@{}rcl@{}} &&{}\int\frac{1}{\sigma^{2(n/2+p/2 + a +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)g_{p} (\upbeta,\sigma^{2})d\sigma^{2}\\ &&{} = \int\frac{1}{\sigma^{2(a^{*}/2 + p/2 +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right) \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}\\ &&{}\exp\left( -\frac{\|\upbeta\|^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right) {\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}}d\sigma^{2} \end{array} $$
(33)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \int\frac{1}{\sigma^{2(a^{*}/2 + p/2 +1)}}\\ &&\exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{0}^{2}}} +b \right)\right) \end{array} $$
(34)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \frac{\Gamma(a^{*}/2+p/2)}{\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{0}^{2}}}+b\right)^{a^{*}/2 + p/2}} \end{array} $$
(35)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \frac{\Gamma(a^{*}/2+p/2)}{\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{{\upbeta}^{T} D_{\tau_{0}}\upbeta}{2}+b\right)^{a^{*}/2 + p/2}} \end{array} $$
(36)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \frac{\Gamma(a^{*}/2+p/2)}{r_{\tau_{0}}^{a^{*}/2 + p/2} \left( 1+\frac{1}{a^{*}}(\upbeta-\mu_{\tau_{0}})^{T} {\Sigma}^{-1}_{\tau_{0}}(\upbeta-\mu_{\tau_{0}})\right)^{a^{*}/2 + p/2}} \end{array} $$
(37)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \frac{\Gamma(a^{*}/2)(a^{*}\pi)^{p/2}\sqrt{|{\Sigma}_{\tau_{0}}|}} {r_{\tau_{0}}^{a^{*}/2 + p/2}} \\&&{}\frac{\Gamma(a^{*}/2+p/2)}{\Gamma(a^{*}/2)(a^{*}\pi)^{p/2}\sqrt{|{\Sigma}_{\tau_{0}}|} \left( 1+\frac{1}{a^{*}}(\upbeta-\mu_{\tau_{0}})^{T} {\Sigma}^{-1}_{\tau_{0}}(\upbeta-\mu_{\tau_{0}})\right)^{a^{*}/2 + p/2}} \end{array} $$
(38)
$$ \begin{array}{@{}rcl@{}} &&{}= {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2}\frac{{\tau_{1}^{p}}}{{\tau_{0}^{p}}}{\prod}_{j}\frac{1-\alpha_{j}}{\alpha_{j}} \frac{\sqrt{|{\Sigma}_{\tau_{0}}|}} {r_{\tau_{0}}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}(\mu_{\tau_{0}}, {\Sigma}_{\tau_{0}}) \end{array} $$
(39)
$$ \begin{array}{@{}rcl@{}} &&{} = {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2} h_{p}(\upbeta). \end{array} $$
(40)
For 1 ≤ k ≤ p − 1, we have
$$ \begin{array}{@{}rcl@{}} &&{} \int\frac{1}{\sigma^{2(n/2+p/2 + a +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)g_{k}(\upbeta, \sigma^{2})d\sigma^{2}\\ &&{} = \int\frac{1}{\sigma^{2(a^{*}/2+p/2 +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}} + b\right)\right)\cdot\\ &&{}\frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}} \exp\left( -\frac{{\upbeta}_{j_{1}}^{2}+{\upbeta}_{j_{2}}^{2}+\cdots+{\upbeta}_{j_{k}}^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)d\sigma^{2} \end{array} $$
(41)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}}\cdot\\ &&{} \int\frac{1}{\sigma^{2(a^{*}/2 + p/2 +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{\|\upbeta\|^{2}}{2{\tau_{1}^{2}}}+b\right)\right) \\&&\exp\left( -\frac{{\upbeta}_{j_{1}}^{2}+{\upbeta}_{j_{2}}^{2}+\cdots+{\upbeta}_{j_{k}}^{2}}{2\sigma^{2}}\left( \frac{1}{{\tau_{0}^{2}}} -\frac{1}{{\tau_{1}^{2}}}\right)\right)d\sigma^{2} \end{array} $$
(42)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}}\cdot\\ &&{} \int\frac{1}{\sigma^{2(a^{*}/2 + p/2 +1)}} \exp\left( -\frac{1}{\sigma^{2}}\left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{{\upbeta}^{T} D_{j_{1},j_{2},\cdots,j_{k}}\upbeta}{2} + b\right)\right)d\sigma^{2} \end{array} $$
(43)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}} \frac{\Gamma(a^{*}/2 + p/2)}{ \left( \frac{\|y-\mathbf{x}\upbeta\|^{2}_{2}}{2} +\frac{{\upbeta}^{T} D_{j_{1},j_{2},\cdots,j_{k}}\upbeta}{2} + b\right)^{a^{*}/2+p/2}} \end{array} $$
(44)
$$ \begin{array}{@{}rcl@{}} &&{}= \frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}} \frac{\Gamma(a^{*}/2)(a^{*}\pi)^{p/2}\sqrt{|{\Sigma}_{j_{1},j_{2},\cdots,j_{k}}|}} {r_{j_{1},j_{2},\cdots,j_{k}}^{a^{*}/2 + p/2}}\cdot\\ &&{} \frac{\Gamma(a^{*}/2+p/2)}{\Gamma(a^{*}/2)(a^{*}\pi)^{p/2} \sqrt{|{\Sigma}_{j_{1},j_{2},\cdots,j_{k}}|} \left( \!1 + \frac{1}{a^{*}}(\upbeta - \mu_{j_{1},j_{2},\cdots,j_{k}})^{T} {\Sigma}^{-1}_{j_{1},j_{2},\cdots,j_{k}}(\upbeta - \mu_{j_{1},j_{2},\cdots,j_{k}})\!\right)^{a^{*}/2 + p/2}} \end{array} $$
(45)
$$ \begin{array}{@{}rcl@{}} &&{}= {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2}\frac{{\tau_{1}^{k}}}{{\tau_{0}^{k}}} {\sum}_{j_{1}<\cdots<j_{k}}\frac{1-\alpha_{j_{1}}}{\alpha_{j_{1}}}\frac{1-\alpha_{j_{2}}}{\alpha_{j_{2}}} \cdots\frac{1-\alpha_{j_{k}}}{\alpha_{j_{k}}} \frac{\sqrt{|{\Sigma}_{j_{1},j_{2},\cdots,j_{k}}|}} {r_{j_{1},j_{2},\cdots,j_{k}}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}\\ &&(\mu_{j_{1},j_{2},\cdots,j_{k}}, {\Sigma}_{j_{1},j_{2},\cdots,j_{k}}) \end{array} $$
(46)
$$ \begin{array}{@{}rcl@{}} &&{}={\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2} h_{k}(\upbeta). \end{array} $$
(47)
□
Then, using identities from lemma B.2, we have
$$ \begin{array}{@{}rcl@{}} P(\upbeta \mid y) &&\overset{\upbeta}{\propto} {\Gamma}(a^{*}/2)(a^{*}\pi)^{p/2}{\sum}_{k=0}^{p} h_{k}(\upbeta) \end{array} $$
(48)
$$ \begin{array}{@{}rcl@{}} &&\overset{\upbeta}{\propto} {\sum}_{k=0}^{p} h_{k}(\upbeta). \end{array} $$
(49)
Now, for 1 ≤ k ≤ p − 1, we can rewrite h(k) so that
$$ h(k) = {\sum}_{\substack{\gamma \\ (\gamma_{j_{1}}=\cdots=\gamma_{j_{k}}=0)}} \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p-{\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|{\Sigma}_{\gamma}|}} {r_{\gamma}^{a^{*}/2 + p/2}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}), $$
(50)
where \(r_{\gamma } = \frac {y^{T}y - y^{T}\mathbf {x}L_{\gamma }\mathbf {x}^{T}y}{2} + b\) and \({\Sigma }_{\gamma } = \frac {a^{*}}{2r_{\gamma }}L^{-1}_{\gamma }\). Therefore,
$$ \begin{array}{@{}rcl@{}} &&{}P(\upbeta \mid y)\\ &&{}\overset{\upbeta}{\propto} {\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|{\Sigma}_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}) \right) \end{array} $$
(51)
$$ \begin{array}{@{}rcl@{}} && {}= \frac{{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|{\Sigma}_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}) \right)} {{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|{\Sigma}_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}}\right)}. \end{array} $$
(52)
This shows that joint posterior of β can be represented as a 2p component mixture of multivariate t-distribution, where each component corresponds to a particular combination of selected variables out of 2p possible combinations.
Now, by simplifying eq. 52, we have the joint posterior of β
$$ \begin{array}{@{}rcl@{}} &&{}P(\upbeta \mid y)\\ &&{} = \frac{{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|{\Sigma}_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}) \right)} {{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|{\Sigma}_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}}\right)} \end{array} $$
(53)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{2^{p/2}r_{\gamma}^{p/2} (a^{*})^{-p/2} \sqrt{|L_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}) \right)} {{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{2^{p/2}r_{\gamma}^{p/2} (a^{*})^{-p/2} \sqrt{|L_{\gamma}|}}{r^{a^{*}/2+p/2}_{\gamma}}\right)} \end{array} $$
(54)
$$ \begin{array}{@{}rcl@{}} &&{} = \frac{{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|L_{\gamma}|}}{r^{a^{*}/2}_{\gamma}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}) \right)} {{\sum}_{\gamma}\left( \left( \frac{\tau_{1}}{\tau_{0}}\right)^{p - {\sum}_{j}\gamma_{j}} {\prod}_{j}\left( \frac{1-\alpha_{j}}{\alpha_{j}}\right)^{1-\gamma_{j}} \frac{\sqrt{|L_{\gamma}|}}{r^{a^{*}/2}_{\gamma}}\right)} \end{array} $$
(55)
$$ \begin{array}{@{}rcl@{}} &&{}=\!\frac{{\sum}_{\gamma}\left( \! \left( {\prod}_{j} \alpha_{j}^{\gamma_{j}} (1 - \alpha_{j})^{1-\gamma_{j}}\right)\left( \frac{\sqrt{| L_{\gamma}|}} {\tau_{1}^{\sum\gamma_{j}} \tau_{0}^{(p-\sum\gamma_{j})}}\right) \frac{1}{\left( b + \frac{y^{T}y - {\mu_{\gamma}}^{T} L_{\gamma}^{-1} \mu_{\gamma}}{2}\right)^{n/2+a}} \mathcal{T}_{a^{*}}(\mu_{\gamma}, {\Sigma}_{\gamma}) \!\right)} {{\sum}_{\gamma}\left( \left( {\prod}_{j} \alpha_{j}^{\gamma_{j}} (1-\alpha_{j})^{1-\gamma_{j}}\right)\left( \frac{\sqrt{| L_{\gamma}|}} {\tau_{1}^{\sum\gamma_{j}} \tau_{0}^{(p-\sum\gamma_{j})}}\right) \frac{1}{\left( b + \frac{y^{T}y - {\mu_{\gamma}}^{T} L_{\gamma}^{-1} \mu_{\gamma}}{2}\right)^{n/2+a}}\right)}. \end{array} $$
(56)
