Appendix 2.1
To prove Eq. (2.12) – (2.14), we first substitute Eqs. (2.10) into (2.11) to obtain
$$ f\left( {v_{\alpha } ,v_{\beta } , \ldots ,v_{\gamma } \left| {u_{i} ,u_{j} , \ldots ,u_{k} } \right.} \right) \propto g\left( {u_{i} ,u_{j} , \ldots ,u_{k} \left| {v_{\alpha } ,v_{\beta } , \ldots ,v_{\gamma } } \right.} \right)g\left( {v_{\alpha } ,v_{\beta } , \ldots ,v_{\gamma } } \right) $$
(2.27)
Because the likelihood function and the prior density are Gaussian probability densities, Eq. (2.27) simplifies to (by the basic properties of conditional densities [5])
$$ f\left( {v_{\alpha } ,v_{\beta } , \ldots ,v_{\gamma } \left| {u_{i} ,u_{j} , \ldots ,u_{k} } \right.} \right) \propto g\left( {v_{\alpha } ,v_{\beta } , \ldots ,v_{\gamma } ,u_{i} ,u_{j} , \ldots ,u_{k} } \right), $$
(2.28)
or, by using Eq. (2.8),
$$ \begin{aligned} & f\left( {v_{\alpha } ,v_{\beta } , \ldots ,v_{\gamma } \left| {u_{i} ,u_{j} , \ldots ,u_{k} } \right.} \right) \\ & \quad \propto \exp \left( { - \frac{1}{2}\left[ {\left( {\begin{array}{*{20}c} {{\mathbf{v}}_{\alpha :\gamma } } \\ {{\mathbf{v}}_{i:k} } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {{\varvec{\upmu}}_{\alpha :\gamma } } \\ {{\varvec{\upmu}}_{i:k} } \\ \end{array} } \right)} \right]^{T} \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma } } & {{\mathbf{K}}_{\alpha :\gamma ,i:k} } \\ {{\mathbf{K}}_{i:k,\alpha :\gamma } } & {{\mathbf{K}}_{i:k,i:k} } \\ \end{array} } \right]^{ - 1} \left[ {\left( {\begin{array}{*{20}c} {{\mathbf{v}}_{\alpha :\gamma } } \\ {{\mathbf{v}}_{i:k} } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {{\varvec{\upmu}}_{\alpha :\gamma } } \\ {{\varvec{\upmu}}_{i:k} } \\ \end{array} } \right)} \right]} \right) \\ \end{aligned} $$
(2.29)
This expression can be simplified using an identity which applies to block matrices (see, Ref. [6])
$$ \begin{aligned} & \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma } } & {{\mathbf{K}}_{\alpha :\gamma ,i:k} } \\ {{\mathbf{K}}_{i:k,\alpha :\gamma } } & {{\mathbf{K}}_{i:k,i:k} } \\ \end{array} } \right]^{ - 1} \\ & \quad = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma } - {\mathbf{K}}_{\alpha :\gamma ,i:k} {\mathbf{K}}_{i:k,i:k}^{ - 1} {\mathbf{K}}_{i:k,\alpha :\gamma } } \right)^{ - 1} } & {\left( {{\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma } - {\mathbf{K}}_{\alpha :\gamma ,i:k} {\mathbf{K}}_{i:k,i:k}^{ - 1} {\mathbf{K}}_{i:k,\alpha :\gamma } } \right)^{ - 1} {\mathbf{K}}_{\alpha :\gamma ,i:k} {\mathbf{K}}_{i:k,i:k}^{ - 1} } \\ { - \left( {{\mathbf{K}}_{i:k,i:k} - {\mathbf{K}}_{i:k,\alpha :\gamma } {\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma }^{ - 1} {\mathbf{K}}_{\alpha :\gamma ,i:k} } \right)^{ - 1} {\mathbf{K}}_{i:k,\alpha :\gamma } {\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma }^{ - 1} } & {\left( {{\mathbf{K}}_{i:k,i:k} - {\mathbf{K}}_{i:k,\alpha :\gamma } {\mathbf{K}}_{\alpha :\gamma ,\alpha :\gamma }^{ - 1} {\mathbf{K}}_{\alpha :\gamma ,i:k} } \right)^{ - 1} } \\ \end{array} } \right] \\ \end{aligned} $$
(2.30)
Substituting Eqs. (2.30) into (2.29) and performing some tedious but straightforward algebraic manipulations yields Eqs. (2.12)–(2.14).
Appendix 2.2
To prove Eq. (2.19), we write
$$ \begin{aligned} EI\left( {r_{\alpha } } \right) & = E_{f} \left[ {\hbox{min} \left( {u_{\hbox{min} } - V\left( {r_{\alpha } } \right),0} \right)} \right] \\ & = \int\limits_{ - \infty }^{{u_{\hbox{min} } }} {\left( {u_{\hbox{min} } - z} \right)\frac{1}{{\sqrt {2\pi K_{\alpha \alpha }^{*} } }}e^{{{{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } {2K_{\alpha \alpha }^{*} }}} \right. \kern-0pt} {2K_{\alpha \alpha }^{*} }}}} dz} \\ & = \underbrace {{u_{\hbox{min} } \int\limits_{ - \infty }^{{u_{\hbox{min} } }} {\frac{1}{{\sqrt {2\pi K_{\alpha \alpha }^{*} } }}e^{{{{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } {2K_{\alpha \alpha }^{*} }}} \right. \kern-0pt} {2K_{\alpha \alpha }^{*} }}}} dz} }}_{A} - \underbrace {{\int\limits_{ - \infty }^{{u_{\hbox{min} } }} {\frac{z}{{\sqrt {2\pi K_{\alpha \alpha }^{*} } }}e^{{{{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } {2K_{\alpha \alpha }^{*} }}} \right. \kern-0pt} {2K_{\alpha \alpha }^{*} }}}} dz} .}}_{B} \\ \end{aligned} $$
(2.31)
The term A on the right-hand side of Eq. (2.31) is equal to
$$ A = u_{\hbox{min} } {\Phi} \left( {\frac{{u_{\hbox{min} } - \mu_{\alpha }^{*} }}{{\sqrt {K_{\alpha \alpha }^{*} } }}} \right) , $$
(2.32)
by the definition of the cumulative normal distribution. As for the term B in Eq. (2.31), we write
$$ \begin{aligned} B & = \int\limits_{ - \infty }^{{u_{\hbox{min} } }} {\frac{{\mu_{\alpha }^{*} + \left( {z - \mu_{\alpha }^{*} } \right)}}{{\sqrt {2\pi K_{\alpha \alpha }^{*} } }}e^{{{{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } {2K_{\alpha \alpha }^{*} }}} \right. \kern-0pt} {2K_{\alpha \alpha }^{*} }}}} dz} . \\ & = \mu_{\alpha }^{*} {\Phi} \left( {\frac{{u_{\hbox{min} } - \mu_{\alpha }^{*} }}{{\sqrt {K_{\alpha \alpha }^{*} } }}} \right) + \underbrace {{\int\limits_{ - \infty }^{{u_{\hbox{min} } }} {\frac{{\left( {z - \mu_{\alpha }^{*} } \right)}}{{\sqrt {2\pi K_{\alpha \alpha }^{*} } }}e^{{{{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {z - \mu_{\alpha }^{*} } \right)^{2} } {2K_{\alpha \alpha }^{*} }}} \right. \kern-0pt} {2K_{\alpha \alpha }^{*} }}}} dz.} }}_{C} \\ \end{aligned} $$
(2.33)
By substituting the variable
$$ h = \frac{{z - \mu_{\alpha }^{*} }}{{\sqrt {2K_{\alpha \alpha }^{*} } }} $$
(2.34)
into the term C in Eq. (2.33) and performing the integration, we obtain
$$ \begin{aligned} C & = \left( {\frac{{2K_{\alpha \alpha }^{*} }}{\pi }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \int\limits_{ - \infty }^{{u_{\hbox{min} } }} {he^{{ - h^{2} }} dh} \\ & = - \left( {\frac{{K_{\alpha \alpha }^{*} }}{2\pi }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} e^{{{{ - \left( {u_{\hbox{max} } - \mu_{\alpha }^{*} } \right)^{2} } \mathord{\left/ {\vphantom {{ - \left( {u_{\hbox{max} } - \mu_{\alpha }^{*} } \right)^{2} } {2K_{\alpha \alpha }^{{}} }}} \right. \kern-0pt} {2K_{\alpha \alpha }^{{}} }}}} \\ & = - \sqrt {K_{\alpha \alpha }^{*} } \phi \left( {\frac{{u_{\hbox{min} } - \mu_{\alpha }^{*} }}{{\sqrt {K_{\alpha \alpha }^{*} } }}} \right) \\ \end{aligned} $$
(2.35)
where the definition of the standard normal probability density was used. We obtain the result after combining Eqs. (2.31), (2.32), (2.33) and (2.35).
Appendix 2.3
To prove Eqs. (2.20) and (2.21), we take the logarithm of the prior probability density in Eq. (2.5) to obtain
$$ \begin{aligned} \log g\left( {u_{i} ,u_{j} , \ldots ,u_{k} } \right) & = - \frac{1}{2}\left( {{\mathbf{u}} - {\varvec{\upmu}}} \right)^{T} \left( {a{\mathbf{R}}} \right)^{ - 1} \left( {{\mathbf{u}} - {\varvec{\upmu}}} \right) - \frac{1}{2}\log \left| {a{\mathbf{R}}} \right| - \frac{s}{2}\log 2\pi \\ & = \underbrace {{ - \frac{1}{2}\left( {{\mathbf{u}} - {\varvec{\upmu}}} \right)^{T} \left( {a{\mathbf{R}}} \right)^{ - 1} \left( {{\mathbf{u}} - {\varvec{\upmu}}} \right)}}_{A} - \underbrace {{\frac{s}{2}\log \left( {a\left| {\mathbf{R}} \right|^{{{1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-0pt} s}}} } \right)}}_{B} - \underbrace {{\frac{s}{2}\log 2\pi }}_{C}. \\ \end{aligned} $$
(2.36)
In the first line of Eq. (2.36), we used the definition of the matrix R in Eq. (2.22) and the fact that a
R = Σ. In the second line, we used the fact that |a
R| = a
s|R|, which follows from the basic properties of determinants. Solving the equation ∂log g(u
i
, u
j
, …, u
k
)/∂a = 0 gives
$$ a = \frac{1}{s}\left( {{\mathbf{u}} - {\varvec{\upmu}}} \right)^{T} {\mathbf{R}}^{ - 1} \left( {{\mathbf{u}} - {\varvec{\upmu}}} \right), $$
(2.37)
which is simply Eq. (2.21). To obtain an expression for L, fist note that the term A reduces to
$$ A = {{ - s} \mathord{\left/ {\vphantom {{ - m} 2}} \right. \kern-0pt} 2} $$
(2.38)
upon substituting Eq. (2.37). Substituting Eq. (2.37) into the term marked B gives
$$ B = - \frac{s}{2}\log \left( {\frac{1}{s}\left| {\mathbf{R}} \right|^{{{1 \mathord{\left/ {\vphantom {1 m}} \right. \kern-0pt} s}}} \left( {{\mathbf{u}} - {\varvec{\upmu}}} \right)^{T} {\mathbf{R}}^{ - 1} \left( {{\mathbf{u}} - {\varvec{\upmu}}} \right)} \right). $$
(2.39)
Substituting Eqs. (2.38) and (2.39) into Eq. (2.34), and noting that the terms A and C are independent of a and L, gives Eq. (2.22). Finally, by noting that maximization of the logarithm of the prior probability density is equivalent to maximizing the prior probability density itself, we arrive at the result.
