Accounting for the uncertainty in the local mean in spatial prediction by Bayesian Maximum Entropy

Abstract

Bayesian Maximum Entropy (BME) has been successfully used in geostatistics to calculate predictions of spatial variables given some general knowledge base and sets of hard (precise) and soft (imprecise) data. This general knowledge base commonly consists of the means at each of the locations considered in the analysis, and the covariances between these locations. When the means are not known, the standard practice is to estimate them from the data; this is done by either generalized least squares or maximum likelihood. The BME prediction then treats these estimates as the general knowledge means, and ignores their uncertainty. In this paper we develop a prediction that is based on the BME method that can be used when the general knowledge consists of the covariance model only. This prediction incorporates the uncertainty in the estimated local mean. We show that in some special cases our prediction is equal to results from classical geostatistics. We investigate the differences between our approach and the standard approach for predicting in this common practical situation.

Appendices

Appendix A

We begin with Eq. 8 for the prior:

$$ \begin{aligned} f_{O} {({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}})} &= {\int {f_{{G.m}} {({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}})}\hbox{d}m}} \\ & \propto {\int {\exp {\left\{{- \frac{1}{2}{({{\mathbf{z}}_{\rm shk} - {\mathbf{1}}_{\rm shk} m})}^{\rm T} {\mathbf{C}}^{-1}_{\rm shk} {({{\mathbf{z}}_{\rm shk} - {\mathbf{1}}_{\rm shk} m})}}\right\}}}}\hbox{d}m \\ &= {\int {\exp {\left\{{- \frac{1}{2}{\left[{{\mathbf{z}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk} - 2m{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk} + m^{2} {\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}\right]}}\right\}}}}\hbox{d}m, \\ \end{aligned} $$

(A1)

We can take the term that is independent of m outside the integral:

$$\begin{aligned} f_{O} {\left({{\mathbf{z}}_{\rm s},{\mathbf{z}}_{\rm h}, z_{\rm k}}\right)} &\propto \exp{\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm shk}{\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}\right\}} \\& {\int {\exp {\left\{{-\frac{{{\mathbf{1}}^{\rm T}_{\rm shk}{\mathbf{C}}^{-1}_{\rm shk}{\mathbf{1}}_{\rm shk}}}{2}{\left[{m^{2} -2m\frac{{{\mathbf{1}}^{\rm T}_{\rm shk}{\mathbf{C}}^{-1}_{\rm shk}{\mathbf{z}}_{\rm shk}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right]}}\right\}}}}\hbox{d}m, \\ \end{aligned} $$

(A2)

and then complete the square to write:

$$ \begin{aligned} f_{O} {\left({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}}\right)} &\propto \exp {\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}\right\}} \\ & {\int {\exp {\left\{{- \frac{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}{2}{\left[{{\left({m - \frac{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right)}^{2} - {\left({\frac{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right)}^{2}}\right]}}\right\}}}}\hbox{d}m. \\ \end{aligned} $$

(A3)

Again, we take out the terms independent of m:

$$ \begin{aligned} f_{O} {\left({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}}\right)} &\propto \exp {\left\{{- \frac{1}{2}{\left[{{\mathbf{z}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk} - \frac{{{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}\right)}^{2}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right]}}\right\}} \\ & {\int {\exp {\left\{{\frac{{- 1}}{{2{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}\right)}^{{- 1}}}}{\left[{m - {\left({\frac{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right)}}\right]}^{2}}\right\}}}}\hbox{d}m.\\ \end{aligned} $$

(A4)

We now notice that the expression inside the integral is proportional to the Gaussian distribution for m with mean ${\frac{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}$ and variance ${\left({\mathbf{1}}_{\rm shk}^{\rm T} {\mathbf{C}}_{\rm shk}^{-1} {\mathbf{1}}_{\rm shk}\right)}^{- 1}$—it is therefore given by:

$${\int {\exp {\left\{{\frac{{- 1}}{{2{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}\right)}^{{- 1}}}}{\left[{m - {\left({\frac{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right)}}\right]}^{2}}\right\}}}}\hbox{d}m = {\left({2\pi}\right)}^{{\frac{N}{2}}} {\left[{{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}\right)}^{{- 1}}}\right]}^{{\frac{1} {2}}},$$

(A5)

where N = n _s + n _h + 1. So

$$\begin{aligned} f_{O} {\left({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}}\right)} & \propto \exp {\left\{{- \frac{1}{2}{\left[{{\mathbf{z}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk} - \frac{{{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{z}}_{\rm shk}}\right)}^{2}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right]}}\right\}}{\left({2\pi}\right)}^{{\frac{N}{2}}} {\left[{{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}\right)}^{{- 1}}}\right]}^{{\frac{1}{2}}} \\ & \propto \exp {\left\{{- \frac{1}{2}{\left[{{\mathbf{z}}^{\rm T}_{\rm shk} {\left({{\mathbf{C}}^{-1}_{\rm shk} - \frac{{{\left({{\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}\right)}{\left({{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk}}\right)}}}{{{\mathbf{1}}^{\rm T}_{\rm shk} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}_{\rm shk}}}}\right)}{\mathbf{z}}_{\rm shk}}\right]}}\right\}} \\ & = \exp {\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm shk} {\mathbf{T}}{\mathbf{z}}_{\rm shk}}\right\}}, \\ \end{aligned} $$

(A6)

where T is given by Eq. 9.

Appendix B

Here, we derive the expression for the conditional distribution, f _O (z _s | z _h, z _k); with a similar expression for f _O (z _k | z _h), this allows us to write Eq. 13. We begin with Eq. 8:

$$f_{O} {\left({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}}\right)} \propto \exp {\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm shk} {\mathbf{T}}{\mathbf{z}}_{\rm shk}}\right\}}.$$

(B1)

Now, since T is symmetric, we can write

$$\begin{aligned} f_{O} {\left({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}}\right)}& \propto \exp {\left\{{- \frac{1}{2}{\left[{{\mathbf{z}}^{\rm T}_{\rm s} {\mathbf{T}}_{\rm s} {\mathbf{z}}_{\rm s} + {\mathbf{z}}^{\rm T}_{\rm hk} {\mathbf{T}}_{\rm hk} {\mathbf{z}}_{\rm hk} + 2{\mathbf{z}}^{\rm T}_{\rm hk} {\mathbf{T}}_{\rm hk,s} {\mathbf{z}}_{\rm s}}\right]}}\right\}} \\& = \exp {\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm hk} {\mathbf{T}}_{\rm hk} {\mathbf{z}}_{\rm hk}}\right\}}\exp {\left\{{- \frac{1}{2}{\left[{{\mathbf{z}}^{\rm T}_{\rm s} {\mathbf{T}}_{\rm s} {\mathbf{z}}_{\rm s} + 2{\mathbf{z}}^{\rm T}_{\rm hk} {\mathbf{T}}_{\rm hk,s} {\mathbf{z}}_{\rm s}}\right]}}\right\}}. \\ \end{aligned} $$

(B2)

Since the matrix T _s is invertible (see Appendix C),

$$\begin{aligned} f_{O} {\left({{\mathbf{z}}_{\rm s}, {\mathbf{z}}_{\rm h}, z_{\rm k}}\right)}& \propto \exp {\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm hk} {\mathbf{T}}_{\rm hk} {\mathbf{z}}_{\rm hk}}\right\}} \\ &\exp {\left\{{- \frac{1}{2}{\left[{{\left({{\mathbf{z}}_{\rm s} + {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk}}\right)}^{\rm T} {\mathbf{T}}_{\rm s} {\left({{\mathbf{z}}_{\rm s} + {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk}}\right)} - {\mathbf{z}}^{\rm T}_{\rm hk} {\mathbf{T}}_{\rm hk,s} {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk}}\right]}}\right\}} \\& = \exp {\left\{{- \frac{1}{2}{\mathbf{z}}^{\rm T}_{\rm hk} {\left[{{\mathbf{T}}_{\rm hk} - {\mathbf{T}}_{\rm hk,s} {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk}}\right]}{\mathbf{z}}_{\rm hk}}\right\}} \\& \exp {\left\{{- \frac{1}{2}{\left({{\mathbf{z}}_{\rm s} + {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk}}\right)}^{\rm T} {\mathbf{T}}_{\rm s} {\left({{\mathbf{z}}_{\rm s} + {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk}}\right)}}\right\}}.\\ \end{aligned} $$

(B3)

Again, because T _s is invertible, the second exponential term is proportional to the multivariate Gaussian distribution for z _s with mean vector, ${{\user2{\upmu}}_{{\rm s}\left| {\rm hk}\right.} = - {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk} },$ and covariance matrix, ${{\user2{\Upsigma}}_{{\rm s}\left| {\rm hk}\right.} = {\mathbf{T}}^{-1}_{\rm s} }.$ Since the first exponential term is independent of z _s, we can write

$$f_{O} {\left({{\mathbf{z}}_{\rm s} \left| {{\mathbf{z}}_{\rm h}, z_{\rm k}} \right.}\right)} = {\rm MVN}_{\rm s} {\left({- {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} {\mathbf{z}}_{\rm hk},{\mathbf{T}}^{{- 1}}_{\rm s}} \right)}.$$

(B4)

Appendix C

Here, we calculate the parameters for the conditional multivariate Gaussian distribution for z _s that is inside the integral in Eq. 13, ${{\rm MVN}_{\rm s} {\left({- {\bf T}_{\rm s}^{-1} {\bf T}_{\rm s,hk} {\bf z}_{\rm hk},{\bf T}_{\rm s}^{{- 1}}}\right)}},$ in terms of the general knowledge covariance matrices. The parameters for the term outside the integral in this equation, ${N_{\rm k} {\left({- T_{\rm k}^{-1} {\bf T}_{\rm k,h} {\bf z}_{\rm h}, T_{\rm k}^{{- 1}}}\right)}},$ follow by a similar argument. To begin with, we calculate C ⁻¹_shk , using the result for the inverse of a block matrix (Searle 1982):

$$\begin{aligned} {\mathbf{C}}^{-1}_{\rm shk} &= {\left[\begin{array}{*{20}l} {\mathbf{C}}_{\rm s} & {\mathbf{C}}_{\rm s,hk}\\ {\mathbf{C}}_{\rm hk,s}& {\mathbf{C}}_{\rm hk}\\ \end{array}\right]}^{{- 1}} \\ &= {\left[\begin{array}{*{20}l} {\left({{\mathbf{C}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}^{{- 1}} & - {\left({{\mathbf{C}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}^{{- 1}} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} \\ - {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\left({{\mathbf{C}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}^{{- 1}}& {\mathbf{C}}^{-1}_{\rm hk} + {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\left({{\mathbf{C}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}^{{- 1}} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk}\\ \end{array}\right]}\\ &= {\left[ \begin{array}{*{20}l} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.}& - {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} \\ -{\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} &{\mathbf{C}}^{-1}_{\rm hk} + {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk} \right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} \\ \end{array}\right]}. \\ \end{aligned} $$

(C1)

Equation 9 states

$${\mathbf{T}} = {\mathbf{C}}^{-1}_{\rm shk} - \frac{{{\left({{\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}}\right)}{\left({{\mathbf{1}}^{\rm T} {\mathbf{C}}^{-1}_{\rm shk}}\right)}}}{{{\mathbf{1}}^{\rm T} {\mathbf{C}}^{-1}_{\rm shk} {\mathbf{1}}}},$$

(C2)

which gives

$$\begin{aligned} {\mathbf{T}}_{\rm s}& = {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} - \frac{{{\left({{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{1}}_{\rm s} - {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right)}{\left({{\mathbf{1}}^{\rm T}_{\rm s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.}}\right)}}}{{{\left[{{\mathbf{1}}^{\rm T}_{\rm s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{1}}_{\rm s} + {\mathbf{1}}^{\rm T}_{\rm hk} {\left({{\mathbf{C}}^{-1}_{\rm hk} + {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk}}\right)}{\mathbf{1}}_{\rm hk} - 2{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{1}}_{\rm s}}\right]}}} \\ &= {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} - \frac{{{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\left({{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right)}{\left({{\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.}}}{{{\left[{{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk} + {\left({{\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\left({{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right)}}\right]}}}. \\ \end{aligned} $$

(C3)

Now, the following identity for the inverse of a matrix (Searle 1982) is often referred to as the Sherman–Morrison–Woodbury identity:

$${\left({A + UGV}\right)}^{{- 1}} = A^{{- 1}} - A^{{- 1}} U{\left({G^{{- 1}} + VA^{{- 1}} U}\right)}^{{- 1}} VA^{{- 1}},$$

(C4)

where A and G are square invertible matrices and U and V matrices of the appropriate size to give UGV the same dimensions as A. Therefore, if we let:

$$\left. \begin{aligned}& A^{{- 1}} = {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} \\& U = {\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk} \\& V = {\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} \\& G^{{- 1}} = {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk} \\ \end{aligned}\right\},$$

(C5)

then we have

$$\begin{aligned} {\mathbf{T}}^{{- 1}}_{\rm s} &= A + UGV \\ &= {\mathbf{C}}_{{\rm s}\left| {\rm hk}\right.} + \frac{{{\left({{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right)}{\left({{\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}}}{{{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}}, \\ \end{aligned} $$

(C6)

which gives the covariance matrix, Eq. 16, for the multivariate Gaussian distribution in the integral of Eq. 14.

Now, for the mean vector, we must calculate T _s,hk; from Eqs. C1 and C2 we have:

$$\begin{aligned} - {\mathbf{T}}_{\rm s,hk} &= {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} \\ &\quad+ \frac{{{\left[{{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{1}}_{\rm s} - {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right]}{\left[{- {\mathbf{1}}^{\rm T}_{\rm s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + {\mathbf{1}}^{\rm T}_{\rm hk} {\left({{\mathbf{C}}^{-1}_{\rm hk} + {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s} {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk}}\right)}}\right]}}}{{{\left[{{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk} + {\left({{\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\left({{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right)}}\right]}}} \\ &= {\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} \\ &\quad- \frac{{{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\left[{{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right]}{\left[{{\left({{\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}}\right]}}}{{{\left[{{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk} + {\left({{\mathbf{1}}^{\rm T}_{\rm s} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{C}}_{\rm hk,s}}\right)}{\mathbf{C}}^{-1}_{{\rm s}\left| {\rm hk}\right.} {\left({{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}\right)}}\right]}}}. \\ \end{aligned} $$

(C7)

We can write this in terms of the matrices in Eq. C5:

$$\begin{aligned} - {\mathbf{T}}_{\rm s,hk}& = A^{{- 1}} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} - \frac{{A^{{- 1}} U{\left[{VA^{{- 1}} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} - {\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}}\right]}}}{{{\left[{G^{{- 1}} + VA^{{- 1}} U}\right]}}} \\ &= {\left\{{A^{{- 1}} - \frac{{A^{{- 1}} UVA^{{- 1}}}}{{{\left[{G^{{- 1}} + VA^{{- 1}} U}\right]}}}}\right\}}{\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + \frac{{A^{{- 1}} U{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}}}{{{\left[{G^{{- 1}} + VA^{{- 1}} U}\right]}}} \\ &= {\mathbf{T}}_{\rm s} {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + \frac{{A^{{- 1}} U{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}}}{{{\left[{G^{{- 1}} + VA^{{- 1}} U}\right]}}}. \\ \end{aligned} $$

(C8)

So, multiplying Eqs. C6 and C8 gives:

$$\begin{aligned} - {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} &= {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + {\mathbf{T}}^{-1}_{\rm s} \frac{{A^{{- 1}} U{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}}}{{{\left({G^{{- 1}} + VA^{{- 1}} U}\right)}}} \\ &= {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + {\left({A + UGV}\right)}\frac{{A^{{- 1}} U}}{{{\left({G^{{- 1}} + VA^{{- 1}} U}\right)}}}{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}, \\ \end{aligned} $$

(C9)

and we can simplify the multiplying term:

$$\begin{aligned} {\left({A + UGV}\right)}\frac{{A^{{- 1}} U}}{{{\left({G^{{- 1}} + VA^{{- 1}} U}\right)}}}& = \frac{{U + UGVA^{{- 1}} U}}{{G^{{- 1}} + VA^{{- 1}} U}} \\ &= UG{\left({\frac{{G^{{- 1}} + VA^{{- 1}} U}}{{G^{{- 1}} + VA^{{- 1}} U}}}\right)} \\& = UG. \\ \end{aligned} $$

(C10)

So, we have

$$ \begin{aligned} - {\mathbf{T}}^{-1}_{\rm s} {\mathbf{T}}_{\rm s,hk} &= {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + UG{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} \\ &= {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} + {\left({\frac{{{\mathbf{1}}_{\rm s} - {\mathbf{C}}_{\rm s,hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}}{{{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk} {\mathbf{1}}_{\rm hk}}}}\right)}{\mathbf{1}}^{\rm T}_{\rm hk} {\mathbf{C}}^{-1}_{\rm hk}, \\ \end{aligned} $$

(C11)

which yields Eq. 15 for the mean of the multivariate Gaussian distribution for z _s.