Prior stage

The goal of the prior stage is to maximize information relative to the general knowledge \(\mathcal{G}\) (Serre, 1999), which begins with a set of assumptions and constraints. Equation (2) represents the mathematical formulation of the prior knowledge in terms of the constraints \[\begin{equation}\label{eqn:E2} \bar{g}_\beta = \int g_\beta\left(\boldsymbol{z}_{map}\right) f_{\mathcal{G}}\left(\boldsymbol{z}_{map}\right) d\boldsymbol{z}_{map},\ \ \beta = 1, 2, \dots, N_{\beta}. \end{equation}\] , N_{}. \end{equation} Here, \(g_{\beta}\) are a set of known functions of \(\boldsymbol{z}_{map}\), and \(f_{\mathcal{G}} \left(\boldsymbol{z}_{map}\right)\) is the prior pdf of \(\boldsymbol{Z}_{map}\) that represents our knowledge before observing the data. The expectations \(\bar{g}_{\beta}\) provide the spatial moments of interest. There are a series of popular choices regarding the form of \(g_{\alpha}\) to represent the prior knowledge. For example, (3) and (4) represent the means, and covariances, where \(|\mathcal{H}|\) and \(|\mathcal{S}|\) are the sizes of hard and soft data, respectively. \[\begin{equation}\label{eqn:cons_mean} g_{\alpha}\left(\boldsymbol{z}_{map}\right) =\boldsymbol{z}_{map,\alpha},\ \ \alpha=1,\cdots,|\mathscr{H}|+|\mathscr{S}|+1, \end{equation}\]s,||+||+1, \end{equation} \[\begin{eqnarray}\label{eqn:cons_cov} g_{\alpha}(\boldsymbol{z}_{map})= \left(\boldsymbol{z}_{map,\alpha}-\bar{\boldsymbol{z}}_{map,\alpha}\right) \left(\boldsymbol{z}_{map,\alpha'}-\bar{\boldsymbol{z}}_{map,\alpha'}\right),\ \ \alpha, \alpha' =1,\cdots,|\mathscr{H}|+|\mathscr{S}|+1, \end{eqnarray}\]athscr{H}|+||+1, \end{eqnarray} Higher spatial orders, if desired, can be formulated similarly. Shannon’s information entropy is used as a measure of uncertainty contained in the prior pdf \(f_{\mathcal{G}}\), which is expressed as
\[\begin{equation}\label{eqn:shannon} \mathbb{E}\left[\text{Info}_\mathcal{G}(\boldsymbol{Z}_{map})\right] = - \int \log\left[{f_\mathcal{G} (\boldsymbol{z}_{map})}\right] \hspace{.2em} f_\mathcal{G} (\boldsymbol{z}_{map}) \hspace{.2em} d \boldsymbol{z}_{map}. \end{equation}\] In general, a more concentrated pdf has lower entropy. To determine the shape of the prior pdf, a process is carried out that maximizes the entropy function (\(\ref{eqn:shannon}\)) subject to the constraints specified in (2) that represent the prior knowledge . In particular, if the constrains are restricted to the first two spatial moments (3)-(4), the \(f_\mathcal{G}\) that maximizes the Shannon’s entropy (\(\ref{eqn:shannon}\)) is the multivariate Gaussian distribution with mean \(\overline{\boldsymbol{z}}_{map}\) and covariance matrix \(\textbf{C}_{map}\) . In our study, since the data is the regression residual and has zero mean, we have \(\overline{\boldsymbol{z}}_{map} = \boldsymbol{0}\), and consequently the prior pdf is given by \[\begin{equation}\label{eqn:prior_pdf} f_\mathcal{G} (\boldsymbol{z}_{map})=(2\pi)^{-(n+1)/2} \rvert{\textbf{C}_{map}\lvert}^{-1/2} \exp\left\{ -\dfrac{1}{2} \boldsymbol{z}_{map}^T \hspace{.3em}\textbf{C}_{map}^{-1}\boldsymbol{z}_{map} \right\}, \end{equation}\] where \(\textbf{C}_{map}\) has the form \[\begin{equation}\label{eqn:C} \textbf{C}_{map} = \begin{bmatrix} \textbf{C}_{h,h} & \textbf{C}_{h,s} & \textbf{C}_{h,k} \\ \textbf{C}_{s,h} & \textbf{C}_{s,s} & \textbf{C}_{s,k} \\ \textbf{C}_{k,h} & \textbf{C}_{k,s} & \textbf{C}_{k,k} \end{bmatrix} \end{equation}\] with regards to the covariances of hard data, soft data, and prediction locations, as well as their cross-covariances.

Pre-posterior stage

This stage considers the specification knowledge \(\mathcal{S}\), mostly the physical data \(\boldsymbol{z}_{data} = \boldsymbol{z} = \{\boldsymbol{z}_h, \boldsymbol{z}_s \}\), to be incorporated in the posterior stage below to provide information leading to the posterior estimation. The process mainly involves collecting and organizing additional information in the forms of hard and soft data suitable for the BME framework. In practice, this is often a nontrivial process and considered more of an art rather than a quantitative task. The soft data in this study are restricted to intervals, but generally they can take a variety of other forms such as probability functions, physical models, and functional data. An important feature of BME, and kriging as well, is that some or all of the data are often used twice in the whole estimation process. More specifically, data are used implicitly in the prior stage to derive the prior pdf, and then used again explicitly in the posterior stage to calculate the posterior estimates. Particularly, in our study, the component blocks \(\boldsymbol{C}_{h,h}, \boldsymbol{C}_{s,s}, \boldsymbol{C}_{k,k}, \boldsymbol{C}_{s,h}, \boldsymbol{C}_{k,h}\) of the prior covariance matrix \(\boldsymbol{C}_{map}\) defined in (\(\ref{eqn:C}\)) are estimated empirically from the data \(\boldsymbol{z}_{data}\) by the corresponding sample covariances. The covariance \(\boldsymbol{C}_{s,s}\) for the sole soft data is computed with an added uncertainty from a prior distribution \(f_s(\boldsymbol{z}_s)\) for the soft data. More precisely, the diagonals of \(\boldsymbol{C}_{s,s}\) are defined as the sum of the usual covariance function at lag \(0\) and the variance of the prior soft data distribution, i.e., \[\begin{equation} diag\left\{\boldsymbol{C}_{s,s}\right\} =C(0)+\sigma^2\left\{f_s(\boldsymbol{z}_s)\right\}, \end{equation}\] where \(C(0)\) is estimated by the sample variance. The value of \(\sigma^2\left\{f_s(\boldsymbol{z}_s)\right\}\) depends on the form of \(f_s\), which is usually assumed to be a uniform distribution without additional prior information, and thus takes the form
\[\begin{equation} \sigma^2\left\{f_s(\boldsymbol{z}_s)\right\}=\frac{1}{12}\left(\boldsymbol{b}-\boldsymbol{a}\right)^2, \end{equation}\] where \(\boldsymbol{b}\) and \(\boldsymbol{a}\) are the upper and lower bounds of the soft-intervals.

Posterior stage

The goal of the posterior stage is to maximize the posterior probability given both \(\mathcal{G}\) and \(\mathcal{S}\), with a focus on coherency. This stage is further divided into two steps. In the first step, the posterior probability density for any realization \(\tilde{\boldsymbol{z}}\) of all locations conditioning on the data \(\boldsymbol{z}(\boldsymbol{x})\) is computed according to the Bayes rule as: \[\begin{equation}\label{eqn:post} \pi(\tilde{\boldsymbol{z}})=A^{-1}l(\boldsymbol{z}|\tilde{\boldsymbol{z}})f_G(\boldsymbol{z}), \end{equation}\] where \(A\) is a normalizing constant. The likelihood function \(l(\boldsymbol{z}|\tilde{\boldsymbol{z}})\) of the data \(\boldsymbol{z}(\boldsymbol{x})\) given a realization \(\tilde{\boldsymbol{z}}\) is the key in this Bayesian formulation, which is defined for three cases. If \(\boldsymbol{z}(\boldsymbol{x}_h)\ne\tilde{\boldsymbol{z}}_h\), then the likelihood is zero. If \(\boldsymbol{z}(\boldsymbol{x}_h)=\tilde{\boldsymbol{z}}_h\) and there are no soft data, then the likelihood is one. The third case is when the hard data coincides exactly with the realization \(\tilde{\boldsymbol{z}}_h\) and there are additional soft data \(\boldsymbol{z}(\boldsymbol{x}_s)\), and the likelihood is \(l(\boldsymbol{z}(\boldsymbol{x})|\tilde{\boldsymbol{z}})=\Pi_{i=1}^{n_s}f_{s,i}(\tilde{z}_{s,i}):=f_s(\tilde{\boldsymbol{z}}_s)\). Namely, \[\begin{equation}\label{eqn:likelihood} l(\boldsymbol{z}(\boldsymbol{x})|\tilde{\boldsymbol{z}})= \begin{cases} f_s(\tilde{\boldsymbol{z}}_s)=\Pi_{i=1}^{n_s}f_{s,i}(\tilde{z}_{s,i}) & \boldsymbol{z}_h=\tilde{\boldsymbol{z}}_h,\ n_s>0,\\ 1, & \boldsymbol{z}_h=\tilde{\boldsymbol{z}}_h,\ n_s=0,\\ 0, & \boldsymbol{z}_h\ne\tilde{\boldsymbol{z}}_h. \end{cases} \end{equation}\] In the second step, the posterior pdf \(\pi(\tilde{z}_k)\) of \(Z(\boldsymbol{x}_k)\) is calculated as the marginal of the posterior joint pdf (\(\ref{eqn:post}\)). For the case when there are both hard and soft data, since the hard data are fixed values, the integration is only with respect to the soft data. More precisely, \[\begin{eqnarray*} \pi(\tilde{z}_k) &=& A^{-1}\int\pi(\tilde{\boldsymbol{z}})d(\tilde{\boldsymbol{z}}_h, \tilde{\boldsymbol{z}}_s) =A^{-1}\int l(\boldsymbol{z}|\tilde{\boldsymbol{z}})f_G(\tilde{\boldsymbol{z}}) d(\tilde{\boldsymbol{z}}_h, \tilde{\boldsymbol{z}}_s)\\ &=&A^{-1}\int_{\tilde{z}_s}f_s(\tilde{\boldsymbol{z}}_s)f_G(\boldsymbol{z}_h,\tilde{\boldsymbol{z}}_s,\tilde{z}_k)d(\tilde{\boldsymbol{z}}_s). \end{eqnarray*}\] For simplification of notation, the above formula for the posterior pdf of \(Z(\boldsymbol{x}_k)\) is equivalently expressed as \[\begin{equation}\label{eqn:post-k} \pi(z_k)=A^{-1}\int f_s(\boldsymbol{z}_s)f_G(\boldsymbol{z}_h, \boldsymbol{z}_s, z_k)d\boldsymbol{z}_s, \end{equation}\] with the note that the \(\boldsymbol{z}_s\) in the integral is not the actual soft data but any realization within the associated domain. For the special case when \(\boldsymbol{z}_s\) are intervals with lower bounds \(\boldsymbol{a}\) and upper bounds \(\boldsymbol{b}\), the posterior pdf (\(\ref{eqn:post-k}\)) takes the form: \[\begin{equation}\label{eqn:post-k-int} \pi(z_k)=A^{-1}\int_{\boldsymbol{a}}^{\boldsymbol{b}}f_G(\boldsymbol{z}_h, \boldsymbol{z}_s, z_k)d\boldsymbol{z}_s, \end{equation}\] where \(\int_{\boldsymbol{a}}^{\boldsymbol{b}}\) is an \(n_s\)-dimensional integral over the intervals \([a_i, b_i]\), \(i=1,\cdots, n_s\).

The updated pdf (\(\ref{eqn:post-k}\)) at the posterior stage is a conditional pdf expressed in terms of the prior pdf and the new data and information considered at the pre-posterior stage as follows \[\begin{equation}\label{E6} f_{\mathcal{K}}(z_k|\boldsymbol{z}_{data}) = A^{-1} \int f(\boldsymbol{z}_s) f_\mathcal{G} (\boldsymbol{z}_{map}) d\boldsymbol{z}_s. \end{equation}\] Equation (\(\ref{E6}\)) represents an application of the Bayes law, where \(\boldsymbol{z}_{data}\) indicates a known knowledge and \(z_k|\boldsymbol{z}_{data}\) represents the potential values \(z_k\) of the unknown knowledge. It is important to recognize that Equation (\(\ref{E6}\)) differs from the traditional Bayesian law by using the mapping prior \(f_{\mathcal{K}}(\boldsymbol{z}_{map})\) rather than the usual prior \(f_{\mathcal{G}}(z_k)\). The mathematical analysis involved in the three stages of the BME approach leads to the posterior pdf at location \(\boldsymbol{x}_k\) expressed as \[\begin{equation}\label{E7} f_{\mathcal{K}}(z_k) = A^{-1} \int^{\boldsymbol{b}}_{\boldsymbol{a}} f_\mathcal{G} (\boldsymbol{z}_s | \boldsymbol{z}_h, z_k) d\boldsymbol{z}_s f_\mathcal{G} (\boldsymbol{z}_h, z_k). \end{equation}\] This posterior pdf, which may not be Gaussian provides a full picture of the prediction situation at the target point. The posterior information, which refers to the updated knowledge after the data \(\boldsymbol{z}(\boldsymbol{x})\) of the pre-posterior stage have been incorporated, is given by \[\begin{equation*} \text{Info}_\mathcal{G}(z_k|\boldsymbol{z}_{data}) = -\log\left[{f_{\mathcal{K}} (z_k|\boldsymbol{z}_{data})}\right]. \end{equation*}\] Interested readers are referred to for a more detailed description of the BME framework.

BME estimate and uncertainty

For a targeted location \(\boldsymbol{x}_k\) that is typically found on a regular grid that includes specific points \(\boldsymbol{x}_i, i = 1,\dots,n\) of the available data \(\boldsymbol{z}_{data}\), the BME prediction is defined as the posterior mean
\[\begin{equation}\label{eqn:post-mean} \bar{z}_{k|\mathcal{K}} = \int z_k f_{\mathcal{K}}(z_k)d z_k, \end{equation}\] which is a nonlinear function of the available data and aims to minimize the squared prediction error. The prediction variance defined as \[\begin{equation}\label{eqn:post-var} \sigma^2_{k|\mathcal{K}} = \int \left(z_k - \bar{z}_{k|K} \right)^2 f_{\mathcal{K}}(z_k)d z_k, \end{equation}\] provides a measure of uncertainty associated with the predicted values. It is of note that the quantity \(\sigma^2_{k|\mathcal{K}}\) corresponds to the variance of the prediction error \(e_k = z_k - \bar{z}_{k|\mathcal{K}}\), not the variance of the actual predictor \(\bar{z}_{k|K}\). It is a commonly used accuracy measure in traditional estimation theory that effectively evaluates the estimation error, especially when the error distribution is not overly complex.