## 6 Alternate Bayesian Approach to Complete the Ad hoc Approach
\right)\sim N(\mu_{\mbox{post}},\Sigma_{\mbox{post}}) \tag{8}\]  
where  
\[\mu_{\mbox{post}}=\ (\Sigma^{-1}+\Sigma_{\mbox{prior}}^{-1})^{-1}(\Sigma^{-1}y+ \Sigma_{\mbox{prior}}^{-1}\mu_{\mbox{prior}})\]  
and  
\[\Sigma_{\mbox{post}}=(\Sigma^{-1}+\Sigma_{\mbox{prior}}^{-1})^{-1} \tag{9}\]  
For example, choosing  
\[\Sigma_{\mbox{prior}} = \left( \begin{array}{cc}100&100(1-.01^{j})\\ 100(1-.01^{j})&100\end{array} \right) \tag{10}\]  
results in \((\mu_{1},\mu_{2})\) estimates of (1.49, 1.00), (1.08, 0.92), and (0.89, 0.89) for \(j=1,2,\) and 3 or larger, respectively, for any moderate value of \(\mu_{\mbox{prior}}\) such as 0.1 to 10. The prior can be chosen such that the resulting estimate of \(\mu_{1}\) is within a small \(\epsilon\) of the estimate of \(\mu_{2}\). Notice that the (0.89, 0.89) values for the \((\mu_{1},\mu_{2})\) estimate agrees with the GLS method for \(j\geq 3\) in Equation 11 as claimed for the ad hoc approach of choosing \((y_{1},y_{2})\) pairs along the diagonal [8]. This Bayesian approach revises the ad hoc recipe and is easily implemented using MCMC for any likelihood. It is sometimes difficult to calculate the likelihood for complicated combinations of distributions involving for example sums and products of random variables (which can arise in modifications of the prescription to generate \(y_{1},y_{2}\) pairs). An appeal of the modified ad hoc approach is that the likelihood does not need to be computed. Instead, one need only be able to generate many \(y_{1},y_{2}\) pairs. This is the same appeal of the ABC approach in our context. To summarize this section, the ad hoc approach in [8] regards simulated \((y_{1},y_{2})\) pairs from any distribution as providing an estimate of \(\mu\) by specifying a small \(\epsilon\) and computing the mean of all scalar-valued \(y_{1}\) and \(y_{2}\) satisfying \(|y_{1}-y_{2}|<\epsilon\) (or by finding the value of \(\mu\) that maximizes the likelihood of the scalar data that arise from concatenating all \(y_{1}\) and \(y_{2}\) values that are similar). This approach has a corresponding formally correct Bayesian interpretation, but is easier to implement than the corresponding correct Bayesian implementation with a particular choice of the prior pdf. For Example 5.1 in Section 5, for which the distributions for \(\epsilon_{R1}\), \(\epsilon_{R2}\), and \(\epsilon_{S}\) are normal, the estimates arising from the Bayesian posterior maximum probability and the posterior mean are 0.89 and 0.89, respectively, both the same as the GLS estimate. * Example 6.1. If the distributions for \(\epsilon_{R1}\), \(\epsilon_{R2}\), and for \(\epsilon_{S}\) are centered and scaled product normals, the estimates for all small values of \(\epsilon\) (\(\epsilon\) less than approximately 0.01) are 1.10 and 1.09, respectively from the Bayesian posterior maximum probability and the posterior mean. If the distributions for \(\epsilon_{R1}\), \(\epsilon_{R2}\) are centered and scaled product normal and the distribution for \(\epsilon_{S}\) is a centered and scaled gamma, the estimates for all small values of \(\epsilon\) are 1.33 and 1.18, from the Bayesian posterior maximum probability and the posterior mean, respectively. The fact that the estimates can lie between \(y_{1}\) and \(y_{2}\) appeals in some way. * Example 6.2 The ad hoc method can be formally defended by the Bayesian framework just described. Therefore, the ad hoc approach is included among the candidates whose RMSEs were given in Example 5.2 from Section 5. In the second part of Example 5.2, there are 100 simulations of 10,000 observations of \((y_{1},y_{2})\) pairs (for which the RMSE of GLS and ABC are 0.22 and 0.15, respectively). The the RMSE of the ad hoc approach is 0.24 (repeatable to within \(\pm 0.01\)). For Example 6.2, although the RMSEs from low to high (low is good) are ABC, GLS, and alternate Bayes, for this example, we do not make any general performance claims ranking GLS, ABC, and the modified ad hoc (\"alternate Bayes\") approaches.