## 6 Alternate Bayesian Approach to Complete the Ad hoc Approach  
Reference [8] presented an ad hoc procedure to estimate \(\mu\) as follows. Using a model of the two experiments that produce \(y_{1},y_{2}\) such as that by [4] in Section 2, generate thousands of a \(y_{1},y_{2}\) pairs but regard them as being \(\mu_{1},\mu_{2}\) pairs. See below for \"justification\" in switching from \(y_{1},y_{2}\) to \(\mu_{1},\mu_{2}\). Because of the strong prior belief that \(\mu_{1}\approx\mu_{2}\), restrict attention to bivariate points \((y_{1},y_{2})\) lying very near the diagonal, defining \(\hat{\mu}\) as the value that maximizes the estimated probability density of the scalar-values of either \(y_{1}\) or \(y_{2}\) (or both \(y_{1}\) and \(y_{2}\)) satisfying \(|y_{1}-y_{2}|<\epsilon\) for some small \(\epsilon\) near 0. See Figure 3. As an important aside, recall [2] showed that PPP cannot occur with this error model (Section 2). However, because of how \(\Sigma\) is estimated (using \(m_{1}\) or \(m_{2}\) to estimate \(\mu_{m}\)), PPP appears to occur with this error model. In one dimension, the analogous ad hoc procedure for a sample of size \(n\) from a normal distribution, is to assume \(\mu\sim N(\overline{y},\sigma^{2}/n)\) in the case of unknown \(\mu\) and known variance \(\sigma^{2}\) with \(\overline{y}=\sum_{i=1}^{n}y_{i}/n\). The formally correct posterior for \(\mu\) depends on the likelihood and on the prior, but for a normal likelihood \(y\sim N(\mu,\sigma^{2})\) and a normal prior \(\mu\sim N(\mu\text{p},\tau^{2})\), as the prior variance \(\tau^{2}\rightarrow\infty\), the prior becomes \"flat,\" and the ad hoc procedure is well known to be correct in an asymptotic sense. Although this ad hoc approach in two dimensions is a recipe that can be followed to produce \(\hat{\mu}\), it is statistically incomplete. It can be completed and justified by adopting a two dimensional prior for \(\mu\). That is, to regard \(y_{1},y_{2}\) pairs as being \(\mu_{1},\mu_{2}\) pairs requires that \(y_{1},y_{2}\) pairs be interpreted as being observations from the posterior distribution for \(\mu_{1},\mu_{2}\) in a Bayesian sense. Second, to formalize the notion of restricting attention to bivariate points \((\mu_{1},\mu_{2})\) lying very near the diagonal, we must define a prior probability distribution. Specifically, assume  
\[\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right)\sim N(\mu,\Sigma) \tag{5}\]  
where  
\[\mu=\left( \begin{array}{c}\mu_{1}\\ \mu_{2}\end{array} \right)\text{ and }\Sigma=\left( \begin{array}{cc}\sigma_{1}^ {2}&\rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{array} \right)\]  
Let the prior for \(\mu\) in Equation 5 be  
\[\left( \begin{array}{c}\mu_{1}\\ \mu_{2}\end{array} \right)\sim N(\mu_{\text{prior}},\Sigma_{\text{prior}}) \tag{6}\]where  
\[\mu_{\mbox{\small{prior}}}=\left( \begin{array}{c}\mu_{\mbox{\small{p}}1}\\ \mu_{\mbox{\small{p}}2}\end{array} \right)\mbox{ and }\Sigma_{\mbox{\small{prior}}}=\left( \begin{array}{cc}\sigma_{\mbox{\small{p}}1}^{2}&\rho\sigma_{\mbox{\small{p}} 1}\sigma_{\mbox{\small{p}}2}\\ \rho\sigma_{\mbox{\small{p}}1}\sigma_{\mbox{\small{p}}2}&\sigma_{\mbox{\small{ p}}2}^{2}\end{array} \right).\]  
The recipe to restrict attention to simulated values satisfying \(y_{1}\approx y_{2}\) is ad hoc, but for a given choice of \(\epsilon\) to define \(y_{1}\approx y_{2}\) as satisfying \(|y_{1}-y_{2}|<\epsilon\), there is a defensible Bayesian posterior corresponding to a choice for \(\Sigma_{\mbox{\small{prior}}}\). And, given  
\[y=\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right) \tag{7}\]  
the posterior distribution is  
Figure 3: Example ABC. The ad hoc “go along the diagonal” approach in [8]. \[\left( \begin{array}{c}\mu_{1}\\ \mu_{2}\end{array}