## 5 ABC for PPP  
This section again uses the slightly modified form of ABC that relies on density estimation rather than using summary statistics computed from the real and simulated data, but MCMC is applied in the context of PPP.  
As mentioned in Section 2, suppose from two experiments \(m_{1}=\mu+\epsilon_{R1}\) and \(m_{2}=\mu+\epsilon_{R2}\), where \(\epsilon_{R1}\) is zero-mean random error in \(m_{1}\) and similarly for \(\epsilon_{R2}\). Then if \(y_{1}=m_{1}+\epsilon_{S}\) and \(y_{2}=m_{2}+\alpha\epsilon_{S}\), where \(\epsilon_{S}\sim N(0,\sigma_{S})\) and \(\alpha\) is any positive scale factor other than 1, the covariance matrix \(\Sigma\) of \((y_{1},y_{2})\) can lie in the PPP region [2].  
To implement our version of ABC, many \((y_{1},y_{2})\) realizations are simulated from each of many values of \(\mu\). Given the observed \((y_{1},y_{2})\) and its known covariance \(\Sigma\), we can again set up good \(L\) and \(U\) values for the grid of candidate \(\mu\) values.  
A key advantage of ABC is that any probability distribution can be easily accommodated for any of \(\epsilon_{R1}\), \(\epsilon_{R2}\), and \(\epsilon_{S}\). Here are two examples.  
* Example 5.1. Assume \(\epsilon_{R1}\), \(\epsilon_{R2}\), and \(\epsilon_{S}\) are all normal, so that \(y_{1},y_{2}\) is bivariate normal. Then as it must, our version of ABC recovers the GLS estimate \(\hat{\mu}=0.89\).
* Example 5.2. Assume \(\epsilon_{R1}\) and \(\epsilon_{R2}\) each have a centered and scaled lognormal distribution and \(\epsilon_{S}\) has a centered and scaled gamma distribution, with parameters chosen so that the covariance matrix \(\Sigma\) of \(y_{1},y_{2}\) is given by Equation 1. Figure 2 plots the log likelihood for \(\mu\) values in the \(\mu\) grid for the correct (true) likelihood for \(y_{1},y_{2}\) and assuming a bivariate normal distribution for \(y_{1},y_{2}\). These log likelihoods are meaningfully different and one might expect the corresponding estimators to behave quite differently. For the single realization of \(y_{1},y_{2}=1.08,1.24\), for which Figure 2 was generated, the GLS and ABC estimate assuming a bivariate normal distribution for \(y_{1},y_{2}\) is 1.27. The ABC estimate assuming the correct bivariate distribution for \(y_{1},y_{2}\) is 1.03. **Figure 2.*
* Example ABC. The solid black curve is for a bivariate normal. The dotted red curve is for a bivariate non-normal (combination of Gamma and lognormal).
* Example 5.2 continued. To investigate whether using the correct likelihood leads to smaller RMSE, we performed 100 simulations for various values of \(\mu\) in this same example 5.2. In each simulation, one new \((y_{1},y_{2})\) pair was generated to represent the observed data pair, and then 10,000 observations of \((y_{1},y_{2})\) pairs were generated to use for density estimation so that the likelihood of each candidate value of \(\mu\) could be approximated. For example, with \(\mu=1.0\), the RMSE for GLS is 0.22 as expected, and the RMSE for ABC using the correct bivariate distribution is 0.15 (repeatable across sets of 100 simulations to \(\pm 0.01\)). So, ABC has smaller RMSE than GLS in this case. However, ABC relies much more strongly on the estimated likelihood which relies on the assumed data distributions, so is not expected to be as robust to misspecifying the likelihood as GLS. Recall that \(\hat{\mu}\) via GLS is always outside the range of \((y_{1},y_{2})\) when PPP occurs, and that some researchers are bothered by this. For the non-normal data just described, even though PPP occurs, we find numerically that \(\hat{\mu}\) via ABC sometimes lies outside the range of \((y_{1},y_{2})\) and sometimes lies within the range of \((y_{1},y_{2})\).