## 4 Approximate Bayesian Computation  
Bayesian analysis requires a prior probability \(p(\theta)\) for the parameters \(\theta\), and a likelihood \(f(D|\theta)\) for the data given the parameter values. The result is a posterior probability \(\pi(\theta|D)\propto f(D|\theta)p(\theta)\) for the parameters \(\theta\). For many priors and likelihoods, calculating the posterior is difficult or impossible, but observations from the posterior can be obtained by Markov chain Monte Carlo (MCMC). MCMC is a Monte Carlo sampling scheme that accepts a candidate value \(\theta^{\prime}\) with probability that depends on the relative probability of the current \(\theta\) value and the candidate \(\theta^{\prime}\) value. The goal of MCMC is to simulate many observations from the posterior, and use summary statistics such as posterior means and standard deviations to characterize the state of knowledge about the parameters. In our examples, \(\theta\) is the scalar-valued mean which we denote throughout as \(\mu\). ABC has arisen relatively recently in applications for which the likelihood \(f(D|\theta)\) is intractable [22, 23], but data can be simulated somehow,often relying on detailed simulations of physical processes. In our context, the simulation is very simple, based on realizations of the measurement error pdfs. And if some subset of the three underlying error distributions described in the background section are non-normal, leading to a complicated bivariate distribution for \(y_{1},y_{2}\) that could be difficult to derive analytically, then ABC is attractive. Predating ABC is a nonBayesian (\"frequentist\") method referred to as \"inference for implicit statistical models\" [21, 24] that was based on density estimation. We use the term ABC to include any Bayesian method in a situation for which an analytical likelihood is not available, but simulations can produce realizations from the likelihood. In the common version of ABC, because data can be simulated from the pdf, candidate values \(\theta^{\prime}\) can be accepted with a probability that depends on how close certain summary statistics such as the first few moments computed from the simulated data having parameter value \(\theta^{\prime}\) are to the corresponding summary statistics from the real data. Some sources [24] reserve the term ABC for methods that rely on such summary statistics, so would not use the term ABC with density estimation. Our combination of MCMC with density estimation appears to be a new combination of ideas, though some might not refer to it as ABC. Regardless of the jargon, the method will be clearly delineated below. As a simple example to motivate our modified form of ABC, suppose \(y_{1},y_{2},\ldots,y_{n}\) are independently and identically distributed as \(N(\mu,\sigma)\) and it is known that \(\sigma=0.1\). Introductory statistics texts usually prove that the sample mean \(\overline{y}\) is the ML estimator (MLE) for \(\mu\) and more advanced texts investigate properties of MLEs. And, Equation (3) with a diagonal \(\Sigma\) (because in this example the \(y_{i}\) are independent) can be used to show that \(\overline{y}\) is the GLS estimator so is also the BLUE and because the data is normal in this simple example (but not in our extended PPP examples below), \(\overline{y}\) is also the MVUE. Specify the prior \(p(\mu)\) to be \(N(\mu_{\mbox{prior}},\sigma_{\mbox{prior}}^{2})\). Then, because the normal prior for \(\mu\) is the \"conjugate\" prior (meaning that the posterior distribution is the same type of distribution as the prior distribution) for the normal likelihood, the posterior \(\pi(\theta|D)\) is \(N(\mu_{\mbox{post}},\sigma_{\mbox{post}}^{2})\), where \(\mu_{\mbox{post}}=\frac{0.1^{2}}{0.1^{2}+\sigma_{\mbox{prior}}^{2}}\mu_{ \mbox{prior}}+\frac{\sigma_{\mbox{prior}}^{2}}{0.1^{2}+\sigma_{\mbox{prior}}^ {2}}\overline{y}\). Our modified ABC that relies on density estimation has the following steps:  
1. Determine a lower (\(L\)) and upper bound (\(U\)) for the parameter \(\mu\).