# Overview of Algorithms for Using Particle Morphology in Pre-Detonation Nuclear Forensics
## 5 ABC
### Model Uncertainty Accommodated by ABC for Inverse Problems in Forensics
For ABC-based model selection effects in the regression context, first a model form such as \(y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}\ x_{2}+\beta_{3}\ x_{3}+\beta_{4}\ x_{1}^{2}+ \beta_{5}\ x_{2}^{2}+\beta_{6}\ x_{3}^{2}+\beta_{7}\ x_{1}x_{2}+\beta_{8}\ x_{1}x_{3}+ \beta_{9}\ x_{2}x_{3}+e\) is specified in the case of \(p=3\) continuous predictors \(x_{1},x_{2},x_{3}\), for each of \(q=5\) measured responses (such as 5 of the 22 MAMA measurements, or transformations of the MAMA measurements). Standard design of experiments assumes that the model form is known in advance, and effective training values of \(x_{1},x_{2},x_{3}\) are selected in order to have good predictions of future y values at future \(x_{1},x_{2},x_{3}\) values. In practice, EDA can suggest other candidate models and to keep this example short and simple, the impact of model selection will be illustrated when the true model is either (1) a 4-parameter model having only linear terms (\(y=\beta_{0}+\beta_{1}x_{1}+\beta_{2}\ x_{2}+\beta_{3}\ x_{3}+e\)) without interactions such as \(\beta_{7}\ x_{1}x_{2}\), or (2) is the full 10-parameter model model with parameters \(\{\beta_{0},\beta_{1},\ldots\beta_{9}\}\). ABC will then be implemented assuming the 4-parameter model, the 10-parameter, or using BIC to choose between the two models. ABC is implemented by simulating possible \(x_{1},x_{2},x_{3}\) values and corresponding \(y_{1},y_{2}\ldots,\ y_{q}\) values for each set of \(\{\beta_{0},\beta_{1},\ldots\beta_{9}\}\) values simulated from their respective prior probability distributions. The measurement error variances in \(x_{1},x_{2},x_{3}\) and \(y_{1},y_{2}\ldots,\ y_{q}\) are assumed here to have been well estimated by using small repeatability studies. The BIC for comparing regression models (1) and (2) fit to the same data is given by  
\[BIC=n\ \log\left(\frac{RSS}{n}\right)-p\ \log\left(n\right) \tag{6}\]  
where RSS in Equation (6) is the residual sum of squares \(RSS=\sum_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}\), \(n\) is the number of observations (\(n=20\) for the central composite design as used to generate Figure 18 as in this example) and \(p\) is the number of model parameters (\(p=10\) for model 1 and \(p=4\) for model 2). Then for a randomly-generated test case, the ABC-based posterior standard deviations (SDs, repeatable to \(\pm 0.001\)) for \(x_{1},x_{2},x_{3}\) are 0.046, 0.050, 0.064, respectively, if model 1 is the true and assumed model; the SDs are 0.054, 0.058, 0.077, respectively, if model 1 is the true and model 2 is the chosen model, and the SDs are 0.068, 0.043, 0.059, respectively, if model 1 is the true model and the model selection process is included in the training data for ABC, so model 1 is assumed with relative probability \(P(M_{1})\propto e^{-\frac{BIC_{1}}{2}}\), which has an average value over 105 simulations of approximately 0.65 for each of the 5 responses in this example. The model selection process can be included in ABC by simulating from model 1 with 10 estimated parameters with the BIC-based model relative probability \(P(M_{1})\propto e^{-\frac{BIC_{1}}{2}}\) and by simulating from model 2 with relative probability \(P(M_{2})\propto e^{-\frac{BIC_{2}}{2}}\). Such a simulation procedure is a representation of our state of knowledge after the \(n=20\) run central composite design, such as a BIC-based approximate 65% probability that model 1 is more appropriate than model 2 when model 1 is the correct model. For the same simulated test case as above, having true values \(x_{1}=0.71,x_{2}=-0.99,x_{3}=-0.10\), Figure 19 plots the ABC-based posterior pdf for the factor \(x_{1}\) ignoring the effects of model selection and including the effects of model selection, for the central composite design with 20 runs. The three cases plotted are: model 1 is the true model and (a) model 1 is always the chosen model, (b) model 2 is always the chosen model, and (c) BIC-based model selection chooses between model 1 and 2 in the ABC training data.