# Application of the Approximate Bayesian Computation Algorithm to Gamma-Ray Spectroscopy
## 6 A New ABC-Based RIID
This is an ad hoc but defensible option to generate bias vectors on the basis of 10 paired comparisons between the real and simulated data. In Figure 10, the modeled bias vector was group-specific, so the five sets of \(\epsilon\) values were only generated one time for the Lu177m isotope group and one time for the WgPu group. For the same example, for one test case, Figure 11 plots the ABC-based probability for model 1, model 2, model 3, and model 4, with the true model being model 1 (a), model 2 (b), model 3 (c), or model 4 (d). One calibration check in this ABC-based RIID is whether the estimated posterior probability of the chosen model provides a good measure of confidence. Using the \(10^{5}\) sets of simulated data that were to train ABC and to produce Figure 11 for one test case, the average probability of the chosen model is 0.91 in 1000 simulated test cases from each of the four groups. The chosen model is model 1 if model 1 has the maximum probability, model 2 if model 2 has the maximum probability, etc. The ABC-based RIID had a good correct classification rate, of 0.96. Because the estimated average confidence (probability) in the predicted group 0.91 is reasonably close to 0.96, this ABC-based RIID is reasonably well calibrated, but there is room for improvement. Note that these percentages are repeatable within \(\pm\)0.01, so 0.91 is clearly distinct and slightly pessimistic compared to the observed 0.96. To illustrate that model bias can matter, in another set of \(10^{5}\) simulations of Lu-177m and WgPu with 10% larger average relative model bias than the previous, the ABC-based RIID had a correct classification rate of 0.65. For the \(10^{5}\) sets of simulated data used to train ABC, the average probability of the chosen model is 0.60, so this ABC-based RIID is fairly well calibrated, but again there is room for improvement because 0.60 is distinct from 0.65. Additionally, 0.65 is far from 1, so future work to improve this ABC-based classifier will experiment with other summary statistics. Figure 12 plots PC2 versus PC1 for the spectra from each of five common naturally occurring radioactive materials (NORMs), using (a) 5 min count times and (b) 0.5 min count times with item-specific modeling of the bias vector. Item-specific bias vectors were generated by drawing a random set of five values of \(\epsilon\) in \(B_{new}=\sum\limits_{1=1}^{5}C_{i}(1+\epsilon)P_{i}\) for each item rather than for each isotope group. Because isotope quantities are not specified in this paper, the 5 min count time is specified to indicate a factor of \(1/10\) smaller random error variance arising from the Poisson count rate than the 0.5 min count time. ABC was implemented for this five-group library using 50,000 training cases; the percent correct classification was 0.93 and the confusion matrix is in Table 1 with the true group in the column and the inferred group in the row. There was no attempt to optimize the summary statistics; instead, the scaled counts in 100 equally spaced energy bins from 50 to 250 keV were used as summary statistics, so improvements are possible. Additionally, regarding calibration, the average ABC-based probability for the chosen group (the group that has the largest membership fraction in the estimated posterior) is 0.89, which compared favorably to the observed success probability of 0.93. Another important isotope to recognize is U-235, whose spectrum can be confused with a few other isotopes, such as Cu-67 or Ga-67 because of nearby strong peaks as seen in Figure 13 in the low energy region. Figure 14a plots PC2 versus PC1 calculated from each of 10 simulated spectra from each of U-235 (indicated as HEU), Cu-67, and Ga-67 for a 5 min count time. Figure 14b is the same, but with a 0.5 min count time and item-specific model bias added as described above.