## 12 Future Directions
### Bayesian Variable Selection  
By \"variable selection,\" we mean to choose the subset of isotopes from the chemical library that are thought to be present in the source. Bayesian Model Averaging (BMA) is one option to implement variable selection. BMA refers to averaging over subsets of candidate radio-isotopes. Variable selection continues to generate research [36, 37] in a wide variety of applications. To infer which isotopes are present in a source, one implementation of BMA consists of the following steps. 1. Define an upper bound \(\mathrm{N_{max}}\) for the number of isotopes that could be present in a source, such as \(\mathrm{N_{max}=5}\). 2. Fit the background spectrum using a library of isotopes that are commonly found in the relevant background. Assume that background measured without the source is negligibly different than  
Figure 7: Example Co-60 spectra from a 1,024 channel NaI detector. (top left) real spectra; (top right, bottom left, bottom right) synthetic spectra from MCNP with large, medium, and small background counts, respectively. the background measured with the source so that the fitted background can effectively be \"subtracted out\" from the background plus source spectrum. Because of background suppression by the source, this \"negligibly different assumption\" is not exactly correct ([5],[18]), and the impact of this assumption should be the subject of additional study. 3. Evaluate the likelihood \(P(M_{j}\,|\,D)\) for each subset \(M_{j}\) consisting of 0, 1, 2, 3, \(\ldots\), N\({}_{\rm max}\) isotopes for given data \(D\). We write \(D\) to represent either the measured spectrum or perhaps a processed spectrum that extracts peak information, and/or smooths the measured spectrum, etc. There are 1+\(\left( \begin{matrix}L\\ 1\end{matrix} \right)+\left( \begin{matrix}L\\ 2\end{matrix} \right)+\ldots+\left( \begin{matrix}L\\ N_{{}_{Max}}\end{matrix} \right)=M\) possible subsets, where L is the size of the isotope library, which is approximately 200 in the broadest challenge problem. The effects of absorbers can be treated using the MBS [13] approach by having a representative library of approximately 3 to 5 absorbers. 4. Apply BMA to evaluate the probability that isotope C is present, using P(C\(|\)D) = \(\sum\limits_{j=1}^{M}I(C\in M_{j})P(M_{j}\,|\,D)\), where \(I(\cdot)\) equals 1 if its argument is true. That is, to estimate P(C\(|\)D), we simply sum the model probabilities for each subset that includes isotope C.  
Although the estimated probabilities \(P(M_{j}|D)\) have varying accuracy depending on the data \(D\), BMA is one of the most effective strategies for variable subset selection, particularly when prior information such as some isotope combinations being unlikely is available. For data \(D\) and a probability model for the data, it would be ideal if we could calculate the exact probability of each isotope subset. By [PERSON] theorem, \(P(M_{i}\,|\,D)\propto P(D\,|\,M_{i})P(M_{i})\), where P(\(M_{1}\)) is the prior probability for subset \(M_{1}\). This calculation requires calculation of \(P(D\,|\,M_{1})=\int P(D\,|\,M_{1},\beta_{1})\pi(\beta_{1}\,|\,M_{1})d\,\beta_{1}\), where \(\beta_{1}\) is the coefficient vector for the isotopes in model \(M_{1}\). Such integrals are notoriously difficult in most real problems requiring either numerical integration, analytical approximation, or Markov chain Monte Carlo (MCMC) methods [36, 37]. Even if the integral could be computed accurately, we would rarely know the exact subset probabilities because real data never follows any probability model exactly. Therefore, various approximations are in common usage, with the BIC (Bayesian Information Criterion) being perhaps the most common [36, 37, 38].