# Machine Learning Subjective Opinions: An Application in Forensic Chemistry
## 2 Background
### Subjective Opinions from Machine Learning  
The research presented here is based on previous work by [PERSON] [30], [PERSON] [2], [PERSON] [31], [PERSON] [5] and [PERSON] [1]. It is useful in forensic science and legal questions to consider two mutually exclusive competing propositions. This approach facilitates taking a probabilistic likelihood ratio approach which is popular in evaluative reporting [15]. The likelihood ratio is equivalent to the slope of the tangent to an ROC curve at any point on the curve. Setting a threshold on the ROC curve is required for making decisions, but not for formulating a subjective opinion or approximating a likelihood ratio. In this work, fire debris samples are classified into two mutually exclusive competing states of containing ILR, denoted state \(x\), or not containing ILR, denoted state \(\overline{x}\), which allows the fusion of the output from multiple ML methods to form a frame of discernment. The binomial subjective opinion that a sample belongs to state \(x\) based on the ML results is denoted as, \(\omega_{x}^{ML}\). The opinion is written as an ordered tuple  
\[\omega_{x}^{ML}\equiv(b_{x}^{ML},d_{x}^{ML},u_{x}^{ML},a_{x}^{ML}) \tag{1}\]  
In Equation (1), \(b_{x}^{ML}\) expresses to the belief that a sample belongs to state \(x\), \(d_{x}^{ML}\) expresses the disbelief in the proposition, \(u_{x}^{ML}\) expresses the uncertainty, and \(a_{x}^{ML}\) represents the base rate for \(x\) in the population. For a binary frame of discernment, \(a_{x}^{ML}\) is typically assumed to have a value of 0.5 (i.e., the uninformative prior) in the absence of other specific information. The belief, disbelief and uncertainty must sum to one, as shown in Equation (2).  
\[1=b_{x}^{ML}+d_{x}^{ML}+u_{x}^{ML} \tag{2}\]  
The values of the belief and disbelief are calculated from the shape parameters derived from fitting the set of ML posterior probabilities that a sample belongs to class \(x\) to a beta distribution. The fitting parameters are \(\alpha\) and \(\beta\), and the equations for calculating the belief and disbelief are given in Equations (3) and (4). (NOTE: [PERSON] states \(\alpha\) and \(\beta>1\). This is not a requirement for a beta distribution, only that the values are positive real numbers.)  
\[b_{x}^{ML}=\frac{\alpha-1}{\alpha+\beta} \tag{3}\]  
\[d_{x}^{ML}=\frac{\beta-1}{\alpha+\beta} \tag{4}\]  
The uncertainty can be calculated from Equation (2), after calculating the belief and disbelief from Equations (3) and (4). The projected probability function for the opinion is given in Equation (5).  
\[P(\omega_{x}^{ML})=b_{x}^{ML}+a_{x}^{ML}u_{x}^{ML} \tag{5}\]  
The projected probability \(P(\omega_{x}^{ML})\) corresponds to \(b_{x}^{ML}\) adjusted by a portion of the base rate, as determined by \(u_{x}^{ML}\)[5]. The value provides an estimate of how the probability of varies from the base rate. The projected probability \"unifies all of the opinion parameters in a single probability distribution and thus enables reasoning about subjective opinions in the classical probability theory\" [5]. It is possible to recover the fitted beta distribution from \(\omega_{x}^{ML}\) using Equation (6). See [1] for examples of how probability distributions correspond to subjective opinions.  
\[\omega_{x}^{ML}\mapsto beta(\frac{2b_{x}^{ML}}{u_{x}^{ML}}+2a_{x}^{ML},\frac{2a _{x}^{ML}}{u_{x}^{ML}}+2(1-a_{x}^{ML})) \tag{6}\]