# An Agent-Ensemble for Thresholded Multi-Target Classification
## 4 Proposed Agent Combiner
### Relationship to the Neyman-Pearson Combiner  
We motivate the combination approach given in (3) by showing that under certain assumptions it is equivalent to the Neyman-Pearson optimal agent combiner. The Neyman-Pearson combiner for the agent score vector \(\mathbf{x}\) is  
\[\hat{y}(\mathbf{x})=\text{sign}(\Lambda_{\text{NP}}(\mathbf{x})-\alpha^{*}), \tag{4}\]  
where the Neyman-Pearson fusion score, \(\Lambda_{\text{NP}}(\mathbf{x})\), is given by the likelihood ratio:  
\[\Lambda_{\text{NP}}(\mathbf{x})=\frac{p(\mathbf{x}|Y\geq 1)}{p(\mathbf{x}|Y=-1)}, \tag{5}\]  
and \(\alpha^{*}\) is set in order to satisfy  
\[P(\Lambda_{\text{NP}}(\mathbf{x})-\alpha^{*}\geq 0|Y=-1)=\text{PFA}^{*}, \tag{6}\]  
where PFA* is the allowable PFA for the system. In Theorem 1 below, we show assumptions under which (3) and (4) are equivalent. First we give the following definitions for non-discriminative and conditionally non-discriminative random variables, which provide the basis for the assumptions of the theorem. Figure 1: Block diagram of the proposed agent combiner. **Definition 1**. : _A random variable \(Z\) is non-discriminative for target-type \(j\) if the conditional pdf of \(Z\) given \(j\) is the same as that for \(Z\) given clutter, \(p(z|Y=j)=p(z|Y=-1).\)_  
**Definition 2**. : _A random variable \(Z_{1}\) is conditionally non-discriminative for target-type \(j\) given random variable \(Z_{2}\) if \(p(z_{1}|z_{2},Y=j)=p(z_{1}|z_{2},Y=-1).\)_  
**Theorem 1**. : _If the agents not trained for target-type \(j\) are jointly conditionally non-discriminative for target-type \(j\) conditioned on the output of agent \(j\) for all \(j\in 1,...,L\), then (3) is equivalent to the optimal Neyman-Pearson combiner given in (4)._  
Proof. : The proof is a straightforward application of the total-probability theorem to the Neyman-Pearson Fusion score given in (5):  
\[\Lambda_{\text{NP}}(\mathbf{x}) =\frac{p(\mathbf{x}|Y\geq 1)}{p(\mathbf{x}|Y=-1)}\] \[=\frac{\sum_{j=1}^{L}p(\mathbf{x}|Y=j)p(Y=j|Y\geq 1)}{p(\mathbf{x} |Y=-1)}\] \[=\sum_{j=1}^{L}\frac{p(\mathbf{x}_{j}|x_{j},Y=j)p(x_{j}|Y=j)p(Y=j |Y\geq 1)}{p(\mathbf{x}_{j}|x_{j},Y=-1)p(x_{j}|Y=-1)},\]  
where \(\mathbf{x}_{j}\) is the vector of agent-outputs with the \(j^{\text{th}}\) agent removed. If the conditionally non-discriminative assumption of the theorem is met, then \(p(\mathbf{x}_{j}|x_{j},Y=j)=p(\mathbf{x}_{j}|x_{j},Y=-1)\) and thus  
\[\Lambda_{\text{NP}}(\mathbf{x})=\sum_{j=1}^{L}p(Y=j|Y\geq 1)\frac{p(x_{j}|Y=j)}{p(x _{j}|Y=-1)}, \tag{7}\]  
which, when substituted into (4) results in the proposed agent combiner given in (3). There are several benefits to implementing (3) as opposed to (4) given that the assumptions of Theorem 1 are met. Implementation of (3) requires estimation of only single variable probability density functions (pdf), as opposed to (4) which requires that we estimate joint pdfs for target and clutter. The standard methods of simplifying or regularizing joint density estimation problems, such as assuming independence or diagonal loading of the covariance matrix of Gaussian distributions, are not appropriate for this problem as some of the underlying agents are likely to be correlated given that they are all functions of the same underlying feature vector \(\mathbf{v}\). This is particularly true for agents trained on targets that are similar in feature space. Furthermore, the fact that (3) requires only one-dimensional pdfs makes it much easier to use non-parametric density estimation methods. Theorem 1 requires the conditionally non-discriminative condition to hold for the agents in a multi-target classification ensemble. The following lemmas give two conditions on the underlying agent-classifiers under which the assumption holds. **Lemma 1**. : _Classifier agents \(x_{j},j\
eq k\) are jointly conditionally non-discriminative for target-type \(k\) given \(x_{k}=g_{k}(\mathbf{v})\) if \(g_{k}(\mathbf{v})\) is the likelihood ratio for target-type \(k\) vs. clutter._  
Proof.