Class-Specific Feature Mixture (CSFM)

To get around the one class, one feature assumption, CSFM assumes that each class $H_k$ is composed of $L$ sub-classes represented by an additive mixture PDF. We assume that class $H_k$ is composed of $L$ subclasses $H_{k,l}, \; 1\leq l \leq L$, that have relative probabilities of occurrence $\alpha_{k,l}$ and individual sub-class PDFs $p({\bf x}\vert H_{k,l})$. The mixture PDF for $H_k$ is given by:

$$p({\bf x}\vert H_k) = \sum_{l=1}^L \; \alpha_{k,l} \; p({\bf x}\vert H_{k,l}),$$ (12.5)

where $\sum_{l=1}^L \; \alpha_{k,l}=1.$ If we assume that each sub-class has a different feature (approximate sufficient statistic) to distinguish it from a sub-class dependent reference hypothesis $H_{0,l}$, we apply (2.2) to get

$$p({\bf x}\vert H_{k}) = \sum_{l=1}^L \; \alpha_{k,l} \; \frac{p({\bf x}\vert H_{0,l})}{p({\bf z}_l\vert H_{0,l})} p({\bf z}_l\vert H_{k,l})$$ (12.6)

Note that each class PDF is represented by the same library of $L$ models. The CSFM classifier is

$$\hat{k}=\arg \max_{k=1}^M \; \left\{ \sum_{l=1}^L \; \alpha_{k,l}... ...l})}{p({\bf z}_l\vert H_{0,l})} p({\bf z}_l\vert H_{k,l}) \right\} \; p(H_{k}),$$ (12.7)

which may be interpreted as a data-specific feature classifier because for each data sample ${\bf x}$, the factor $p({\bf x}\vert H_{0,l})/p({\bf z}_l\vert H_{0,l})$ has a dominant effect, effectively picking one feature to classfy the sample.