$$ y={argmax} _{c_{j}\in C}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(T|h_{i})P(h_{i})} $$

$$ ={argmax} _{c_{j}\in C}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(T,h_{i})} $$

$$= {argmax}_{c_{j}\in C}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(h_{i}|T)}$$

\propto doesn't work well.