I'm reading this paper http://eccc.hpi-web.de/report/2016/044/download/, at page 7 you can see this equation:$$ \sum_t \mathbb{E}_{w \in W} [|p_t - p_{t|w,w}|] = 2 \sum_t \mathbb{E}_{w \in W} [\max(0,p_t - p_{t|w,w})]$$I think what we're using here is this:$$a+b+|a-b| = 2 \max(a,b)$$but with $a=0$ and $b=p_t-p_{t|w,w}$ we get:$$p_t - p_{t|w,w} + |p_t - p_{t|w,w}| = 2\max(0,p_t - p_{t|w,w})$$I know it's a bit of a silly question but I just can't figure it out, maybe I'm missing some fact about probabilities?
Issue with inequalities involving probablities
pr.probability