Before revision:  We focus on the inference tasks of learning (density estimation) and identity testing ( goodness-of-fit ) for discrete distributionsunder total variation distance, and establish general lower bounds that take into account the local constraints modeled as a channel family.
After revision:  We focus on the tasks of goodness-of-fit ) for discrete distributionsunder total variation distance, and establish general lower bounds that take into account the local constraints modeled as a channel family.