An “estimator” is a function that takes a dataset as input and outputs an estimate. We discuss this statistics terminology more in Section 2.4.

That’s the math for why we need the positivity assumption, but what’s the intuition? Well, if we have a positivity violation, that means that within some subgroup of the data, everyone always receives treatment or everyone always receives the control. It wouldn’t make sense to be able to estimate a causal effect of treatment vs. control in that subgroup since we see only treatment or only control. We never see the alternative in that subgroup