This paper establishes a general equivalence between discrete choice and rational inattention models. Matejka and McKay (2015, AER) showed that when information costs are modelled using the Shannon entropy function, the resulting choice probabilities in the rational inattention model take the multinomial logit form. By exploiting convex-analytic properties of the discrete choice model, we show that when information costs are modelled using a class of generalized entropy functions, the choice probabilities in any rational inattention model are observationally equivalent to some additive random utility discrete choice model and vice versa. Thus any additive random utility model can be given an interpretation in terms of boundedly rational behavior. This includes empirically relevant specifications such as the probit and nested logit models.