The prior independent framework for algorithm design considers how well an algorithm that does not know the distribution of its inputs approximates the expected performance of the optimal algorithm for this distribution. This paper gives a method that is agnostic to problem setting for proving lower bounds on the prior independent approximation factor of any algorithm. The method constructs a correlated distribution over inputs that can be described both as a distribution over i.i.d. good-for-algorithms distributions and as a distribution over i.i.d. bad-for-algorithms distributions. We call these two descriptions equivocal blends. Prior independent algorithms are upper-bounded by the optimal algorithm for the latter distribution even when the true distribution is the former. Thus, the ratio of the expected performances of the Bayesian optimal algorithms for these two decompositions is a lower bound on the prior independent approximation ratio.

We apply this framework to give new lower bounds on canonical prior independent mechanism design problems. For one of these problems, we also exhibit a near-tight upper bound. Towards solutions for general problems, we give distinct descriptions of two large classes of correlated-distribution "solutions" for the technique, depending respectively on an order-statistic separability property and a paired inverse-distribution property. We exhibit that equivocal blends do not generally have a Blackwell ordering, which puts this paper outside of standard information design.