Bounded-ratio analytic bounds for TV distance between Bernoulli product measures

Develop simple, analytically tractable upper and lower bounds UB(p,q) and LB(p,q) for the total variation distance between the product Bernoulli distributions Ber(p) and Ber(q), where Ber(p)=Ber(p1)⊗⋯⊗Ber(pn) and Ber(q)=Ber(q1)⊗⋯⊗Ber(qn), such that the ratio UB(p,q)/LB(p,q) is bounded by an absolute constant independent of n and of the parameter vectors p and q.

Background

The paper studies tensorization of total variation (TV) distance for product measures and provides lower bounds in terms of the marginal TV sequence, showing a fundamental √n gap between worst-case upper and lower bounds based on marginals. Computing exact TV distance between two n-fold product distributions is #P-complete, and while there exist efficient (deterministic) algorithms for multiplicative approximation, these are algorithmic rather than simple, closed-form analytic bounds.

Motivated by both hardness and the desire for tractable analysis, the authors pose the problem of finding simple, analytic upper and lower bounds for TV distance between Ber(p) and Ber(q) whose ratio is uniformly bounded by a constant independent of the dimension n and parameters p and q. This would yield practically useful two-sided estimates with a dimension-free bounded-ratio property.