Optimal e-variables under constraints

E-variables enable safe and anytime-valid inference, with log-optimal e-variables given by the likelihood ratio of the least favorable distributions (LFDs) when they exist in composite settings. While this unconstrained theory is well understood, one may need/wish to impose additional structural constraints, including differential privacy, quantization, boundedness, or moment restrictions. We show that under these constraints, log-optimal constrained e-variables can often be constructed by a simple optimize-then-constrain principle: first compute the unconstrained log-optimal e-variable, then impose the constraint via an appropriate transformation. Thus, the constrained growth-rate optimization problem does not require solving for a different LFD pair; the constrained optimal solution is just a post-processing of the unconstrained optimal solution.