While the minimum is a valid lower bound of the causal effect, it represents a biased estimate that can lead to missed causal discoveries.
Estimating R min is by no means an easy task, so that most of the previous work focused on the lower bound of R min, which is also a valid lower bound of the true number of recombination events occurred.
Finally, Fig. 6 demonstrates that the lower bound of Theorem 4 is valid, not only for random selection but also for uniform and consecutive selections.
People have focused on estimating the lower bound of R min, because it is also a valid lower bound for the true number of recombination events occurred.
Bafna and Bansal [ 11] has proved that R I is a valid lower bound of R min given a particular order of the sequences.
In practice, the bound of q ( A ∘ B − 1 ) can give a rough estimate before actually solving it and can serve as a check of whether the solution technique for it actually resulted in valid solution.
For a broad class of noise distributions the presented bound is valid whenever the temperature parameter β of the EWA is larger than or equal to 4σ2, where σ2 is the noise variance.
The lower bound is only valid for the case where the number of sources is equal to the number of relays, and where each relay only helps two sources.
Under the assumption that m is (p,C -smooth (i.e., roughly sp,C -smoothis p-times continuously differentiable) it is shown that the optimal minimax rate of convergence of the L1 error is n−p/d, where the upper bound is valid even if the support of the design measure is unbounded but the design measure satisfies some moment condition.e
See Appendix 3. Theorem 3 provides a general expression of the capacity upper bound, which is valid for arbitrary number of antenna, CCI and feedback delay.
