Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking

Frequentist and Bayesian Quantum Phase Estimation

“…However, because of the different operational meanings of the frequentist and the Bayesian paradigms, there is no reason for Equation (24) to fulfill the Cramér-Rao bound: indeed, it does not, as we show below.…”

“…The ChRB is always stricter than the CRLB and we obtain the last inequality in the chain (7). Notice that the CRLB requires the probability distribution p(µ|θ 0 ) to be differentiable [24]-a condition that can be dropped for the ChRB and the more general BB. Even if the distribution is regular, the above derivation shows that the ChRB, and more generally the BB, provide tighter error bounds than the CRLB.…”

“…A detailed derivation of the Ziv-Zakai bound [24,58,59] is provided in Appendix B. The final result reads MSE(θ est ) µ,θ 0 ≥ ∆ 2 θ ZZB , where…”

“…where the first term is the Fisher information F(θ 0 ), defined by Equation (10), averaged over p(θ 0 ), and the second term can be interpreted as a Fisher information of the prior [24]. Asymptotically in the number of measurements m and for regular distributions p(θ 0 ), the first term in Equation (34) dominates over the second one.…”

“…There is a vast amount of literature dealing with the parameter estimation problem that has been mostly developed following two different approaches [22][23][24]: frequentist and Bayesian. Both approaches have been investigated in the context of quantum phase estimation [18,20,[25][26][27][28][29][30][31] and implemented/tested experimentally [32][33][34][35][36].…”