Conditional Posterior Cramér–Rao Lower Bounds for Nonlinear Sequential Bayesian Estimation

The posterior CramérRao lower bound (PCRLB) for sequential Bayesian estimators, which was derived by Tichavsky in 1998, provides a performance bound for a general nonlinear filtering problem. However, it is an offline bound whose corresponding Fisher information matrix (FIM) is obtained by taking the expectation with respect to all the random variables, namely the measurements and the system states. As a result, this unconditional PCRLB is not well suited for adaptive resource management for dynamic systems. The new concept of conditional PCRLB is proposed and derived in this paper, which is dependent on the actual observation data up to the current time, and is implicitly dependent on the underlying system state. Therefore, it is adaptive to the particular realization of the underlying system state and provides a more accurate and effective online indication of the estimation performance than the unconditional PCRLB. Both the exact conditional PCRLB and its recursive evaluation approach including an approximation are derived. Further, a general sequential Monte Carlo solution is proposed to compute the conditional PCRLB recursively for nonlinear non-Gaussian sequential Bayesian estimation problems. The differences between this new bound and existing measurement dependent PCRLBs are investigated and discussed. Illustrative examples are also provided to show the performance of the proposed conditional PCRLB.