Engineering risk analysis is concerned with the likelihood of failure and statistical scenarios when it occurs. Decades of Monte Carlo research have revealed the relevance of failure samples (scenarios) to the success of an algorithm. In the context of Subset Simulation, a theory has been recently developed [1] that expresses the correlation between successive Markov Chain Monte Carlo (MCMC) failure samples in terms of the ‘failure mixing rate’ (FMR), a new measure that opens up opportunities for optimizing MCMC. General formulas for the first two derivatives of FMR with respect to MCMC hyperparameters were obtained in terms of those of the candidate response. ‘Neighborhood estimators’ were proposed for the first derivative of FMR, resolving the difficulty from conditioning on zero-probability event at the expense of introducing a second order bias from non-zero neighborhood probability. Proper estimator for the second derivative was not available, however. As a sequel to [1], this work 1) proposes a neighborhood estimator for the second derivative of FMR, keeping the bias to second order; 2) derives for first passage problems the second derivative of candidate response, discovering a non-trivial additional term that accounts for the random nature of peak response time; 3) derives for general dynamical systems the governing equations for state derivatives that are indispensable for candidate response derivatives, offering a semi-analytical means for accurate solution. The second derivatives of FMR for the same examples in [1] are also reported.
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