Estimating failure probability by simulation

Use simulation to estimate the probability of the specified failure event and to check local approximations. The cost depends on the failure probability, limit-state evaluation cost and precision required. Begin with the comparison in Simulation methods for reliability analysis.

Budget a direct Monte Carlo run

SimulationOptions(n_samples=...) sets the maximum sample budget for crude Monte Carlo; target_cov can end the run earlier when its precision criterion is met. Pass rng a seed for a reproducible run; NumPy’s global random state is neither used nor changed. Record the seed and actual evaluation count in a study.

Set target_cov=0 to use the whole sample budget. A successful fixed-budget run reports status="completed" and retains its measured cov; completion alone does not establish adequate precision. Invalid limit-state evaluations raise AnalysisError before a probability is reported.

import numpy as np
import pystra as ra

model = ra.StochasticModel()
model.add_variable(ra.Normal("R", 3.0, 1.0))
model.add_variable(ra.Normal("S", 0.0, 1.0))
options = ra.SimulationOptions(n_samples=20_000, target_cov=0.05)
analysis = ra.CrudeMonteCarlo(
    model, ra.LimitState(lambda R, S: R - S), options=options, rng=2026
)
result = analysis.run()
assert result.status in ("completed", "precision_not_met")
assert 0 < result.failure_probability < 1

Check precision, not just probability

For a fixed number of independent direct samples, the approximate relative standard error is \(\sqrt{(1-p_f)/(N p_f)}\). Rare failures therefore need many evaluations. A zero failure count is insufficient evidence for a zero failure probability; see Troubleshooting reliability analyses.

Importance sampling concentrates samples using a proposal distribution and weights their contributions. The proposal must cover the important failure regions. The traditional ImportanceSampling uses a FORM-based center; the separate active-learning estimator supports explicit proposal components, as shown in PC-Kriging, bootstrap voting and importance sampling.

Line sampling uses an important direction and searches for intersections along lines. Subset simulation reaches a rare event through intermediate events and conditional Markov chains. Inspect the thresholds and repeat independent runs to assess stability: the current subset-simulation cov ignores chain correlation and is a lower-bound precision diagnostic, not a calibrated confidence interval.

Importance and line sampling stay accurate for very small probabilities. See Working at high reliability for the ranges covered and for reading a probability that underflows.

Continue: Interpreting results and convergence · Reliability algorithms and results · Simulation methods