Propagation of Input Uncertainty
Forward uncertainty propagation pushes distributions over uncertain inputs through a model to characterize the distribution of its outputs.
The forward problem
Given probability distributions describing uncertain inputs and a model mapping inputs to outputs, forward propagation determines the output distribution: its mean, variance, quantiles, and failure probabilities. It is the core operation of predictive UQ and the consumer of surrogates, sampling schemes, and variance-reduction methods.
Method families
- Monte Carlo: sample inputs, evaluate the model, aggregate outputs; robust but slow to converge
- Quasi-Monte Carlo: low-discrepancy sequences for faster convergence in moderate dimension
- Stochastic collocation and polynomial chaos: spectral methods for smooth responses
- Surrogate-based: replace the model with a fast emulator, then sample abundantly
- Moment methods: propagate means and covariances through a linearization
Choosing a method
The right method depends on input dimension, response smoothness, the statistic of interest, and the model's cost. Smooth low-dimensional problems favor spectral methods; rough or high-dimensional problems favor sampling with variance reduction; rare-event tails demand importance sampling or subset simulation rather than crude Monte Carlo.
Input characterization first
Propagation is only as good as the input distributions fed to it. Correlations between inputs, distribution shape, and bounds all affect the output, and assuming independence or normality when it does not hold biases the result. Eliciting or estimating input distributions carefully is as important as the propagation method itself.
Reporting
Report which statistic was estimated, the method and sample size, and a convergence diagnostic such as the estimator's standard error. For decision-relevant outputs, quantiles and tail probabilities are usually more useful than the mean alone. Distinguish contributions from aleatoric input variability and from surrogate or model error so a reader knows what could be reduced with more work.