Working definition
Uncertainty propagation estimates the distribution or standard uncertainty of an output produced by uncertain inputs. Linearized covariance propagation, interval methods, and Monte Carlo simulation answer different questions and require declared distributions, dependencies, and model equations.
Notation
Σᵧ ≈ JΣₓJᵀy = g(x)Assumptions
- Input uncertainty models are defensible.
- Dependencies and covariance are represented.
- The forward model covers material effects.
Invariants
- Units and covariance dimensions remain consistent.
- Perfectly shared error must not be counted as independent.
- Output uncertainty is conditional on the model.
Reproducible procedure
- Inventory uncertain inputs and correlations.
- Select linearized, interval, or simulation propagation.
- Check sensitivity and convergence, then report included components.
Error and boundary controls
- Linearization fails for strong nonlinearity or boundaries.
- Unknown systematics remain outside the budget.
- Monte Carlo sampling error must be quantified.
What this does not establish
A quantified uncertainty budget does not certify that omitted variables, interpretations, or causal assumptions are correct.
Explicit applications
3 cross-domain bridges
Metrology uncertainty budget
Propagate instrument, calibration, sampling, and model components into a reported process measurement.
Inputs
- instrument readings
- calibration covariance
- sampling model
Outputs
- measurand estimate
- combined uncertainty
- dominant sensitivities
Transformation: Combine correlated uncertainty components through the measurement equation.
Limit: Unmodeled systematic effects remain outside the reported budget.
Open connected system →Astrometric covariance propagation
Carry correlated position and motion uncertainty to another epoch or frame.
Inputs
- state estimate
- covariance matrix
- transformation Jacobian
Outputs
- transformed state
- output covariance
- sensitivity diagnostics
Transformation: Propagate covariance through the declared astrometric model.
Limit: Linear covariance propagation may fail across nonlinear or multimodal uncertainty.
Open connected system →Daśā timing under uncertain birth time
Propagate a birth-time interval through lunar mansion position and period-boundary calculations.
Inputs
- birth-time interval
- Moon longitude function
- daśā convention
Outputs
- period date ranges
- boundary sensitivity
- stable and unstable labels
Transformation: Evaluate boundary ranges across the admissible input interval.
Limit: Timing ranges quantify input sensitivity, not predictive validity.
Open connected system →Authoritative references
- [1]NIST Technical Note 1297: Guidelines for Evaluating and Expressing Measurement Uncertainty · National Institute of Standards and Technology
Establishes: A measurement framework for identifying uncertainty components, combining standard uncertainties, and reporting expanded uncertainty with declared coverage.
Boundary: Reported uncertainty describes the measurement model and included components. It is not a guarantee that all systematic errors or model inadequacies were found.
- [2]NIST/SEMATECH e-Handbook of Statistical Methods · National Institute of Standards and Technology
Establishes: Methods for uncertainty analysis, calibration, time-series modeling, process monitoring, experimental design, reliability, and statistical comparison.
Boundary: Statistical procedures quantify evidence under a design and model; they do not repair biased sampling, outcome leakage, post-hoc hypotheses, or unmeasured confounding.