precisionweighted

Precisionweighted is a method for combining estimates by weighting each source according to its precision, where precision is the reciprocal of uncertainty (variance). In practice, sources with smaller variance contribute more to the final result, reflecting greater reliability.

Mathematically, if a set of independent estimates θi comes with variances σi^2, the weights are wi = 1/σi^2. The precision-weighted (or inverse-variance) mean is θ̂ = (∑i wi θi) / (∑i wi), and the variance of the combined estimate is Var(θ̂) = 1 / ∑i wi. This approach minimizes the variance of the combined estimator under the assumption of known, independent variances.

In Bayesian terms, precision weighting appears when combining information from different sources, such as a prior distribution and observed data. The posterior mean is a precision-weighted average of prior and data-driven estimates. In engineering and statistics, precision weighting is central to Kalman filters and related information-form methods, where updates are performed in terms of information (the inverse covariance) rather than direct values.

Applications of precisionweighted include meta-analysis, where study results are combined using inverse-variance weighting; sensor fusion and data assimilation, where measurements with different uncertainties are integrated; and forecasting ensembles that blend predictions according to their reported confidence. It is also used in decision-making frameworks that fuse multiple expert opinions or measurements.

Limitations include the need for accurate variance estimates and the assumption of independence. If some sources are biased or variances are misestimated, precision weighting can mislead. In cases with substantial heterogeneity, fixed-variance weights may be inappropriate, and random-effects or robust approaches may be preferred.