In statistics, inverse-variance weighting is a method of aggregating two or more random variables to minimize the variance of the weighted average.
Each random variable is weighted in inverse proportion to its variance (i.e., proportional to its precision).
Given a sequence of independent observations yi with variances σi2, the inverse-variance weighted average is given by[1] The inverse-variance weighted average has the least variance among all weighted averages, which can be calculated as If the variances of the measurements are all equal, then the inverse-variance weighted average becomes the simple average.
Inverse-variance weighting is typically used in statistical meta-analysis or sensor fusion to combine the results from independent measurements.
Suppose an experimenter wishes to measure the value of a quantity, say the acceleration due to gravity of Earth, whose true value happens to be
A careful experimenter makes multiple measurements, which we denote with
If they are all noisy but unbiased, i.e., the measuring device does not systematically overestimate or underestimate the true value and the errors are scattered symmetrically, then the expectation value
The scatter in the measurement is then characterised by the variance of the random variables
, and if the measurements are performed under identical scenarios, then all the
measurements, a typical estimator for
Note that this empirical average is also a random variable, whose expectation value
If the individual measurements are uncorrelated, the square of the error in the estimate is given by
are equal, then the error in the estimate decreases with increase in
, thus making more observations preferred.
repeated measurements with one instrument, if the experimenter makes
different instruments with varying quality of measurements, then there is no reason to expect the different
from a simple pendulum, from analysing a projectile motion etc.
The simple average is no longer an optimal estimator, since the error in
Instead of discarding the noisy measurements that increase the final error, the experimenter can combine all the measurements with appropriate weights so as to give more importance to the least noisy measurements and vice versa.
, which for the optimal choice of the weights become
Consider a generic weighted sum
is given by (see Bienaymé's identity) For optimality, we wish to minimise
which can be done by equating the gradient with respect to the weights of
to enforce the constraint, we express the variance: For
, The individual normalised weights are: It is easy to see that this extremum solution corresponds to the minimum from the second partial derivative test by noting that the variance is a quadratic function of the weights.
Thus, the minimum variance of the estimator is then given by: For normally distributed random variables inverse-variance weighted averages can also be derived as the maximum likelihood estimate for the true value.
Furthermore, from a Bayesian perspective the posterior distribution for the true value given normally distributed observations
and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance
For potentially correlated multivariate distributions an equivalent argument leads to an optimal weighting based on the covariance matrices
: For multivariate distributions the term "precision-weighted" average is more commonly used.