“Inverse-variance weighting”的意思、由来-开放百科全书

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.

Given a sequence of independent observations {{math|y_i}} with variances {{math|σ_i²}}, 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 to combine the results from independent measurements.

Context

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

random variables

. 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

are the same, which we shall refer to by

. Given the

measurements, a typical estimator for

, denoted as

, is given by the simple average

. Note that this empirical average is also a random variable, whose expectation value

but also has a scatter. If the individual measurements are uncorrelated, the square of the error in the estimate is given by

. Hence, if all the

are equal, then the error in the estimate decreases with increase in

, thus making the more number of observations preferred.

Instead of

repeated measurements with one instrument, if the experimenter makes

of the same quantity with

different instruments with varying quality of measurements, then there is no reason to expect the different

to be the same. Some instruments could be noisier than others. In the example of measuring the acceleration due to gravity, the different "instruments" could be measuring

from a simple pendulum, from analysing a projectile motion etc.

The simple average is no longer an optimal estimator, since the error in

might actually exceed the error in the least noisy measurement if different measurements have very different errors. 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. Given the knowledge of

, an optimal estimator to measure

would be a weighted mean of the measurements

, for the particular choice of the weights

The variance of the estimator

, which for the optimal choice of the weights become

Note that since

, the estimator has a scatter smaller than the scatter in any individual measurement. Furthermore, the scatter in

decreases with adding more measurements, however noisier those measurements may be.

Derivation

For optimality, we wish to minimise

which can be done by equating the gradient with respect to the weights of

to zero, while maintaining the constraint that

. Using a Lagrange multiplier

to enforce the constraint, we express the variance

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.

Context

Derivation

See also

References