Inverse-variance weighting

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 ${\displaystyle \mu }$. A careful experimenter makes multiple measurements, which we denote with ${\displaystyle n}$ random variables ${\displaystyle X_1,X_2,...,X_n}$. 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 ${\displaystyle E[X_i]=\mu }$ ${\displaystyle \mathrm{\forall }i}$. The scatter in the measurement is then characterised by the variance of the random variables ${\displaystyle Var(X_i):={\sigma }_i^2}$, and if the measurements are performed under identical scenarios, then all the ${\displaystyle {\sigma }_i}$ are the same, which we shall refer to by ${\displaystyle \sigma }$. Given the ${\displaystyle n}$ measurements, a typical estimator for ${\displaystyle \mu }$, denoted as ${\displaystyle \hat{\mu }}$, is given by the simple average ${\displaystyle \overline{X}=\frac{1}{n}\sum _i{X}_i}$. Note that this empirical average is also a random variable, whose expectation value ${\displaystyle E[\overline{X}]}$ is ${\displaystyle \mu }$ but also has a scatter. If the individual measurements are uncorrelated, the square of the error in the estimate is given by ${\displaystyle Var(\overline{X})=\frac{1}{n^2}\sum _i{\sigma }_i^2={\left(\frac{\sigma }{\sqrt{n}}\right)}^2}$. Hence, if all the ${\displaystyle {\sigma }_i}$ are equal, then the error in the estimate decreases with increase in ${\displaystyle n}$ as ${\displaystyle 1/\sqrt{n}}$, thus making more observations preferred.

Instead of ${\displaystyle n}$ repeated measurements with one instrument, if the experimenter makes ${\displaystyle n}$ of the same quantity with ${\displaystyle n}$ different instruments with varying quality of measurements, then there is no reason to expect the different ${\displaystyle {\sigma }_i}$ 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 ${\displaystyle g}$ from a simple pendulum, from analysing a projectile motion etc. The simple average is no longer an optimal estimator, since the error in ${\displaystyle \overline{X}}$ 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 ${\displaystyle {\sigma }_1^2,{\sigma }_2^2,...,{\sigma }_n^2}$, an optimal estimator to measure ${\displaystyle \mu }$ would be a weighted mean of the measurements ${\displaystyle \hat{\mu }=\frac{\sum _i{w}_i{X}_i}{\sum _i{w}_i}}$, for the particular choice of the weights ${\displaystyle w_i=1/{\sigma }_i^2}$. The variance of the estimator ${\displaystyle Var(\hat{\mu })=\frac{\sum _i{w}_i^2{\sigma }_i^2}{{\left(\sum _i{w}_i\right)}^2}}$, which for the optimal choice of the weights become ${\displaystyle Var({\hat{\mu }}_{\text{opt}})={\left(\sum _i{\sigma }_i^{-2}\right)}^{-1}.}$

Note that since ${\displaystyle Var({\hat{\mu }}_{\text{opt}})<\underset{j}{min}{\sigma }_j^2}$, the estimator has a scatter smaller than the scatter in any individual measurement. Furthermore, the scatter in ${\displaystyle {\hat{\mu }}_{\text{opt}}}$ decreases with adding more measurements, however noisier those measurements may be.

Derivation

Uncorrelated measurements

Consider a generic weighted sum ${\displaystyle Y=\sum _i{w}_i{X}_i}$, where the weights ${\displaystyle w_i}$ are normalised such that ${\displaystyle \sum _i{w}_i=1}$. If the ${\displaystyle X_i}$ are all independent, the variance of ${\displaystyle Y}$ is given by (see Bienaymé's identity)

${\displaystyle Var(Y)=\sum _i{w}_i^2{\sigma }_i^2.}$

For optimality, we wish to minimise ${\displaystyle Var(Y)}$ which can be done by equating the gradient with respect to the weights of ${\displaystyle Var(Y)}$ to zero, while maintaining the constraint that ${\displaystyle \sum _i{w}_i=1}$. Using a Lagrange multiplier ${\displaystyle w_0}$ to enforce the constraint, we express the variance:

${\displaystyle Var(Y)=\sum _i{w}_i^2{\sigma }_i^2-w_0(\sum _i{w}_i-1).}$

For ${\displaystyle k>0}$,

${\displaystyle 0=\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_k}Var(Y)=2w_k{\sigma }_k^2-w_0,}$

which implies that:

${\displaystyle w_k=\frac{w_0/2}{{\sigma }_k^2}.}$

The main takeaway here is that ${\displaystyle w_k\propto 1/{\sigma }_k^2}$. Since ${\displaystyle \sum _i{w}_i=1}$,

${\displaystyle \frac{2}{w_0}=\sum _i\frac{1}{{\sigma }_i^2}:=\frac{1}{{\sigma }_0^2}.}$

The individual normalised weights are:

${\displaystyle w_k=\frac{1}{{\sigma }_k^2}{\left(\sum _i\frac{1}{{\sigma }_i^2}\right)}^{-1}.}$

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:

${\displaystyle Var(Y)=\sum _i\frac{{\sigma }_0^4}{{\sigma }_i^4}{\sigma }_i^2={\sigma }_0^4\sum _i\frac{1}{{\sigma }_i^2}={\sigma }_0^4\frac{1}{{\sigma }_0^2}={\sigma }_0^2=\frac{1}{\sum _{i}1/{\sigma }_i^2}.}$

Correlated measurements

Normal distributions

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 ${\displaystyle y_i}$ and a flat prior is a normal distribution with the inverse-variance weighted average as a mean and variance ${\displaystyle Var(Y)}$.

Multivariate case

For potentially correlated multivariate distributions an equivalent argument leads to an optimal weighting based on the covariance matrices ${\displaystyle {\mathbf{C}}_i}$ of the individual vector-valued estimates ${\displaystyle {\mathbf{x}}_i}$:

${\displaystyle \hat{\mathbf{x}}=\hat{\mathbf{C}}\sum _i{\mathbf{C}}_i^{-1}{\mathbf{x}}_i}$

${\displaystyle \hat{\mathbf{C}}={\left(\sum _i{\mathbf{C}}_i^{-1}\right)}^{-1}}$

For multivariate distributions the term "precision-weighted" average is more commonly used.

The Kalman filter gain minimizes the determinant of the posterior covariance (i.e. the generalized variance) of the estimated mean. This step of the Kalman filter, therefore, employs inverse-variance weighting, where more precise (i.e., lower variance) sources of information are weighted more heavily in the update step of the Kalman filter.