Simultaneous Confidence Intervals for Linear Estimates of Linear Functionals
This note presents three ways of constructing simultaneous confidence intervals for linear estimates of linear functionals in inverse problems, including ``Backus-Gilbert'' estimates. Simultaneous confidence intervals are needed to compare estimates, for example, to find spatial variations in a distributed parameter. The notion of simultaneous confidence intervals is introduced using coin tossing as an example before moving to linear inverse problems. The first method for constructing simultaneous confidence intervals is based on the Bonferroni inequality, and applies generally to confidence intervals for any set of parameters, from dependent or independent observations. The second method for constructing simultaneous confidence intervals in inverse problems is based on a ``global'' measure of fit to the data, which allows one to compute simultaneous confidence intervals for any number of linear functionals of the model that are linear combinations of the data mappings. This leads to confidence intervals whose widths depend on percentage points of the chi-square distribution with $n$ degrees of freedom, where $n$ is the number of data. The third method uses the joint normality of the estimates to find shorter confidence intervals than the other methods give, at the cost of evaluating some integrals numerically.