A Simplified Version of the Tomography Problem Can Help to Estimate the Errors of Indirect Measurements
In many real-life situations, it is very difficult or even impossible to directly measure the quantity y in which we are interested: e.g., we cannot directly measure a distance to a distant galaxy or the amount of oil in a given well. Since we cannot measure such quantities directly, we can measure them indirectly: by first measuring some relating quantities x1,...,xn, and then by using the known relation between xi and y to reconstruct the value of the desired quantity y.
In practice, it is often very important to estimate the error of the resulting indirect measurement. In this paper, we show that in a natural statistical setting, the problem of estimating the error of indirect measurement can be formulated as a simplified (finite-dimensional) version of a tomography problem (reconstructing an image from its projections).
In this paper, we use the ideas of invariance (natural in statistical setting) to find the optimal algorithm for solving this simplified tomography problem, and thus, for solving the statistical problem of error estimation for indirect measurements.