A Simpliied Version of the Tomography Problem Can Help to Estimate the Errors of Indirect Measurements. Error Estimation for Indirect Measurements { a Statistical Computational Problem: Formulation of the Problem and Known Methods of Solving It

In many real-life situations, it is very diicult 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 rst measuring some relating quantities x 1 ; : : :; x n , and then by using the known relation between x i 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 simpliied ((nite-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 nd the optimal algorithm for solving this simpliied tomography problem, and thus, for solving the statistical problem of error estimation for indirect measurements. What are indirect measurements. In many real-life situations, it is diicult or even impossible to directly measure the quantity y in which we are interested. For example, it is, at present, practically impossible to directly measure a distance to a distant quasar, or the amount of oil in a given area. Since we cannot measure such quantities directly, we have to measure them indirectly: Namely, we measure some other quantities x 1 ; : : :; x n which are related to y by a known dependence y = f(x 1 x n of measuring x i. For example, to determine the amount of oil y in a given area, we measure the results x i of sending ultrasound signals between the two parallel wells, and then estimate y by solving the appropriate system of partial diierential equations (in this example, f(x 1 ; : : :; x n) is an algorithm for solving this system). In general, such a two-stage procedure (measurement followed by computations) is called an indirect measurement, and the value e y = f(e x 1 ; : : :; e x n) resulting from this two-stage procedure is called the result of indirect measurement. Error estimation for indirect measurements: a real-life problem. Measurements are never 100% accurate; hence, the result e x i of …