BAYESIAN CONFIDENCE LIMITS OF ELECTRON SPECTRA OBTAINED THROUGH REGULARIZED INVERSION OF SOLAR HARD X-RAY SPECTRA

Many astrophysical observations are characterized by a single, non-repeatable measurement of a source brightness or intensity, from which we are to construct estimates for the true intensity and its uncertainty. For example, the hard X-ray count spectrum from transient events such as solar flares can be observed only once, and from this single spectrum one must determine the best estimate of the underlying source spectrum I (is an element of), and hence the form of the responsible electron spectrum F(E). Including statistical uncertainties on the measured count spectrum yields a "confidence strip" that delineates the boundaries of electron spectra that are consistent with the observed photon spectrum. In this short article, we point out that the expectation values of the source brightness and its variance in a given photon energy bin are in general not (as has been assumed in prior works) equal to n, the number of counts observed in that energy bin. Rather, they depend both on n and on prior knowledge of the overall photon spectrum. Using Bayesian statistics, we provide an explicit procedure and formulas for determining the "confidence strip" (Bayesian credible region) for F(E), thus providing rigorous bounds on the intensity and shape of the accelerated electron spectrum.