Maximum entropy μSR analysis II: the search for truthful errors

In muon spin rotation (μSR) measurements, the details of the local field distribution P ( b ) are of interest, so either a fast Fourier transform (FFT) or the maximum entropy method (MEM) is applied to extract P ( b ) from the μSR time spectra. Ideally, one would like to analyze the resulting P ( b ) as if it were a data set in its own right: fitting P ( b ) to appropriate theoretical functions and calculating its mean, second moment and third moment. This is only possible if there are rigorous errors associated with P ( b ). In this paper, we show that the maximum entropy method finds its maximum in a way analogous to χ 2 fitting. Calculation of a covariance matrix naturally follows. The covariance matrix demonstrates that the maximum entropy (ME) result P ( b ) has intrinsically correlated , not independent, errors. With proper care, P ( b ) can be fit and have its moments calculated. This method is not limited to μSR applications, it can be applied to other cases involving Fourier transforms of data, such as NMR.