Fully Adaptive Bayesian Algorithm for Data Analysis, FABADA
  The aim of this paper is to describe a novel non-parametric noise reduction
technique from the point of view of Bayesian inference that may automatically
improve the signal-to-noise ratio of one- and two-dimensional data, such as
e.g. astronomical images and spectra. The algorithm iteratively evaluates
possible smoothed versions of the data, the smooth models, obtaining an
estimation of the underlying signal that is statistically compatible with the
noisy measurements. Iterations stop based on the evidence and the $\chi^2$
statistic of the last smooth model, and we compute the expected value of the
signal as a weighted average of the whole set of smooth models. In this paper,
we explain the mathematical formalism and numerical implementation of the
algorithm, and we evaluate its performance in terms of the peak signal to noise
ratio, the structural similarity index, and the time payload, using a battery
of real astronomical observations. Our Fully Adaptive Bayesian Algorithm for
Data Analysis (FABADA) yields results that, without any parameter tuning, are
comparable to standard image processing algorithms whose parameters have been
optimized based on the true signal to be recovered, something that is
impossible in a real application. State-of-the-art non-parametric methods, such
as BM3D, offer slightly better performance at high signal-to-noise ratio, while
our algorithm is significantly more accurate for extremely noisy data (higher
than $20-40\%$ relative errors, a situation of particular interest in the field
of astronomy). In this range, the standard deviation of the residuals obtained
by our reconstruction may become more than an order of magnitude lower than
that of the original measurements. The source code needed to reproduce all the
results presented in this report, including the implementation of the method,
is publicly available at https://github.com/PabloMSanAla/fabada
