Aug 2 – 6, 2021
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Unfolding by Folding: a Bayesian resampling approach to the problem of matrix inversion without actually inverting any matrix

Aug 5, 2021, 9:20 AM


Parallel contribution H. Statistical Methods for Physics Analysis in the XXI Century Parallels Track H


Pietro Vischia (Universite Catholique de Louvain (UCL) (BE))


Matrix inversion problems are often encountered in experimental physics, and in particular in high-energy
particle physics, under the name of unfolding. The true spectrum of a physical quantity is deformed by
the presence of a detector, resulting in an observed spectrum. If we discretize both the true and observed
spectra into histograms, we can model the detector response via a matrix. Inferring a true spectrum
starting from an observed spectrum requires therefore inverting the response matrix. Many methods exist
in literature for this task, all starting from the observed spectrum and using a simulated true spectrum as
a guide to obtain a meaningful solution in cases where the response matrix is not easily invertible.
In this Manuscript, I take a different approach to the unfolding problem. Rather than inverting the
response matrix and transforming the observed distribution into the most likely parent distribution in
generator space, I sample many distributions in generator space, fold them through the original response
matrix, and pick the generator-level distribution which yields the folded distribution closest to the data
distribution. Regularization schemes can be introduced to treat the case where non-diagonal response
matrices result in high-frequency oscillations of the solution in true space, and the introduced bias is
The algorithm performs as well as traditional unfolding algorithms in cases where the inverse problem
is well-defined in terms of the discretization of the true and smeared space, and outperforms them in
cases where the inverse problem is ill-defined—when the number of truth-space bins is larger than that of
smeared-space bins. These advantages stem from the fact that the algorithm does not technically invert
any matrix and uses only the data distribution as a guide to choose the best solution.
The algorithm is also extended, in analogy to Bayesian approaches such as Metropolis Hastings, to a full description of the posterior distribution of each bin yield.

Primary author

Pietro Vischia (Universite Catholique de Louvain (UCL) (BE))

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