Ways of choosing \(\alpha\)

The regularization of the misfit \(\chi^2\) with an entropy \(S\) introduces the ad-hoc parameter \(\alpha\). The way to choose \(\alpha\) marks various varieties of the MaxEnt approach:

• Historic MaxEnt: \(\chi^2\) equal to number of data points

• Probability \(p(\alpha | G, D)\) based:

  • Classic: determine maximum of \(p\)

  • Bryan: average over \(A(\alpha)\) with weights \(p\)

• Kink in \(\log(\chi^2)\) vs. \(\log(\alpha)\)

  • \(\Omega\)-MaxEnt: use \(\alpha\) at maximum curvature

  • Line fit: fit two lines and use intersection for optimal \(\alpha\)

A disadvantage of the Historic MaxEnt and the probabilistic methods is that the resulting \(A\) is strongly dependent on the provided covariance matrix. If the statistical error of Monte Carlo measurements, for example, is not estimated accurately, the data could be over- or under-fitted.

Methods analyzing the dependence of \(\log(\chi^2)\) on \(\log(\alpha)\) can overcome this problem by searching for the cross-over point from the noise-fitting (small \(\alpha\)) to the information-fitting (intermediate \(\alpha\)) regime. In the noise-fitting regime \(\chi^2\) is essentially constant, while in the information-fitting region it behaves linearly.

One can either select the point of maximum curvature (\(\Omega\)-MaxEnt), or fit two straight lines and select the \(\alpha\) at the intersection.

In this package, different ways of determining \(\alpha\) are implemented, and with one run of the code the solutions of different MaxEnt flavors can be obtained.