Editing Ridge regression (section)

=== Application to existing fit results ===

Since Tikhonov Regularization simply adds a quadratic term to the objective function in optimization problems,
it is possible to do so after the unregularised optimisation has taken place.
E.g., if the above problem with <math>\Gamma = 0</math> yields the solution <math>\hat\mathbf{x}_0</math>,
the solution in the presence of <math>\Gamma \ne 0</math> can be expressed as:
<math display="block">\hat\mathbf{x} = B \hat\mathbf{x}_0,</math>
with the "regularisation matrix" <math>B = \left(A^\mathsf{T} A + \Gamma^\mathsf{T} \Gamma\right)^{-1} A^\mathsf{T} A</math>.

If the parameter fit comes with a covariance matrix of the estimated parameter uncertainties <math>V_0</math>,
then the regularisation matrix will be
<math display="block">B = (V_0^{-1} + \Gamma^\mathsf{T}\Gamma)^{-1} V_0^{-1},</math>
and the regularised result will have a new covariance
<math display="block">V = B V_0 B^\mathsf{T}.</math>

In the context of arbitrary likelihood fits, this is valid, as long as the quadratic approximation of the likelihood function is valid. This means that, as long as the perturbation from the unregularised result is small, one can regularise any result that is presented as a best fit point with a covariance matrix. No detailed knowledge of the underlying likelihood function is needed. <ref>{{cite journal|arxiv=2207.02125 |doi=10.1088/1748-0221/17/10/P10021 |title=Post-hoc regularisation of unfolded cross-section measurements |date=2022 |last1=Koch |first1=Lukas |journal=Journal of Instrumentation |volume=17 |issue=10 |pages=10021 |bibcode=2022JInst..17P0021K }}</ref>