Editing Ridge regression (section)

==Relation to probabilistic formulation==
The probabilistic formulation of an [[inverse problem]] introduces (when all uncertainties are Gaussian) a covariance matrix <math> C_M</math> representing the ''a priori'' uncertainties on the model parameters, and a covariance matrix <math> C_D</math> representing the uncertainties on the observed parameters.<ref>{{cite book |last1=Tarantola |first1=Albert |title=Inverse Problem Theory and Methods for Model Parameter Estimation |date=2005 |publisher=Society for Industrial and Applied Mathematics (SIAM) |location=Philadelphia |isbn=0-89871-792-2 |edition=1st |url=http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/SIAM/index.html |access-date=9 August 2018 |ref=ATarantolaSIAM2004}}</ref> In the special case when these two matrices are diagonal and isotropic, <math> C_M = \sigma_M^2 I </math> and <math> C_D = \sigma_D^2 I </math>, and, in this case, the equations of inverse theory reduce to the equations above, with <math> \alpha = {\sigma_D}/{\sigma_M} </math>.<ref>{{cite journal | last1 = Huang | first1 = Yunfei. | display-authors = etal | year = 2019 | title = Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells | journal = Scientific Reports | volume = 9 | number = 1| page = 537 | doi = 10.1038/s41598-018-36896-x | pmid = 30679578 | doi-access = free | pmc = 6345967 | arxiv = 1810.05848 | bibcode = 2019NatSR...9..539H }}</ref><ref>{{cite journal | last1 = Huang | first1 = Yunfei |  last2 = Gompper | first2 =  Gerhard |  last3 = Sabass | first3 = Benedikt |year = 2020 | title = A Bayesian traction force microscopy method with automated denoising in a user-friendly software package | journal = Computer Physics Communications | volume = 256 | page = 107313 | doi = 10.1016/j.cpc.2020.107313 | arxiv = 2005.01377 | bibcode = 2020CoPhC.25607313H }}</ref>