class TMVA::MethodLikelihood: public TMVA::MethodBase

Likelihood analysis ("non-parametric approach")

Also implemented is a "diagonalized likelihood approach", which improves over the uncorrelated likelihood ansatz by transforming linearly the input variables into a diagonal space, using the square-root of the covariance matrix

The method of maximum likelihood is the most straightforward, and certainly among the most elegant multivariate analyser approaches. We define the likelihood ratio, R_L, for event i, by:

Here the signal and background likelihoods, L_S, L_B, are products of the corresponding probability densities, p_S, p_B, of the N_var discriminating variables used in the MVA:
and accordingly for L_B. In practise, TMVA uses polynomial splines to estimate the probability density functions (PDF) obtained from the distributions of the training variables.

Note that in TMVA the output of the likelihood ratio is transformed by

to avoid the occurrence of heavy peaks at R_L=0,1. Decorrelated (or "diagonalized") Likelihood

The biggest drawback of the Likelihood approach is that it assumes that the discriminant variables are uncorrelated. If it were the case, it can be proven that the discrimination obtained by the above likelihood ratio is optimal, ie, no other method can beat it. However, in most practical applications of MVAs correlations are present.

Linear correlations, measured from the training sample, can be taken into account in a straightforward manner through the square-root of the covariance matrix. The square-root of a matrix C is the matrix C′ that multiplied with itself yields C: C=C′C′. We compute the square-root matrix (SQM) by means of diagonalising (D) the covariance matrix:

and the linear transformation of the linearly correlated into the uncorrelated variables space is then given by multiplying the measured variable tuple by the inverse of the SQM. Note that these transformations are performed for both signal and background separately, since the correlation pattern is not the same in the two samples.

The above diagonalisation is complete for linearly correlated, Gaussian distributed variables only. In real-world examples this is not often the case, so that only little additional information may be recovered by the diagonalisation procedure. In these cases, non-linear methods must be applied.