The focus is within the context of the dimension reduction for conditional extreme values. More specifically, the focus is on the case where the extreme values of a response variable may be explained by non-linear functions of some linear projections of the input random vector. The estimation of the projection directions has been investigated in the extreme-PLS (EPLS) method, an adaptation of the original PLS method to the extreme-value framework. A new interpretation of the EPLS direction is introduced as a maximum likelihood estimator based on the von Mises-Fisher distribution on hyperballs. The Bayesian paradigm then makes it possible to introduce prior information on the dimension reduction direction. The maximum a posteriori estimator is derived in two particular cases and interpreted as a shrinkage of the EPLS estimator. Its asymptotic behavior is established as the sample size tends to infinity. A simulated data study shows that the proposed method is effective for moderate data problems in high dimensional settings. An illustration of the effectiveness of the proposed method is provided on French farm income data from which 259 dimensions have been considered in the descriptor.