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The measurement of the position of the chiral critical end point (CEP) in theQCD phase diagram is under debate. While it is possible to predict its positionby using effective models specifically built to reproduce some of the featuresof the underlying theory (QCD), the quality of the predictions (\textit{e.g.},the CEP position) obtained by such effective models, depends on whether solvingthe model equations constitute a well- or ill-posed inverse problem.  Considering these predictions as being inverse problems provides tools toevaluate if the problem is ill-conditioned, meaning that infinitesimalvariations of the inputs of the model can cause comparatively large variationsof the predictions. If it is ill-conditioned, it has major consequences becauseof finite variations that could come from experimental and/or theoreticalerrors.  In the following, we shall apply such a reasoning on the predictions of aparticular Nambu--Jona-Lasinio model within the mean field + ringapproximations, with special attention to the prediction of the chiral CEPposition in the $(T-\mu)$ plane. We find that the problem is ill-conditioned(\idest very sensitive to input variations) for the $T$-coordinate of the CEP,whereas, it is well-posed for the $\mu$-coordinate of the CEP. As aconsequence, when the chiral condensate varies in a $10$ MeV range, $\mu_{CEP}$varies far less.  As an illustration to understand how problematic this could be, we show thatthe main consequence when taking into account finite variation of the inputs,is that the existence of the CEP itself cannot be predicted anymore: for adeviation as low as 0.6 \% with respect to vacuum phenomenology (well withinthe estimation of the first correction to the ring approximation) the CEP mayor may not exist.

Sensitivity of predictions in an effective model -- application to the chiral critical end point position in the Nambu--Jona-Lasinio model