Ja. D. Shirman, V. M. Orlenko

The fact of discussion between the two most prominent academicians and mathematicians A.N. Kolmogorov and S.N. Bernshtein in 1940-1942 is not widely known. In the discussion A.N. Kolmogorov was defending the «maximum likelihood» approximation in statistics proposed by R. Fisher. In his turn, S.N. Bernshtein objected against this approximation as not fully corresponding to Bayesian theory. Now, even the former proponents of Fisher's theory such as J. Neyman and C. Rao together with their followers have changed their attitude towards the Bayessian methods. The early lack of accumulated material prevented revealing the inadvisability of Fisher's approximation. Direct harm was done to the theory of space-time signal processing, which has been studied since 1950-60s. The harm is illustrated by comparing the difference between Fisher's and Bayes approaches with reference to single-channel CFAR detector and covariance matrix estimate. It is shown that use of maximum likelihood approach in CFAR processing may not provide stabilization of false alarm probability in case of arbitrary interference and may lead to corresponding underestimation of adaptation losses. As applied to covariance matrix estimate, the use of maximum likelihood approach leads to estimates that yield singular covariance matrix when sample size is less than matrix dimension without any regard to the matrix structure. This comes into direct contradiction with the practical experience of jamming cancellation from few jammers using correlation side-lobe cancellers. Discussion of the appropriateness of maximum likelihood approach in signal processingis at the center position in the paper. As a result of paradoxes revealed for signal processing, further question arises as of appropriateness of the maximum likelihood ap-proach use in different branches of science, such as cybernetics, economics, medicine, quantum physics. At the end of the paper, the ways are discussed of how to handle the situation in hand.