  The mystery of the infamous sign problem in quantum Monte Carlo (QMC)
simulations mightily restricts applications of the method in fermionic and
frustrated systems. A recent work [Science \textbf{375}, 418 (2022)] made a
remarkable breakthrough in the sign problem by pointing out that the sign can
be used to probe phase transition. In this work, we proposed a general argument
based on the definition of the sign that is related to the difference in free
energy between the original and reference systems to clarify that the sign
problem and phase transition cannot always be strictly related. The sign can
exactly probe phase transition only if the free energy in the reference system
is flat under variable parameters, which is almost impossible to design.
Generally speaking, the conclusion that the sign can probe phase transition is
survivorship bias without universality. To solve this problem, we define a
modified sign that excludes the influence of the reference system, which can
probe the phase transition strictly. The work gives an unbiased solution for
detecting phase transition by the new modified sign.
