Note that the desired expectation value is now a ratio where the numerator and denominator are expectation values that both use a positive weighting function, p [ σ ] {\displaystyle p[\sigma ]} . However, the phase exp ⁡ ( i θ [ σ ] ) {\displaystyle \exp(i\theta [\sigma ])} is a highly oscillatory function in the configuration space, so if one uses Monte Carlo methods to evaluate the numerator and denominator, each of them will evaluate to a very small number, whose exact value is swamped by the noise inherent in the Monte Carlo sampling process. The "badness" of the sign problem is measured by the smallness of the denominator ⟨ exp ⁡ ( i θ [ σ ] ) ⟩ p {\displaystyle \langle \exp(i\theta [\sigma ])\rangle _{p}} : if it is much less than 1 then the sign problem is severe.