Calculating the EFT likelihood via saddle-point expansion

In this paper, we extend the functional approach for calculating the EFT likelihood by applying the saddle-point expansion. We demonstrate that, after suitable reformulation, the likelihood expression is consistent with the path integral required to be computed in the theory of false vacuum decay. In contrast to the saddle-point approximation, the application of the saddle-point expansion necessitates more nuanced considerations, particularly concerning the treatment of the negative eigenvalues of the second derivative of the action at the saddle point. We illustrate that a similar issue arises in the likelihood calculation, which requires approximating the original integral contour through the combination of the steepest descent contours in the field space. As a concrete example, we focus on calculating the EFT likelihood under a Gaussian distribution and propose a general procedure for computing the likelihood using the saddle-point expansion method for arbitrary partition functions. Precise computation of the likelihood will benefit Bayesian forward modeling, thereby enabling more reliable theoretical predictions.