  It is well-known that the critical gravitational collapse produces continuous
self-similar solutions characterized by the Choptuik critical exponent,
$\gamma$. We examine the solutions in the domains of the linear perturbation
equations, considering the numerical measurement errors. Specifically, we study
quantum perturbation theory for the four-dimensional Einstein-axion-dilaton
system of the elliptic class of $\text{SL}(2,\mathbb{R})$ transformations. We
develop a novel artificial neural network-assisted Metropolis-Hastings
algorithm based on quantum perturbation theory to find the distribution of the
critical exponent in a Bayesian framework. Unlike existing methods, this new
probabilistic approach identifies the available deterministic solution and
explores the range of physically distinguishable critical exponents that may
arise due to numerical measurement errors.
