One of the most pressing questions in modern cosmology pertains to the physical processes governing the early universe and the origins of cosmic structure. Primordial signals are manifest in various probes of the large-scale cosmic structure, such as the higher-order statistics of the density field and the scale-dependent bias effect. Detecting and measuring non-Gaussian primordial signals would shed light on the potential shape of the inflaton field, the hypothetical particle responsible for cosmic inflation. In the near future, next-generation galaxy surveys will begin operation, aiming to constrain the non-linearity parameter fNL to the degree of uncertainty necessary for identifying feasible inflationary models. Nevertheless, accomplishing this objective necessitates modern statistical data analysis tools to accurately account for stochastic and systematic uncertainties when extracting these subtle signals from observations.

In this thesis, I describe a novel approach for measuring primordial non-Gaussianity in galaxy redshift surveys that I have developed. The method is based on a Bayesian field-level inference technique, which includes the full field to constrain fNL. In this way, the method is able to go beyond current state-of-the-art methods, which employ a limited set of summary statistics, to capture the full information content of the three-dimensional cosmic structure. The method uses a physical forward model that translates any set of initial conditions to a predicted observable. The space of plausible initial conditions and cosmological parameters are sampled with the help of a Bayesian framework utilizing a Hamiltonian Monte Carlo approach. The method accounts for the gravitational formation of the three-dimensional cosmic structure, and inherently and fully self-consistently accounts for all stochastic uncertainties and systematic effects associated with selection effects, galaxy biasing, and survey geometries. The method is able to account for multiple probes of primordial non-Gaussianity, e.g. the higher-order correlation functions, galaxy mass distributions, peculiar velocity fields, and the scale-dependent bias effect. I showcase highlights of the development process, and present work in inferring primordial non-Gaussianity in galaxy survey data sets. Lastly, necessary preparation for next-generation galaxy redshift surveys is discussed.