  We derive, using functional methods and the bias expansion, the conditional
likelihood for observing a specific tracer field given an underlying matter
field. This likelihood is necessary for Bayesian-inference methods. If we
neglect all stochastic terms apart from the ones appearing in the auto
two-point function of tracers, we recover the result of Schmidt et al., 2018.
We then rigorously derive the corrections to this result, such as those coming
from a non-Gaussian stochasticity (which include the stochastic corrections to
the tracer bispectrum) and higher-derivative terms. We discuss how these
corrections can affect current applications of Bayesian inference. We comment
on possible extensions to our result, with a particular eye towards primordial
non-Gaussianity. This work puts on solid theoretical grounds the EFT-based
approach to Bayesian forward modeling.
