Global 21-cm signal extraction from foreground and instrumental effects
  IV: Accounting for realistic instrument uncertainties and their overlap with
  foreground and signal models
  All 21-cm signal experiments rely on electronic receivers that affect the
data via both multiplicative and additive biases through the receiver's gain
and noise temperature. While experiments attempt to remove these biases, the
residuals of their imperfect calibration techniques can still confuse signal
extraction algorithms. In this paper, the fourth and final installment of our
pipeline series, we present a technique for fitting out receiver effects as
efficiently as possible. The fact that the gain and global signal, which are
multiplied in the observation equation, must both be modeled implies that the
model of the data is nonlinear in its parameters, making numerical sampling the
only way to explore the parameter distribution rigorously. However,
multi-spectra fits, which are necessary to extract the signal confidently as
demonstrated in the third paper of the series, often require large numbers of
foreground parameters, increasing the dimension of the posterior distribution
that must be explored and therefore causing numerical sampling inefficiencies.
Building upon techniques in the second paper of the series, we outline a method
to explore the full parameter distribution by numerically sampling a small
subset of the parameters and analytically marginalizing over the others. We
test this method in simulation using a type-I Chebyshev band-pass filter gain
model and a fast signal model based on a spline between local extrema. The
method works efficiently, converging quickly to the posterior signal parameter
distribution. The final signal uncertainties are of the same order as the noise
in the data.
