  LISA Pathfinder (LPF), ESA's precursor mission to a gravitational wave
observatory, will measure the degree to which two test-masses can be put into
free-fall, aiming to demonstrate a residual relative acceleration with a power
spectral density (PSD) below 30 fm/s$^2$/Hz$^{1/2}$ around 1 mHz. In LPF data
analysis, the measured relative acceleration data series must be fit to other
various measured time series data. This fitting is required in different
experiments, from system identification of the test mass and satellite dynamics
to the subtraction of noise contributions from measured known disturbances. In
all cases, the background noise, described by the PSD of the fit residuals, is
expected to be coloured, requiring that we perform such fits in the frequency
domain. This PSD is unknown {\it a priori}, and a high accuracy estimate of
this residual acceleration noise is an essential output of our analysis. In
this paper we present a fitting method based on Bayesian parameter estimation
with an unknown frequency-dependent background noise. The method uses noise
marginalisation in connection with averaged Welch's periodograms to achieve
unbiased parameter estimation, together with a consistent, non-parametric
estimate of the residual PSD. Additionally, we find that the method is
equivalent to some implementations of iteratively re-weighted least-squares
fitting. We have tested the method both on simulated data of known PSD, and to
analyze differential acceleration from several experiments with the LISA
Pathfinder end-to-end mission simulator.
