  Galactic dust emission is often accounted for in cosmic microwave background
(CMB) analyses by fitting a simple two-parameter modified blackbody (MBB) model
in each pixel, which nominally accounts for the temperature and opacity of the
dust. While this may be a good approximation for individual dust clouds,
typically a number of such clouds are found along each line of sight and within
each angular pixel, resulting in a superposition of their spectra. In this
paper, we study the effects of this superposition on pixel-based foreground
fitting strategies by modelling the spectral energy distribution (SED) in each
pixel as the integral of individual MBB spectra over various
physically-motivated statistical distributions of dust cloud properties. We
show that fitting these SEDs with the simple two-parameter MBB model generally
results in unbiased estimates of the CMB Stokes Q and U amplitudes in each
pixel, unless there are significant changes in both the dust SED and
polarization angle along the line of sight, in which case significant ($ >
10\sigma$) biases are observed in an illustrative model. We also find that the
best-fit values of the dust temperature, $T_d$, and spectral index, $\beta_d$,
are significantly biased away from the mean/median of the corresponding
statistical distributions when the distributions are broad, suggesting that MBB
model fits can give an unrepresentative picture of the physical properties of
the dust at microwave wavelengths if not interpreted carefully. Using a Fisher
matrix analysis, we also determine the experimental sensitivity required to
recover the parameters of the $T_d$ and $\beta_d$ distributions themselves by
fitting a probabilistic MBB model, finding that only the parameters of broad
distributions can be measured by SED fitting on a single line of sight.
