  We present a Bayesian method for estimating instrumental noise parameters and
propagating noise uncertainties within the global BeyondPlanck Gibbs sampling
framework, and apply this to Planck Low Frequency Instrument (LFI) time-ordered
data. Following previous literature, we initially adopt a $1/f$ model for the
noise power spectral density (PSD), but find the need for an additional
lognormal component in the noise model for the 30 and 44\,GHz bands. We
implement an optimal Wiener-filter (or constrained realization) gap-filling
procedure to account for masked data. We then use this procedure to both
estimate the gapless correlated noise in the time-domain, $n_\mathrm{corr}$,
and to sample the noise PSD parameters, $\xi^n = \{\sigma_0, f_\mathrm{knee},
\alpha, A_\mathrm{p}\}$. In contrast to previous \textit{Planck} analyses, we
assume piecewise stationary noise only within each pointing period (PID), not
throughout the full mission, but we adopt the LFI Data Processing Center (DPC)
results as priors on $\alpha$ and $f_\mathrm{knee}$. On average, we find
best-fit correlated noise parameters that are mostly consistent with previous
results, with a few notable exceptions. However, a detailed inspection of the
time-dependent results reveals many important findings. First and foremost, we
find strong evidence for statistically significant temporal variations in all
noise PSD parameters, many of which are directly correlated with satellite
housekeeping data. Second, while the simple $1/f$ model appears to be an
excellent fit for the LFI 70 GHz channel, there is evidence for additional
correlated noise not described by a $1/f$ model in the 30 and 44 GHz channels,
including within the primary science frequency range of 0.1--1 Hz. (Abridged)
