  In a number of data-driven applications such as detection of arrhythmia,
interferometry or audio compression, observations are acquired indistinctly in
the time or frequency domains: temporal observations allow us to study the
spectral content of signals (e.g., audio), while frequency-domain observations
are used to reconstruct temporal/spatial data (e.g., MRI). Classical approaches
for spectral analysis rely either on i) a discretisation of the time and
frequency domains, where the fast Fourier transform stands out as the
\textit{de facto} off-the-shelf resource, or ii) stringent parametric models
with closed-form spectra. However, the general literature fails to cater for
missing observations and noise-corrupted data. Our aim is to address the lack
of a principled treatment of data acquired indistinctly in the temporal and
frequency domains in a way that is robust to missing or noisy observations, and
that at the same time models uncertainty effectively. To achieve this aim, we
first define a joint probabilistic model for the temporal and spectral
representations of signals, to then perform a Bayesian model update in the
light of observations, thus jointly reconstructing the complete (latent) time
and frequency representations. The proposed model is analysed from a classical
spectral analysis perspective, and its implementation is illustrated through
intuitive examples. Lastly, we show that the proposed model is able to perform
joint time and frequency reconstruction of real-world audio, healthcare and
astronomy signals, while successfully dealing with missing data and handling
uncertainty (noise) naturally against both classical and modern approaches for
spectral estimation.
