  New telescopes like the Square Kilometre Array (SKA) will push into a new
sensitivity regime and expose systematics, such as direction-dependent effects,
that could previously be ignored. Current methods for handling such systematics
rely on alternating best estimates of instrumental calibration and models of
the underlying sky, which can lead to inadequate uncertainty estimates and
biased results because any correlations between parameters are ignored. These
deconvolution algorithms produce a single image that is assumed to be a true
representation of the sky, when in fact it is just one realization of an
infinite ensemble of images compatible with the noise in the data. In contrast,
here we report a Bayesian formalism that simultaneously infers both systematics
and science. Our technique, Bayesian Inference for Radio Observations (BIRO),
determines all parameters directly from the raw data, bypassing image-making
entirely, by sampling from the joint posterior probability distribution. This
enables it to derive both correlations and accurate uncertainties, making use
of the flexible software MEQTREES to model the sky and telescope
simultaneously. We demonstrate BIRO with two simulated sets of Westerbork
Synthesis Radio Telescope data sets. In the first, we perform joint estimates
of 103 scientific (flux densities of sources) and instrumental (pointing
errors, beamwidth and noise) parameters. In the second example, we perform
source separation with BIRO. Using the Bayesian evidence, we can accurately
select between a single point source, two point sources and an extended
Gaussian source, allowing for 'super-resolution' on scales much smaller than
the synthesized beam.
