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the problem of extracting scientific parameters from dirty ( dominated by artifacts ) interferometric radio images has resulted in many deconvolution algorithms being developed . however
, none of these solve the problem of incorporating instrumental errors as a source of uncertainty when making measurements from radio data .
deconvolution algorithms can only produce one image , which the scientist must then assume is correct before extracting any science ( for example , a catalogue of source fluxes ) from it .
algorithms such as clean ( @xcite ) , the most popular algorithm in use , can not reliably produce any uncertainties ( @xcite ) , making it impossible to propagate the uncertainties from instrumental errors to the scientific parameters .
further , if these parameters are correlated , as they likely are , even correcting the data for instrumental errors may lead to biased scientific results , as the measurement of the instrumental error may be wrong . these instrumental effects will become more important as more sensitive telescopes such as the square kilometre array come online .
we propose in @xcite , a completely different approach , whereby we model the sky and all known sources of instrumental error simultaneously using the radio interferometry measurement equation ( rime ) ( @xcite ) .
the rime can be written as ( @xcite ) : @xmath0 where @xmath1 is the visibility matrix ( the radio data ) , @xmath2 is the @xmath3th jones matrix for antenna @xmath4 ( containing instrumental effects ) and @xmath5 is the brightness matrix ( containing the sky model , and hence all scientific parameters ) .
we use the software package meqtrees ( @xcite ) , which implements the rime , to model our radio field and any known instrumental effects . we can then estimate the parameters of this model , both scientific and instrumental , in a bayesian context using a sampling method such as mcmc ( @xcite ) .
we assume uncorrelated gaussian noise on the visibilities which leads to a simple gaussian likelihood for @xmath6 . with this approach ,
we are able to determine the full posterior for the problem , obtaining not only the best fits for all parameters , but also their uncertainties and correlations .
figure [ fig : field ] shows the mock wsrt field which we tested biro on .
this field , simulated using meqtrees and based on a real field , consists of 17 point sources .
we also applied pointing errors to each antenna as an example of a source of instrumental error , which wsrt ( and many other radio telescopes ) have had to deal with in the past ( @xcite ) .
a mispointed antenna will observe a point source through the edge rather than the centre of its beam .
thus , in general , we would expect pointing errors and fluxes to be correlated and we apply our method to determine these correlations , as well as the parameters themselves .
the instrumental parameters for this simulated data consist of the pointing errors , the width of the primary beam and the noise on the visibilities , which we assume to be gaussian , as is widely considered a good approximation .
we allow the pointing errors to vary in time as second ordr polynomial functions , resulting in 84 pointing error parameters , 3 parameters for each direction for each antenna .
our scientific parameters include the flux for each source , and the shape parameters of the extended gaussian source .
we compared biro with the standard , commonly used clean algorithm combined with a source extraction algorithm on the cleaned image ( we call this clean+se ) for this dataset .
figure [ fig : compare ] illustrates this comparison for the estimated fluxes from biro and clean+se . with no knowledge of the pointing errors , it is no surprise that clean+se returns biased fluxes and fails to find several sources .
the danger is that the error bars , estimated using only the surrounding flux of the point source in this source extraction algorithm , do not and can not take into account the additional source of uncertainty from the instrumental errors .
in contrast , the biro estimates are unbiased and the error bars are larger , correctly propagating the instrumental errors , and determining the correlations between parameters ( figure [ fig : covmat ] ) .    , due to the strong correlations between pointing error and flux . ]
-plane , while the pointing errors are correlated simply because every pointing error affects every source . of particular interest is the complex way in which pointing errors correlate with the fluxes , which would be very difficult to determine from first principles.,scaledwidth=55.0% ]
we have introduced biro , a bayesian approach to the deconvolution problem of radio interferometry observations .
figure [ fig : covmat ] highlights the importance of fully propagating the uncertainty on instrumental errors , as they can be highly correlated with the scientific parameters and hence bias scientific results . due to the fully bayesian nature of biro
, it allows for very elegant extensions to this simple scenario .
for example , the choice of model for the field can be selected for using the bayesian evidence ( see @xcite ) .
biro can be useful in any scenario where reliable statistics are required for the science extracted and may be essential to fully exploit the sensitivity of the ska .
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ml and jz are grateful to the south africa national research foundation square kilometre array project for financial support .
ml acknowledges support from the university of cape town and resources from the african institute for mathematical sciences .
os is supported by the south african research chairs initiative of the department of science and technology and national research foundation .
in acknowledges the meerkat hpc for radio astronomy programme .
part of the computations were performed using facilities provided by the university of cape town s icts high performance computing team : http://hpc.uct.ac.za .
radio interferometers suffer from the problem of missing information in their data , due to the gaps between the antennas . 
 this results in artifacts , such as bright rings around sources , in the images obtained . 
 multiple deconvolution algorithms have been proposed to solve this problem and produce cleaner radio images . however , these algorithms are unable to correctly estimate uncertainties in derived scientific parameters or to always include the effects of instrumental errors . 
 we propose an alternative technique called bayesian inference for radio observations ( biro ) which uses a bayesian statistical framework to determine the scientific parameters and instrumental errors simultaneously directly from the raw data , without making an image . 
 we use a simple simulation of westerbork synthesis radio telescope data including pointing errors and beam parameters as instrumental effects , to demonstrate the use of biro .