  We describe an analysis of the First International Pulsar Timing Array Data
Challenge. We employ a robust, unbiased Bayesian framework developed by van
Haasteren to study the three Open and Closed datasets, testing various models
for each dataset and using MultiNest to recover the evidence for the purposes
of Bayesian model-selection. The parameter constraints of the favoured model
are confirmed using an adaptive MCMC technique. Our results for Closed1
favoured a gravitational-wave background with strain amplitude at f=1 yr-1, A,
of (1.1 +/- 0.1) x 10^{-14}, power spectral-index gamma=4.30 +/- 0.15 and no
evidence for red-timing noise or single-sources. The evidence for Closed2
favours a gravitational-wave background with A=(6.1 +/- 0.3) x 10^{-14},
gamma=4.34 +/- 0.09 with no red-timing noise or single-sources. Finally, the
evidence for Closed3 favours the presence of red-timing noise and a
gravitational-wave background, with no single-sources. The properties of the
background were A=(5 +/- 1) x 10^{-15} and gamma=4.23 +/- 0.35, while the
properties of the red-noise were N_{red}=(12 +/- 4) ns and gamma_{red}=1.5 +/-
0.3. In all cases the redness of the recovered background is consistent with a
source-population of inspiraling supermassive black-hole binaries. We also
investigate the effect that down-sampling of the datasets has on parameter
constraints and run-time. Finally we provide a proof-of-principle study of the
ability of the Bayesian framework used in this paper to reconstruct the angular
correlation of gravitational-wave background induced timing-residuals,
comparing this to the Hellings and Downs curve.
