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Observations of stratospheric ozone from multiple instruments now span three decades; combining these into composite datasets allows long-term ozone trends to be estimated. Recently, several ozone composites have been published, but trends disagree by latitude and altitude, even between composites built upon the same instrument data. We confirm that the main causes of differences in decadal trend estimates lie in (i) steps in the composite time series when the instrument source data changes and (ii) artificial sub-decadal trends in the underlying instrument data. These artefacts introduce features that can alias with regressors in multiple linear regression (MLR) analysis; both can lead to inaccurate trend estimates. Here, we aim to remove these artefacts using Bayesian methods to infer the underlying ozone time series from a set of composites by building a joint-likelihood function using a Gaussian-mixture density to model outliers introduced by data artefacts, together with a data-driven prior on ozone variability that incorporates knowledge of problems during instrument operation. We apply this Bayesian self-calibration approach to stratospheric ozone in 10 degrees bands from 60 degrees S to 60 degrees N and from 46 to 1 hPa (similar to 21-48 km) for 1985-2012. There are two main outcomes: (i) we independently identify and confirm many of the data problems previously identified, but which remain unaccounted for in existing composites; (ii) we construct an ozone composite, with uncertainties, that is free from most of these problems - we call this the BAyeSian Integrated and Consolidated (BASIC) composite. To analyse the new BASIC composite, we use dynamical linear modelling (DLM), which provides a more robust estimate of long-term changes through Bayesian inference than MLR. BASIC and DLM, together, provide a step forward in improving estimates of decadal trends. Our results indicate a significant recovery of ozone since 1998 in the upper stratosphere, of both northern and southern midlatitudes, in all four composites analysed, and particularly in the BASIC composite. The BASIC results also show no hemispheric difference in the recovery at midlatitudes, in contrast to an apparent feature that is present, but not consistent, in the four composites. Our overall conclusion is that it is possible to effectively combine different ozone composites and account for artefacts and drifts, and that this leads to a clear and significant result that upper stratospheric ozone levels have increased since 1998, following an earlier decline.