Table 3 Posterior predictive distributions

In Bayesian modelling, it is possible to derive a full posterior predictive distribution for any new individual. Thus the uncertainty associated with a prediction is captured through a probability distribution, from which point estimates can be derived, or alternatively, probabilistic questions can be asked, such as those described in the Probabilistic predictions using the Bayesian treatment selection model section. The definition of a posterior predictive distribution for an individual with complete predictor information is:

    Thus, the predictive distribution integrates (or averages) over the posterior distribution for the parameters and thus naturally incorporates the uncertainties about the parameters as well as those arising from the underlying model. The examples in this paper present the posterior predictive distribution for the expected outcome, with a given set of characteristics defined by:

    We are free to use either distribution (4–5), dependent on the context, although in practice, most treatment selection models use an analogous approach to (5), focusing on the conditional average treatment response. Please see the Supplementary Material for more information about how different predictive distributions can be derived and sampled from (including when predictors are missing, in which case we integrate the missing information modelled using the DPMM).