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Developing the full update rules for the variational inference model is beyond the scope of this paper. What we have opted to do instead is attempt to reformulate approaches used for similar models. The work of Wang et al. \cite{Wang2011} and Bryant et al.\cite{Bryant2012} on variational inference is a good first step. They have developed a variational framework for a specific realisation of the HDP. It differs from the topic model underlying our TLG in a few ways (how hierarchy is represented, whether temporality is considered etc). However, it is an excellent proof-of-concept for applying VI methods to the timeline generation problem.  For an example of the optimisation update-rules, see (Fig. 4). **(Fig. 4)**.  The complete procedure can be found in the original paper \cite{Bryant2012}.