  When using Markov chain Monte Carlo (MCMC) algorithms to perform inference
for Bayesian clustering models, such as mixture models, the output is typically
a sample of clusterings (partitions) drawn from the posterior distribution. In
practice, a key challenge is how to summarise this output. Here we build upon
the notion of the posterior similarity matrix (PSM) in order to suggest new
approaches for summarising the output of MCMC algorithms for Bayesian
clustering models. A key contribution of our work is the observation that PSMs
are positive semi-definite, and hence can be used to define
probabilistically-motivated kernel matrices that capture the clustering
structure present in the data. This observation enables us to employ a range of
kernel methods to obtain summary clusterings, and otherwise exploit the
information summarised by PSMs. For example, if we have multiple PSMs, each
corresponding to a different dataset on a common set of statistical units, we
may use standard methods for combining kernels in order to perform integrative
clustering. We may moreover embed PSMs within predictive kernel models in order
to perform outcome-guided data integration. We demonstrate the performances of
the proposed methods through a range of simulation studies as well as two real
data applications. R code is available at
https://github.com/acabassi/combine-psms.
