  We develop a non-parametric Bayesian prior for a family of random probability
measures by extending the Polya tree ($PT$) prior to a joint prior for a set of
probability measures $G_1,\dots,G_n$, suitable for meta-analysis with event
time outcomes. In the application to meta-analysis $G_i$ is the event time
distribution specific to study $i$. The proposed model defines a regression on
study-specific covariates by introducing increased correlation for any pair of
studies with similar characteristics. The desired multivariate $PT$ model is
constructed by introducing a hierarchical prior on the conditional splitting
probabilities in the $PT$ construction for each of the $G_i$. The hierarchical
prior replaces the independent beta priors for the splitting probability in the
$PT$ construction with a Gaussian process prior for corresponding (logit)
splitting probabilities across all studies. The Gaussian process is indexed by
study-specific covariates, introducing the desired dependence with increased
correlation for similar studies. The main feature of the proposed construction
is (conditionally) conjugate posterior updating with commonly reported
inference summaries for event time data. The construction is motivated by a
meta-analysis over cancer immunotherapy studies.
