  This paper is concerned with Bayesian inferential methods for data from
controlled branching processes that account for model robustness through the
use of disparities. Under regularity conditions, we establish that estimators
built on disparity-based posterior, such as expectation and maximum a
posteriori estimates, are consistent and efficient under the posited model.
Additionally, we show that the estimates are robust to model misspecification
and presence of aberrant outliers. To this end, we develop several fundamental
ideas relating minimum disparity estimators to Bayesian estimators built on the
disparity-based posterior, for dependent tree-structured data. We illustrate
the methodology through a simulated example and apply our methods to a real
data set from cell kinetics.
