  Distance-based hierarchical clustering (HC) methods are widely used in
unsupervised data analysis but few authors take account of uncertainty in the
distance data. We incorporate a statistical model of the uncertainty through
corruption or noise in the pairwise distances and investigate the problem of
estimating the HC as unknown parameters from measurements. Specifically, we
focus on single linkage hierarchical clustering (SLHC) and study its geometry.
We prove that under fairly reasonable conditions on the probability
distribution governing measurements, SLHC is equivalent to maximum partial
profile likelihood estimation (MPPLE) with some of the information contained in
the data ignored. At the same time, we show that direct evaluation of SLHC on
maximum likelihood estimation (MLE) of pairwise distances yields a consistent
estimator. Consequently, a full MLE is expected to perform better than SLHC in
getting the correct HC results for the ground truth metric.
