We study the problem of identifying shape classes in point clouds. These clouds contain sampled points along contours and are corrupted by clutter and observation noise. Taking an analysis-by-synthesis approach, we simulate high-probability configurations of sampled contours using models learned from training data to evaluate the given test data. To facilitate simulations, we develop statistical models for sources of (nuisance) variability: 1) shape variations within classes, 2) variability in sampling continuous curves, 3) pose and scale variability, 4) observation noise, and 5) points introduced by clutter. The variability in sampling closed curves into finite points is represented by positive diffeomorphisms of a unit circle. We derive probability models on these functions using their square-root forms and the Fisher-Rao metric. Using a Monte Carlo approach, we simulate configurations from a joint prior on the shape-sample space and compare them to the data using a likelihood function. Average likelihoods of simulated configurations lead to estimates of posterior probabilities of different classes and, hence, Bayesian classification.