  Existing techniques to reconstruct tree models of progression for
accumulative processes, such as cancer, seek to estimate causation by combining
correlation and a frequentist notion of temporal priority. In this paper, we
define a novel theoretical framework called CAPRESE (CAncer PRogression
Extraction with Single Edges) to reconstruct such models based on the notion of
probabilistic causation defined by Suppes. We consider a general reconstruction
setting complicated by the presence of noise in the data due to biological
variation, as well as experimental or measurement errors. To improve tolerance
to noise we define and use a shrinkage-like estimator. We prove the correctness
of our algorithm by showing asymptotic convergence to the correct tree under
mild constraints on the level of noise. Moreover, on synthetic data, we show
that our approach outperforms the state-of-the-art, that it is efficient even
with a relatively small number of samples and that its performance quickly
converges to its asymptote as the number of samples increases. For real cancer
datasets obtained with different technologies, we highlight biologically
significant differences in the progressions inferred with respect to other
competing techniques and we also show how to validate conjectured biological
relations with progression models.
