  Spectral clustering views the similarity matrix as a weighted graph, and
partitions the data by minimizing a graph-cut loss. Since it minimizes the
across-cluster similarity, there is no need to model the distribution within
each cluster. As a result, one reduces the chance of model misspecification,
which is often a risk in mixture model-based clustering. Nevertheless, compared
to the latter, spectral clustering has no direct ways of quantifying the
clustering uncertainty (such as the assignment probability), or allowing easy
model extensions for complicated data applications. To fill this gap, we
propose the Bayesian forest model as a generative graphical model for spectral
clustering. This is motivated by our discovery that the posterior connecting
matrix in a forest model has almost the same leading eigenvectors, as the ones
used by normalized spectral clustering. To induce a distribution for the
forest, we develop a ``forest process'' as a graph extension to the urn
process, while we carefully characterize the differences in the partition
probability. We derive a simple Markov chain Monte Carlo algorithm for
posterior estimation, and demonstrate superior performance compared to existing
algorithms. We illustrate several model-based extensions useful for data
applications, including high-dimensional and multi-view clustering for images.
