  There is a wide variety of models in which the dimension of the parameter
space is unknown. For example, in factor analysis the number of latent factors
is typically not known and has to be inferred from the observed data. Although
classical shrinkage priors are useful in these contexts, increasing shrinkage
priors can provide a more effective option, which progressively penalizes
expansions with growing complexity. In this article we propose a novel
increasing shrinkage prior, named the cumulative shrinkage process, for the
parameters controlling the dimension in over-complete formulations. Our
construction has broad applicability, simple interpretation, and is based on a
sequence of spike and slab distributions which assign increasing mass to the
spike as model complexity grows. Using factor analysis as an illustrative
example, we show that this formulation has theoretical and practical advantages
over current competitors, including an improved ability to recover the model
dimension. An adaptive Markov chain Monte Carlo algorithm is proposed, and the
methods are evaluated in simulation studies and applied to personality traits
data.
