  Assigning weights to a large pool of objects is a fundamental task in a wide
variety of applications. In this article, we introduce the concept of
structured high-dimensional probability simplexes, in which most components are
zero or near zero and the remaining ones are close to each other. Such
structure is well motivated by (i) high-dimensional weights that are common in
modern applications, and (ii) ubiquitous examples in which equal weights --
despite their simplicity -- often achieve favorable or even state-of-the-art
predictive performance. This particular structure, however, presents unique
challenges partly because, unlike high-dimensional linear regression, the
parameter space is a simplex and pattern switching between partial constancy
and sparsity is unknown. To address these challenges, we propose a new class of
double spike Dirichlet priors to shrink a probability simplex to one with the
desired structure. When applied to ensemble learning, such priors lead to a
Bayesian method for structured high-dimensional ensembles that is useful for
forecast combination and improving random forests, while enabling uncertainty
quantification. We design efficient Markov chain Monte Carlo algorithms for
implementation. Posterior contraction rates are established to study large
sample behaviors of the posterior distribution. We demonstrate the wide
applicability and competitive performance of the proposed methods through
simulations and two real data applications using the European Central Bank
Survey of Professional Forecasters data set and a data set from the UC Irvine
Machine Learning Repository (UCI).
