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Assigning weights to a large pool of objects is a fundamental task in a widevariety of applications. In this article, we introduce the concept ofstructured high-dimensional probability simplexes, in which most components arezero or near zero and the remaining ones are close to each other. Suchstructure is well motivated by (i) high-dimensional weights that are common inmodern applications, and (ii) ubiquitous examples in which equal weights --despite their simplicity -- often achieve favorable or even state-of-the-artpredictive performance. This particular structure, however, presents uniquechallenges partly because, unlike high-dimensional linear regression, theparameter space is a simplex and pattern switching between partial constancyand sparsity is unknown. To address these challenges, we propose a new class ofdouble spike Dirichlet priors to shrink a probability simplex to one with thedesired structure. When applied to ensemble learning, such priors lead to aBayesian method for structured high-dimensional ensembles that is useful forforecast combination and improving random forests, while enabling uncertaintyquantification. We design efficient Markov chain Monte Carlo algorithms forimplementation. Posterior contraction rates are established to study largesample behaviors of the posterior distribution. We demonstrate the wideapplicability and competitive performance of the proposed methods throughsimulations and two real data applications using the European Central BankSurvey of Professional Forecasters data set and a data set from the UC IrvineMachine Learning Repository (UCI).

Double spike Dirichlet priors for structured weighting