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We propose a novel Bayesian model for estimating the rank of the coefficient matrix, which obviates the need for post-processing steps and allows for uncertainty quantification. Our method employs a mixture prior on the regression coefficient matrix along with a global-local shrinkage prior on its low-rank decomposition. Then, we rely on the Signal Adaptive Variable Selector to perform sparsification and define two novel tools: the Posterior Inclusion Probability uncertainty index and the Relevance Index. The validity of the method is assessed in a simulation study, and then its advantages and usefulness are shown in real-data applications on the chemical composition of tobacco and on the photometry of galaxies.<\/jats:p>","DOI":"10.1007\/s11222-025-10629-3","type":"journal-article","created":{"date-parts":[[2025,6,4]],"date-time":"2025-06-04T03:52:45Z","timestamp":1749009165000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Uncertainty Quantification in Bayesian Reduced-Rank Sparse Regressions"],"prefix":"10.1007","volume":"35","author":[{"given":"Maria F.","family":"Pintado","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matteo","family":"Iacopini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luca","family":"Rossini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander Y.","family":"Shestopaloff","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,4]]},"reference":[{"key":"10629_CR1","volume-title":"Statistical Theory in Research","author":"RL Anderson","year":"1952","unstructured":"Anderson, R.L., Bancroft, T.A.: Statistical Theory in Research. 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