View Submission - CMStatistics

B1168
**Title: **Projection-type priors for structured orthogonal matrices
**Authors: **Michael Jauch - Florida State University (United States) **[presenting]**

**Abstract: **A family of projection-type priors for structured (sparse or smooth) orthogonal matrices is introduced that leads to tractable posterior inference for a wide variety of statistical models built from matrix decompositions. Let $\bm{Z}=(z_{i,j})$ be a $p \times k$ matrix with i.i.d. real entries having mean zero, unit variance, and finite fourth moments; let $\bm{\Omega}$ be a $p \times p$ correlation matrix, and set $\bm{X} = \bm{\Omega}^{1/2}\bm{Z}.$ The proposed prior is the distribution of $\bm{Q}_X,$ the projection of $\bm{X}$ onto the Stiefel manifold obtained via the polar decomposition. It turns out that features of the distribution of $\bm{X}$ are inherited by the distribution of $\bm{Q}_X.$ Most significantly, the distribution of finitely many elements of $\sqrt{p}\bm{Q}_X$ converges weakly to the distribution of the corresponding elements of $\bm{X}$ as $p,k \to \infty$ with $k/p \to 0.$ Thus, if we want the prior distribution for a tall and skinny orthogonal matrix parameter to reflect structural assumptions, we can build these features into the distribution of $\bm{X}.$ We illustrate the proposed prior through applications to real data and make connections to recent literature on projection-type priors as well as high-dimensional probability and random matrix theory.