A Mass-Shifting Phenomenon of Truncated Multivariate Normal Priors

Shuang Zhou, Pallavi Ray, Debdeep Pati, Anirban Bhattacharya

Research output: Contribution to journalArticlepeer-review

Abstract

We show that lower-dimensional marginal densities of dependent zero-mean normal distributions truncated to the positive orthant exhibit a mass-shifting phenomenon. Despite the truncated multivariate normal density having a mode at the origin, the marginal density assigns increasingly small mass near the origin as the dimension increases. The phenomenon accentuates with stronger correlation between the random variables. This surprising behavior has serious implications toward Bayesian constrained estimation and inference, where the prior, in addition to having a full support, is required to assign a substantial probability near the origin to capture flat parts of the true function of interest. A precise quantification of the mass-shifting phenomenon for both the prior and the posterior, characterizing the role of the dimension as well as the dependence, is provided under a variety of correlation structures. Without further modification, we show that truncated normal priors are not suitable for modeling flat regions and propose a novel alternative strategy based on shrinking the coordinates using a multiplicative scale parameter. The proposed shrinkage prior is shown to achieve optimal posterior contraction around true functions with potentially flat regions. Synthetic and real data studies demonstrate how the modification guards against the mass shifting phenomenon while retaining computational efficiency. Supplementary materials for this article are available online.

Original languageEnglish (US)
JournalJournal of the American Statistical Association
DOIs
StateAccepted/In press - 2022
Externally publishedYes

Keywords

  • Basis expansion
  • Bayesian
  • Comparison inequality
  • Constrained estimation
  • Gaussian process
  • Shrinkage

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Fingerprint

Dive into the research topics of 'A Mass-Shifting Phenomenon of Truncated Multivariate Normal Priors'. Together they form a unique fingerprint.

Cite this