  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. A precise quantification
characterizing the role of the dimension as well as the dependence is provided.
This surprising behavior has serious implications towards 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 at parts of the true function of interest. Without further
modification, we show that truncated normal priors are not suitable for
modeling at regions and propose a novel alternative strategy based on shrinking
the coordinates using a multiplicative scale parameter. The proposed shrinkage
prior is empirically shown to guard against the mass shifting phenomenon while
retaining computational efficiency.
