On truncated multivariate normal priors in constrained parameter spaces
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We show that any lower-dimensional marginal density obtained from truncating dependent multivariate normal distributions to the positive orthant exhibits a mass-shifting phenomenon. Despite the truncated multivariate normal having a mode at the origin, the marginal density assigns increasingly small mass near the origin as the dimension increases. The phenomenon is accentuated as the correlation between the random variables increases; in particular we show that the univariate marginal assigns vanishingly small mass near zero as the dimension increases provided the correlation between any two variables is greater than 0.8. En-route, we develop precise comparison inequalities to estimate the probability near the origin under the marginal distribution of the truncated multivariate normal. This surprising behavior has serious repercussions in the context of 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 modifications, we show that commonly used tMVN 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 guards against the mass shifting phenomenon while retaining computational efficiency.
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