A flexible class of prior distributions is proposed, for the covariance matrix of a multivariate normal distribution, yielding much more general hierarchical and empirical Bayes smoothing and inference, when compared with a conjugate analysis involving an inverted Wishart distribution. A likelihood approximation is obtained for the matrix logarithm of the covariance matrix, via Bellman's iterative solution to a Volterra integral equation. Exact and approximate Bayesian, empirical and hierarchical Bayesian estimation and finite sample inference techniques are developed. Some risk and asymptotic frequency properties are investigated. A subset of the Project Talent American High School data is analyzed. Applications and extensions to multivariate analysis, including a generalized linear model for covariance matrices, are indicated.