An empirical Bayes approach to smoothing in backcalculation of HIV infection rates

Backcalculation is a methodology to reconstruct the past human immunodeficiency virus (HIV) infection rates from the AIDS incidence data and incubation distribution by deconvolution. Smoothing has proved important in backcalculation, and a key question is how to choose the amount of smoothing. This paper proposes an empirical Bayes approach in which the smoothing parameter is estimated from the data. We introduce a family of priors that reflect the notion of closeness of neighboring infection rates. The variance parameter in the prior family plays the role of the smoothing parameter and is estimated by a method similar to the residual maximum likelihood in linear random effects model through an efficient EM (expectation/maximization) algorithm. A number of penalized likelihood functions that have been used in backcalculation have an empirical Bayes formulation. A bootstrap confidence interval for the infection rates is proposed. The methodology is illustrated with United States AIDS incidence data.