The Value of Informative Priors in Bayesian Inference with Sparse Data

Abstract

Informative priors that reflect the structure of the model can improve estimation when data are sparse, while “standard,” noninformative priors can have unintended consequences. First, the authors discuss selecting informative priors for variances and introduce a conjugate prior for covariance matrices. The proposed prior is more flexible than the inverse Wishart without increasing computations. Second, the authors investigate the impact of priors for the covariance of parameter heterogeneity when the predictor variables are qualitative. Estimates of the omitted effects are spurious with the standard prior. The authors propose an effects prior that treats all effects symmetrically. Third, the authors consider willingness to pay. These ratio estimators magnify uncertainty in the price coefficients and can give unreasonable values for price-insensitive consumers. The authors show that estimation of willingness to pay can be greatly improved by restricting the parameters without distorting them. In all three cases, the standard, noninformative priors perform well for nonsparse data. However, for sparse data, the proposed prior distributions better match the structures of the marketing research problems and improve inferences.