According to Hansen, Madow and Tepping [J. Amer. Statist. Assoc. 78 (1983) 776–793], “Probability sampling designs and randomization inference are widely accepted as the standard approach in sample surveys.” In this article, reasons are advanced for the wide use of this design-based approach, particularly by federal agencies and other survey organizations conducting complex large scale surveys on topics related to public policy. Impact of Bayesian methods in survey sampling is also discussed in two different directions: nonparametric calibrated Bayesian inferences from large samples and hierarchical Bayes methods for small area estimation based on parametric models.
"Impact of Frequentist and Bayesian Methods on Survey Sampling Practice: A Selective Appraisal." Statist. Sci. 26 (2) 240 - 256, May 2011. https://doi.org/10.1214/10-STS346