Bayes-based non-Bayesian inference on finite populations from non-representative samples: A unified approach: *Based on S. N. Roy Memorial Lecture in the symposium.

Classical inference on finite populations is based on probability samples drawn from the target population with predefined selection probabilities. The target population parameters are either descriptive statistics such as totals or proportions, or parameters of statistical models assumed to hold for the population values. Familiar examples of estimation of models include the estimation of income elasticities from household surveys, comparisons of pupils’ achievements from educational surveys, and the study of causal relationships between risk factors and disease prevalence from health surveys. Models are also routinely used to account for measurement errors and for small area estimation with small samples in at least some of the areas.In practice, the samples selected are often not representative of the finite populations from which they are taken. This is so because the sample selection probabilities might be correlated with the model target values, known as informative sampling, or that observations are missing because of not missing at random (NMAR) nonresponse. Sometimes, the samples are subject to mode effects resulting from the use of different answering methods for different sample units, and in more extreme cases, the samples are drawn from sub-populations such as in web-based surveys or in observational studies.The focus of this article is to discuss and illustrate how all these diverse scenarios can be handled in a unified manner by use of Bayes theorem. The use of Bayes theorem allows relating the model holding for the observed data with the model holding for the missing data and the model operating in the target population. I discuss different estimation procedures and review articles that illustrate their performance