Fully Bayesian Estimation Under Informative Sampling

TL;DR

This paper introduces a fully Bayesian method for estimation under informative sampling, improving uncertainty quantification and avoiding weight normalization issues common in traditional pseudo-likelihood approaches.

Contribution

It proposes a novel Bayesian adjustment method that performs weight smoothing and joint parameter estimation, with theoretical guarantees under certain sampling design conditions.

Findings

Better posterior uncertainty estimates than pseudo-likelihood methods

No need for weight normalization calibration

Effective application to health survey data

Abstract

Bayesian estimation is increasingly popular for performing model based inference to support policymaking. These data are often collected from surveys under informative sampling designs where subject inclusion probabilities are designed to be correlated with the response variable of interest. Sampling weights constructed from marginal inclusion probabilities are typically used to form an exponentiated pseudo likelihood that adjusts the population likelihood for estimation on the sample due to ease-of-estimation. We propose an alternative adjustment based on a Bayes rule construction that simultaneously performs weight smoothing and estimates the population model parameters in a fully Bayesian construction. We formulate conditions on known marginal and pairwise inclusion probabilities that define a class of sampling designs where $L_{1}$ consistency of the joint posterior is guaranteed. We compare performances between the two approaches on synthetic data, which reveals that our fully Bayesian approach better estimates posterior uncertainty without a requirement to calibrate the normalization of the sampling weights. We demonstrate our method on an application concerning the National Health and Nutrition Examination Survey exploring the relationship between caffeine consumption and systolic blood pressure.