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Bayesian Integration of Probability and Non-Probability Samples

Author

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  • Qi Chen

    (Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA)

  • Balgobin Nandram

    (Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA)

Abstract

Non-probability samples (NPS) are increasingly used in practice because they are relatively inexpensive and often contain the study outcome variable. However, NPS may suffer from selection bias and usually do not provide survey weights, making finite-population inference difficult. Probability samples (PS), on the other hand, provide survey weights under known sampling designs but may not contain the study outcome variable. This paper develops a Bayesian approach for integrating probability and non-probability samples under a superpopulation model. The proposed method jointly models latent survey weights and missing probability-sample outcomes within a hierarchical Bayesian framework. The method is compared with a naive Bayesian bootstrap estimator and a weighted regression estimator. A BMI data application and simulation studies show that the proposed method reduces bias relative to the naive estimator while accounting for uncertainty in latent weights and missing outcomes.

Suggested Citation

  • Qi Chen & Balgobin Nandram, 2026. "Bayesian Integration of Probability and Non-Probability Samples," Stats, MDPI, vol. 9(4), pages 1-20, July.
    • Handle: RePEc:gam:jstats:v:9:y:2026:i:4:p:73-:d:1984112
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