Non-representative surveys are commonly used and widely available but suffer from selection bias that generally cannot be entirely eliminated using weighting techniques. Instead, we propose a Bayesian method to synthesize longitudinal representative unbiased surveys with non-representative biased surveys by estimating the degree of selection bias over time. We show using a simulation study that synthesizing biased and unbiased surveys together out-performs using the unbiased surveys alone, even if the selection bias may evolve in a complex manner over time. Using COVID-19 vaccination data, we are able to synthesize two large sample biased surveys with an unbiased survey to reduce uncertainty in now-casting and inference estimates while simultaneously retaining the empirical credible interval coverage. Ultimately, we are able to conceptually obtain the properties of a large sample unbiased survey if the assumed unbiased survey, used to anchor the estimates, is unbiased for all time-points.

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