Synthetic Priors

Abstract: Bayesian inference in generalized linear models requires a prior on the coefficient vector $β$. Practitioners naturally reason about response probabilities at specific covariate values, not about abstract log-odds parameters. We develop synthetic priors: informative Bayesian priors for GLMs grounded in Good's device of imaginary observations -- the principle that every conjugate prior is equivalent to a likelihood on pseudo-data from the same exponential family. The conditional means prior of Bedrick (1996) elicits independent Beta priors on the conditional mean response at $p$ expert-chosen design points; the induced prior on $β$ is a product of binomial likelihoods at synthetic data points. Combined with Pólya-Gamma data augmentation \citep{polson2013}, the posterior admits an exact conjugate Gibbs sampler -- no tuning, no Metropolis step -- by treating the augmented dataset as a standard logistic regression. We show that ridge regression and catalytic priors \citep{huang2020} are instances of Good's device, and identify prediction-powered inference \citep{angelopoulos2023ppi} as a structural analogue in the frequentist setting -- all three mediate a variance-bias tradeoff through a single informativeness parameter. We illustrate the approach on two benchmark problems: the Challenger O-ring data \citep{dalal1989}, where the BCJ prior provides a more moderate posterior predictive at the 31°F launch temperature; and a Phase~II atopic dermatitis dose-finding trial ($n = 300$), where the synthetic prior narrows 95\% credible intervals by 3-6\% and raises decision probabilities by up to 2 percentage points relative to a flat prior.