Bayesian analysis requires prior distributions over parameters. But practitioners think in terms of outcomes, not parameters. A clinician can say “I expect about 30% of patients to respond at this dose” far more naturally than “the log-odds coefficient should be Normal(−0.8, 0.5).” The translation from beliefs about responses to beliefs about parameters is where most prior elicitation fails.
Franks et al. revive Good's device of imaginary observations: express prior beliefs as synthetic data points from the same model family, then let the likelihood of the synthetic data define the prior. The practitioner specifies independent Beta priors on the conditional mean response at a few chosen covariate values. The induced prior on regression coefficients is equivalent to the likelihood of a binomial experiment at those design points. No abstract parameter reasoning required.
The computational benefit is exact: combined with Pólya-Gamma data augmentation, the posterior admits a conjugate Gibbs sampler with no tuning and no Metropolis step. Ridge regression and catalytic priors turn out to be special cases of this device. Prediction-powered inference — using machine learning predictions as a prior data source — is the frequentist analogue.
The through-claim is about the direction of specification. Standard Bayesian practice asks: what do you believe about the parameters? Synthetic priors ask: what would you expect to observe? The second question is answerable because it's in the same space as the data. The prior isn't an opinion about abstraction — it's a fictional experiment whose results encode the opinion. The imaginary observation makes the prior empirical.