We propose BayesPIM, a Bayesian prevalence-incidence mixture model for estimating time- and covariate-dependent disease incidence from screening and surveillance data. The method is particularly suited to settings where some individuals may have the disease at baseline, baseline tests may be missing or incomplete, and the screening test has imperfect sensitivity. Building on the existing PIMixture framework, which assumes perfect sensitivity, BayesPIM accommodates uncertain test accuracy by incorporating informative priors. By including covariates, the model can quantify heterogeneity in disease risk, thereby informing personalized screening strategies. We motivate the model using data from high-risk familial colorectal cancer (CRC) surveillance through colonoscopy, where adenomas - precursors of CRC - may already be present at baseline and remain undetected due to imperfect test sensitivity. We show that conditioning incidence and prevalence estimates on covariates explains substantial heterogeneity in adenoma risk. Using a Metropolis-within-Gibbs sampler and data augmentation, BayesPIM robustly recovers incidence times while handling latent prevalence. Informative priors on the test sensitivity stabilize estimation and mitigate non-convergence issues. Model fit can be assessed using information criteria and validated against a non-parametric estimator. In this way, BayesPIM enhances estimation accuracy and supports the development of more effective, patient-centered screening policies.

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