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Women infected by the Human papilloma virus are at an increased risk to develop cervical intraepithalial neoplasia lesions (CIN). CIN are classified into three grades of increasing severity (CIN-1, CIN-2, and CIN-3) and can eventually develop into cervical cancer. The main purpose of screening is detecting CIN-2 and CIN-3 cases which are usually surgically removed. Screening data from the POBASCAM trial involving 1,454 HPV-positive women is analyzed with two objectives: estimate (a) the transition time from HPV diagnosis to CIN-3; and (b) the transition time from CIN-2 to CIN-3. The screening data have two key characteristics. First, the CIN state is monitored in an interval-censored sequence of screening times. Second, a woman's progression to CIN-3 is only observed, if the woman progresses to, both, CIN-2 and from CIN-2 to CIN-3 in the same screening interval. We propose a Bayesian accelerated failure time model for the two transition times in this three-state model. To deal with the unusual censoring structure of the screening data, we develop a Metropolis-within-Gibbs algorithm with data augmentation from the truncated transition time distributions., Comment: 22 pages (Manuscript), 34 pages (Supplemental Material)