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The false positive rate is $\mathbb{P}(\text{positive result}\mid \text{truly negative class})$. In screening, it measures erroneous positives among genuinely disease-free individuals.

Let $ϵ$ be the false positive rate for a genuinely cancer-free person, encompassing the harmless-tumour misclassification, and assume screening errors are conditionally independent given the latent states. A pre-clinical case tests positive with probability one; a clinical case is separately recognized. Thus a negative screen identifies latent state $1$, whereas a positive pre-clinical screen can arise from state $1$ or $2$. This is a Hidden Markov model with a screening misclassification model.

Conditional on the negative baseline screen, the state at year $0$ is $1$. The negative screen at year $2$ requires staying in state $1$ for two years and then testing negative. From that state, the positive screen at year $4$ can arise either with probability $ϵ$ while still disease-free or through pre-clinical disease. Consequently the conditional likelihood isIf the study instead specifies an initially disease-free patient and includes the baseline screening result in the likelihood function, its probability contributes a further factor $1-ϵ$, givingThe formulas use the positive emission for a pre-clinical screen, with no clinical diagnosis in this record. A persistent harmless-tumour subtype producing correlated screening errors would need an additional latent-state or emission specification; its likelihood is not determined by the single parameter $ϵ$.

In a screening misclassification model, true disease states are latent and screen results are emissions. A cancer-free state may emit a positive result with a false positive rate, while a perfectly sensitive pre-clinical state always emits a positive result. Conditional independence of emissions given latent states is an additional model assumption, not a consequence of a marginal false-positive rate.