 Symptom equivalence does not establish mechanistic identity

Let $Z$ index either of two candidate pathways and suppose $P(Y=1\mid Z=1)=P(Y=1\mid Z=2)=p$ for a measured symptom $Y$. For any mixing probability $\pi =P(Z=1)$, marginalization gives $P(Y=1)=\pi p+(1-\pi )p=p$. An independent sample with $k$ positive observations out of $n$ therefore has likelihood $p^k(1-p{)}^{n-k}$, independent of $\pi$. The symptom alone does not identify the mixture proportion or pathway. This is an identifiability counterexample, not an empirical claim that real clinical pathways necessarily have identical symptom distributions. Additional indicators or interventions may distinguish them.