= Solution
Use broad proper uniform priors for $\Delta t$, $\Delta m$, and $c$ over physically plausible ranges, and broad log-uniform priors for the positive scales $A_f$ and $\tau_f$. Then
$$
\boxed{p(\theta\mid y_1,y_2)\propto
p(y_1,y_2\mid\theta)\,p(\Delta t)p(\Delta m)p(c)p(A_f)p(\tau_f).}
$$
A <Random-walk Metropolis algorithm> can update $(\Delta t,\Delta m,c,\log A_f,\log\tau_f)$ with a multivariate Gaussian <proposal distribution>. Initialize several dispersed chains near plausible cross-correlation delays and near the marginal-likelihood optimum; reject proposals outside the prior bounds; discard warm-up while adapting only the proposal scale and covariance; then freeze the kernel and retain a long run. Evaluate trace plots, <autocorrelations>, acceptance rates, between-chain agreement, and the <effective sample size of a Markov chain>. Posterior predictive <quasar light curves> provide a model check.
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