= Solution
With $D=(y_1,y_2,t)$ and $\nu=(\Delta m,c,A,\tau)$,
$$
p(\theta\mid D)\propto
p(y_1,y_2\mid t,\theta)
\exp\left(-\frac{\Delta t^2}{2\sigma_{\mathrm{prior}}^2}\right)p(\nu).
$$
A <Random-walk Metropolis algorithm> proposes $\theta'=\theta+Z$ from a symmetric multivariate normal increment, conveniently using logarithmic coordinates for $A,\tau>0$, and accepts with probability
$$
\min\left\{1,
\frac{p(\theta'\mid D)}{p(\theta\mid D)}
\right\}.
$$
After burn-in, retain a suitably long chain and assess convergence and <effective sample size of a Markov chain>.
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