= Solution
Center the log-time predictor and the baseline log measurement near their sample means, so the intercept describes a typical predictor configuration. This reduces intercept-slope posterior correlation and can improve <Markov chain Monte Carlo> mixing. If exactly the original model is required, transform the intercept prior consistently when changing coordinates rather than silently assigning a different independent prior.
For weakly informed child effects, a <non-centered Gaussian random-effect parameterization> can also help:
$$
z_i\sim N(0,1),\qquad\boxed{\alpha_i=\delta+\tau z_i.}
$$
This retains the same conditional <prior distribution> for $\alpha_i$ but removes its prior-scale dependence from the sampled $z_i$. It can reduce strong dependence between the effects and $\tau$. A centered parameterization can be better when each child's effect is very accurately observed, so improvement is not automatic. Blocking strongly correlated coefficients is another option. Check the resulting chains with <Markov chain Monte Carlo convergence diagnostics> and <effective sample size of a Markov chain>, rather than judging convergence from the number of iterations alone.
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