In Gelman & Hill (2007)'s book (Data Analysis Using Regression and Multilevel/Hierarchical Models), the authors claim that including redundant mean parameters can help speed up MCMC.
The given example is a non-nested model of "flight simulator" (Eq 13.9):
$$ \begin{align} y_i &\sim N(\mu + \gamma_{j[i]} + \delta_{k[i]}, \sigma^2_y) \\ \gamma_j &\sim N(0, \sigma^2_\gamma) \\ \delta_k &\sim N(0, \sigma^2_\delta) \end{align} $$
They recommend a reparameterization, adding the mean parameters $\mu_\gamma$ and $\mu_\delta$ as follows:
$$ \begin{align} \gamma_j \sim N(\mu_\gamma, \sigma^2_\gamma) \\ \delta_k \sim N(\mu_\delta, \sigma^2_\delta) \end{align} $$
The only justification offered is that (p. 420):
It is possible for the simulations to get stuck in a configuration where the entire vector $\gamma$ (or $\delta$) is far from zero (even though they are assigned a distribution with mean 0). Ultimately, the simulations will converge to the correct distribution, but we do not want to have to wait.
How do the redundant mean parameters help with this problem?
It seems to me that the non-nested model is slow mainly because of $\gamma$ and $\delta$ are negatively correlated. (Indeed, if one goes up, the other has to go down, given that their sum is "fixed" by the data). Do the redundant mean parameters help with reducing the correlation between $\gamma$ and $\delta$, or something else entirely?
