{\bf Left} The plot of temperature-spectral index $\beta$ showing the results from $\chi^2$-based fitting and from hierarchical Bayesian-fitting, for the 128$\times$128 box. The red dots are randomly drawn from the Monte Carlo chains, with each chain corresponding to one pixel on the map. The distributions of the Monte Carlo chains provide a natural estimate of the uncertainty of the fit. {\bf Right} The same plot for the 42 65$\times$65 boxes, after mosaiced together. Notice that in this plot, since we use the mean values of the Monte Carlo chains to create the maps, the red dots here are showing mean values shifted by the mosaicing process. This is why the scatter in the data seems smaller.