Upper panels: difference between the conventional and the Bayesian mean posterior estimates of p and ψ as a function of the conventional estimate. Lower panels: Bayesian mean posterior estimates of σ_p and σ_ψ as a function of the conventional estimate. The dashed red lines show where the two methods give the same result. Each plot shows the density of points in log-scale for the Planck data at 1° resolution. The dotted line in the lower right plot shows the value for pure noise. The colour scale shows the pixel density on a log₁₀ scale.