Diagnostics for insufficiencies of posterior calculations in Bayesian signal inference

We present an error-diagnostic validation method for posterior distributions in Bayesian signal inference, an advancement of a previous work. It transfers deviations from the correct posterior into characteristic deviations from a uniform distribution of a quantity constructed for this purpose. We show that this method is able to reveal and discriminate several kinds of numerical and approximation errors, as well as their impact on the posterior distribution. For this we present four typical analytical examples of posteriors with incorrect variance, skewness, position of the maximum, or normalization. We show further how this test can be applied to multidimensional signals.

Figure 1

Influence of an insufficient posterior on the DIP distribution $P\left(x\right)$. The (a) upper [(b) lower] panel shows the effect of calculating $P\left(x\right)$ from a posterior with wrong variance (skewness) as described by Eq. (5) [Eq. (7)].

Figure 2

Influence of an insufficient posterior on the DIP distribution $P\left(x\right)$. The panel is showing the effect of calculating $P\left(x\right)$ from a posterior with wrong maximum position as described by Eq. (10).

Figure 3

Distributions of the numerically calculated $x$ values. The (a) left [(b) middle, (c) right] histogram shows the unnormalized distribution of 500 (500, 50000) $x$ values within eight (eight, fifty) bins as calculated from the posterior with correct (wrong, wrong) maximum position. The standard deviation interval (1$\sigma$) around the expectation value as calculated from Poissonian statistics is also shown.

Figure 4

DIP test in two dimensions with related contour plot, according to Eq. (21). $\varphi$ denotes the azimuth (of the $t_x$-$t_y$ plane), where the two-dimensional posterior $P^{\epsilon }\left(s\right|d)$ intersects with a plane, given by $t_y=t_x\tan \left(\varphi \right)$. The DIP distribution of the resulting one-dimensional posterior is denoted by $P_{\varphi }\left(x\right)$. The labeling of the color bar refers to the $\varphi$ coordinate.

Figure 5

Scheme of the Bayesian hierarchical model.

Figure 6

Distributions of the numerically calculated $x$ values. The (a) left [(b) middle, (c) right] histogram shows the unnormalized distribution of 500 $x$ values within eight bins as calculated from the $m$- ($m$-, ${\sigma }^2$-) marginalized posterior with correct (wrong, wrong) $\alpha$ parameter. The standard deviation interval (1$\sigma$) around the expectation value as calculated from Poissonian statistics is also shown.