A numerical example

The numerical feasibility of a solution of Eq. (7) has been tested for a density estimation problem with one-dimensional $x$ and $y$ variables and the choice $\phi(x,y)$ = $\ln P(x,y)$, for which ${\bf P}^\prime$ = ${\bf P}$. Choosing a true field $\phi_{\rm true}$ (Fig. 1b), 50 data points have been sampled (Fig. 2a) from the corresponding true conditional likelihood $P_{\rm true}$ = $\exp[L_{\rm true}]$ (Fig. 1a) with uniform $p(x)$. Including a Gaussian prior term for $\phi$ with zero mean and inverse covariance $-\frac{\partial^2}{\partial x^2}$ $-\frac{1}{2}\frac{\partial^2}{\partial y^2}$ Eq. (7) has been solved on a mesh with 10 points in $x$-direction and 15 points in $y$-direction by standard iteration methods. (For technical details see [4].) The reconstructed conditional likelihood $P$ is shown in Fig. 2b, true and reconstructed regression functions are compared in Fig. 3.