Example: Distribution functions

Instead in terms of the probability density function, one can formulate the prior in terms of its integral, the distribution function. The density $P$ is then recovered from the distribution function $\phi$ by differentiation,

$$P(\phi) = \prod_k^{d_y} \frac{\partial \phi}{\partial y_k} ... ... \bigotimes_k^{d_y} {\bf R}_k^{-1} \phi. = {\bf R}^{-1} \phi,$$ (203)

resulting in a non-diagonal ${\bf P}^\prime$. The inverse of the derivative operator ${\bf R}^{-1}$ is the integration operator ${\bf R}$ = $\bigotimes_k^{d_y} {\bf R}_k P$ with matrix elements

$${\bf R} (x,y;x^\prime,y^\prime) = \delta (x-x^\prime) \theta (y-y^\prime),$$ (204)

i.e.,

$${\bf R}_k (x,y;x^\prime,y^\prime) = \delta (x-x^\prime) \theta (y_k-y^\prime_k) \prod_{l\ne k}\delta (y_l-y_l^\prime).$$ (205)

Thus, (203) corresponds to the transformation of ($x$-conditioned) density functions $P$ in ($x$-conditioned) distribution functions $\phi$ = ${\bf R} P$, i.e., $\phi(x,y)$ = $\int_{-\infty}^y P(x,y^\prime ) dy^\prime$. Because ${\bf R^T} {{\bf K}} {\bf R}$ is positive (semi-)definite if ${{\bf K}}$ is, a specific prior which is Gaussian in the distribution function $\phi$ is also Gaussian in the density $P$. ${\bf P}^\prime$ becomes

$${\bf P}^\prime (x,y;x^\prime,y^\prime ) = \frac{\delta \le... ...ta (x-x^\prime ) \prod_k^{d_y}\delta^\prime (y_k-y_k^\prime ).$$ (206)

Here the derivative of the $\delta$-function is defined by formal partial integration

$$\int_{-\infty}^\infty \!dy^\prime\, f(y^\prime) \delta^\prim... ...) \delta(y^\prime -y)\vert _{-\infty}^\infty - f^\prime (y) .$$ (207)

Fixing $\phi(x,-\infty) = 0$ the variational derivative $\delta/( \delta \phi (x, -\infty) )$ is not needed. The normalization condition for $P$ becomes for the distribution function $\phi$ = ${\bf R} P$ the boundary condition $\phi(x,\infty) = 1$, $\forall x\in X$. The non-negativity condition for $P$ corresponds to the monotonicity condition $\phi(x,y) \ge \phi(x,y^\prime)$, $\forall y\ge y^\prime$, $\forall x\in X$ and to $\phi(x,-\infty) \ge 0$, $\forall x\in X$.