Unrestricted variation

To get a first understanding of the approach (432) let us consider the extreme example of completely unrestricted $t$-variations. In that case the template function $t(x,y)$ itself represents the hyperparameter. (Such function hyperparameters or hyperfields are also discussed in Sect. 5.6.) Then, ${\bf t}^\prime = {\bf I}$ and Eq. (435) gives ${{\bf K}}(\phi -t)$ = $0$ (which for invertible ${{\bf K}}$ is solved uniquely by $t=\phi$), resulting according to Eq. (229) in

$$\Lambda_X = N_X .$$ (441)

The case of a completely free prior mean $t$ is therefore equivalent to a situation without prior. Indeed, for invertible ${\bf P}^\prime$, projection of Eq. (436) into the $x$-data space by ${\bf I}_D$ of Eq. (260) yields

$$P_D = {\bf\Lambda}_{X,D}^{-1} N ,$$ (442)

where ${\bf\Lambda}_{X,D}$ = ${\bf I}_D {\bf\Lambda}_{X}{\bf I}_D$ is invertible and $P_D = {\bf I}_D P$. Thus for $x_i$ for which $y_i$ are available

$$P(x_i,y_i) = \frac{N(x_i,y_i)}{N_X (x_i)}$$ (443)

is concentrated on the data points. Comparing this with solutions of Eq. (228), for $\theta$ = $t$ fixed, we see that adaptive means tend to lower the influence of prior terms.