Local masses and gauge theories

The Bayesian analog of a mass term in quantum field theory is a term proportional to the identity matrix ${\bf I}$ in the inverse prior covariance ${\bf K}_0$. Consider, for example,

$${\bf K}_0 = \theta^2 \, {\bf I} - \Delta ,$$ (464)

with $\theta$ real (so that $\theta^2\ge 0$) representing a mass parameter. For large masses $\phi$ tends to copy the template $t$ locally, and longer range effects of data points following from smoothness requirements become less important. A constant mass can be replaced by a mass function $\theta(x)$. This allows to adapt locally that interplay between ``template copying'' and smoothness related influence of training data. As hyperprior, one may use a smoothness constraint on the mass function $\theta(x)$, e.g.,

$$\frac{1}{2} (\phi-t,\; {\bf M}^2 (\phi-t) ) -\frac{1}{2} (\p... ...mbda\, (\theta,\; {\bf K}_\theta \theta) +\ln Z_\phi(\theta) ,$$ (465)

where ${\bf M}$ denotes the diagonal mass operator with diagonal elements $\theta(x)$.

Functional hyperparameters like $\theta(x)$ represent, in the language of physicists, additional fields entering the problem (see also Sect. 5.6). There are similarities for example to gauge fields in physics. In particular, a gauge theory-like formalism can be constructed by decomposing $\theta(x)$ = $\sum_i \theta_i(x)$, so that the inverse covariance

$${\bf K}_0 = \sum_i \left( {\bf M}^2_i - \partial_i^2\right)... ...eft( {\bf M}_i - \partial_i\right) = \sum_i D^\dagger_i D_i ,$$ (466)

can be expressed in terms of a ``covariant derivative'' $D_i$ = $\partial_i + \theta_i$. Next, one may choose as hyperprior for $\theta_i(x)$

$$\frac{1}{2}\left( \sum_i^{m_x} \big(\theta_i,\; -\Delta \,\t... ...} \theta_j \big) \right) = \frac{1}{4}\sum_{ij}^{m_x} F_{ij}^2$$ (467)

which can be expressed in terms of a ``field strength tensor'' (for Abelian fields),

$$F_{ij} = \partial_i \theta_j - \partial_j \theta_i ,$$ (468)

like, for example, the Maxwell tensor in quantum electrodynamics. (To relate this, as in electrodynamics, to a local $U(1)$ gauge symmetry $\phi\rightarrow e^{i\alpha}\phi$ one can consider complex functions $\phi$, with the restriction that their phase cannot be measured.) Notice, that, due to the interpretation of the prior as product $p(\phi\vert\theta) p(\theta)$, an additional $\theta$-dependent normalization term $\ln Z_\phi(\theta)$ enters the energy functional. Such a term is not present in quantum field theory, where one relates the prior functional directly to $p(\phi,\theta)$, so the norm is independent of $\phi$ and $\theta$.