lin-0146
11.2 The BRIDGE learning sketch
For a regime \(e\) with \(P^{(e)}\ll P^{(0)}\), define the Radon–Nikodym ratio
\[ \rho _e(z)=\frac{dP^{(e)}}{dP^{(0)}}(z), \qquad \mathbb E_{P^{(e)}}f = \mathbb E_{P^{(0)}}[\rho _ef]. \]
This identity supplies a statistical interface. It does not identify a causal effect. A differentiable density-ratio or transport model converts the finite regime change into a response field \(v_e\).
The sketch contains:
observational and regime-specific probability objects;
measure-change or transport arrows from the reference regime;
coordinate observations of every response field;
the visible distribution \(\mathcal D_V=\operatorname {span}\{ v_1,\ldots ,v_q\} \); and
a downstream graph or equivalence-class learner restricted by the geometric mask.
