lin-0079

5.7 Linearized copy and discard

The intervention fields above live on a parameter manifold. Copy and discard live on sample objects. To compare them, take a differentiable path of finite-state stochastic kernels

\[ K_\alpha :\mathbb R^X\longrightarrow \mathbb R^Y, \qquad K_0=K, \]

and define its signed derivative

\[ D=\left.\frac{d}{d\alpha }\right|_{\alpha =0}K_\alpha . \]

Proposition 5.11 Infinitesimal normalization

Every differentiable path of stochastic kernels satisfies

\[ !_Y\circ D=0. \]

Proof ▼

Differentiate the stochasticity identity \(!_Y\circ K_\alpha =!_X\).

If \(X=Y\), \(K_0=\operatorname {id}\), and the path preserves copying to first order, differentiation of

\[ \Delta _XK_\alpha = (K_\alpha \otimes K_\alpha )\Delta _X \]

gives the coderivation identity

\[ \Delta _XD = (D\otimes \operatorname {id} +\operatorname {id}\otimes D)\Delta _X. \]

Definition 5.12 Copy defect

The first-order copy defect is

\[ B_D = \Delta _XD - (D\otimes \operatorname {id} +\operatorname {id}\otimes D)\Delta _X. \]

Discard compatibility is automatic for a valid stochastic path. Copy compatibility is much stronger: in a Markov category, generic stochastic maps do not preserve copying. It should therefore be tested only where the application declares deterministic, copyable information.