Analogy as Strategy Transfer

This note records the analogy framing that is present in the abstract-strategy discussion but not developed in the later long-form Feb 25 document.

Related notes:

In this framing, analogy means enforcing or transferring a particular strategy $\bar{z}$ for a target instance $x$ based on its success on a related source instance $\bar{x}$ .

The point is not that the model lacks the ability to use $\bar{z}$ . The point is that

$\pi_{z \mid x}(\cdot \mid x)$ may assign $\bar{z}$ low probability on the target task,
while $\pi_{z \mid x}(\cdot \mid \bar{x})$ assigns $\bar{z}$ high probability on a related source task,
and the model already knows how to realize that strategy successfully when it is activated.

Operational picture

To reason by analogy for a target task $x$ :

Find a related source problem $\bar{x}$ on which the model succeeds.
Obtain a successful source strategy $\bar{z}$ for that source problem.
Apply or condition on $\bar{z}$ when attempting to solve the target problem $x$ .

Key idea

Analogy is not just retrieving a similar example. It is transferring a high-level method or motif that succeeded on the source problem and trying to instantiate it on the target problem.

Examples

Electrostatics to gravitation

Source problem: compute the electric field inside a uniformly charged spherical shell.

Source strategy: exploit symmetry through Gauss's law.

Target problem: compute the gravitational force inside a hollow spherical shell of mass.

Transfer: recognize the common inverse-square-law structure and transfer the symmetry argument.

Brownian motion to option pricing

Source problem: model the random motion of particles in a fluid.

Source strategy: Brownian motion / diffusion equation / heat-equation reasoning.

Target problem: model the evolution of an option price.

Transfer: recognize the shared stochastic-diffusion structure and transfer the PDE-based strategy.

Why this note matters

This gives a concrete semantic meaning to latent strategy variables. If a strategy representation is real and useful, one natural test is whether it can be transferred across superficially different problems that share the same high-level method.