Context

Problem Statement

We want to understand when a disentangled latent factorization of the form

$$p(y \mid x) = \sum_z p(y \mid x, z) p(z \mid x)$$

is not only useful, but identifiable in the sense that latent values correspond to stable high-level strategies.

The central tension is:

• a latent can be predictive and useful without meaning the same thing across examples,
• but the research goal is for z to represent a strategy that is consistent enough to be interpreted, intervened on, and compared across inputs.

Why This Matters

The disentanglement notes already motivate a CVAE-style recovery procedure, but they also expose the main obstacle: multiple predictive factorizations can explain the same p(y|x). A model can therefore achieve good reconstruction and still fail to recover semantically stable strategy variables.

This makes identifiability a distinct question from optimization success. Learnability asks whether training can find a good factorization; identifiability asks whether the target factorization is uniquely recoverable, up to the expected symmetries.

Relevant Context

• methodology/cvae-disentangling
• methodology/concepts/token_weighted_excess_reconstruction
• methodology/cvae-objective-and-losses
• path:.tasks/doing/2026-03-15-theory-latent-space-exploration/spec.md
• ideation/concepts/abstract-strategy-vs-solution

Working Hypothesis

The useful notion may not be a single identifiability definition. It may instead be a hierarchy:

• task-wise identifiability for a fixed x,
• problem-wise identifiability across a task family,
• globally consistent identifiability across many x.

The theory thread should clarify which of these is attainable under the current modeling assumptions and which metrics can detect the difference.

Next

Next: Core questions