Three Geometries of Agency — Crooks, Wasserstein, and Schrödinger Bridges
Overview
The IEI module treats intelligent agency as transport of probability mass and distinguishes three geometries that must not be collapsed: (1) Fisher–Rao / Crooks thermodynamic length (near-equilibrium, dissipation bounded by $\mathcal{L}^2/\tau$); (2) Wasserstein (minimum ground-cost mass transport); (3) Schrödinger bridge (maximum-entropy interpolation; discrete MaxEnt coupling via Sinkhorn). Machine learning has made Schrödinger bridges practical generative tools (Sinkhorn bridges with statistical rates; LightSB-M; SB flow for unpaired translation), usually without asking which geometry explains an agent’s belief updates under metabolic or information cost. This project treats the three geometries as competing scientific explanations of agency, not as interchangeable samplers.
Prerequisite
L172 Information, Energy and Intelligence (IEI), or equivalent preparation in information theory, maximum entropy, and information geometry.
FAQs
- What are the prerequisites?
L172 Information, Energy and Intelligence (IEI), or equivalent preparation in information theory, maximum entropy, and information geometry.
- What will I learn in this Project?
You will study three geometries of changing probability: Fisher–Rao / Crooks thermodynamic length, Wasserstein optimal transport, and Schrödinger bridges (entropic OT, Sinkhorn). You will fit competing interpolations to logged belief trajectories and perform model comparison under interventions on information cost.
- What is the objective of the project?
(1) Log belief or latent states of a tractable agent (Bayesian agent, small controlled LLM agent, or both) across a multi-step task. (2) Fit Fisher–Rao, Wasserstein, and Schrödinger-bridge interpolations between successive belief states; score by likelihood / predictive quality and decision-relevance. (3) Intervene on cost (temperature, token budget, compute, noise) and test which geometry’s predictions move correctly. (4) State a comparative, falsifiable thesis claim — e.g. under bandwidth stress SB fits; under near-equilibrium fine-tuning Fisher–Rao fits — and report where all three fail. Deliverables: fitting pipeline, intervention study, and thesis chapter.
- How does this fit into the bigger picture?
The inaccessible game attempts to give mathematical teath to the notion of information topography. We need more than metaphors for “information flow.” Distinguishing geometries clarifies what kind of cost shapes communication structure in engineered agents and connects IEI’s closing lectures to current generative-modelling practice (seel Welling et al. on generative AI and stochastic thermodynamics).
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