GOALS AS REVERSE-TIME ACTIVE-INFERENCE AGENTS
A Schrödinger-Bridge Formulation for Bidirectional Control
Thomas Orr Anderson · Preprint · Published July 13, 2025 · Version v1
DOI: 10.5281/zenodo.1609681
Overview
This paper shows that goal-directed Active Inference and the continuous-time Schrödinger bridge solve the same optimization: steer a noisy system from an initial belief distribution to a terminal goal distribution while deviating as little as possible from baseline dynamics. The bridge is governed by a single scalar potential whose gradient splits into two complementary drifts—one pushes the present toward the goal, and the other propagates the goal backward through time. Under this construction, the goal distribution itself can be interpreted as a reverse-time Active-Inference agent.
A key structural result is that Markov-blanket factorization survives in both temporal directions when the time-directed potentials factorize into internal–active and sensory–external parts, with diffusion block-diagonal in the same partition. Within the standing assumptions (constant diffusion and smooth strictly positive endpoints), the framework yields a practical consequence: any continuous-time Schrödinger-bridge solver can be used as a turnkey Active-Inference controller.
Abstract
Active-Inference and the continuous-time Schrödinger bridge solve the same optimization: steer a noisy system from an initial probability cloud to a target one while straying as little as possible from its natural dynamics. The bridge is governed by a single scalar potential whose gradient splits into two complementary drifts—one pushes the present toward the goal, the other lets the goal propagate backward through time. Because both drifts minimize the same free-energy functional, the target distribution itself behaves as a reverse-time Active-Inference agent. If the potential factorizes into an internal–active part and a sensory–external part, the familiar Markov-blanket partition remains intact in both temporal directions.
At a glance
• Schrödinger-bridge reformulation of goal-directed Active Inference as entropic optimal transport.
• Closed-form bridge potential coupling a forward diffusion to its adjoint.
• Complementary forward and reverse drift corrections whose gap quantifies task difficulty.
• Markov-blanket symmetry criterion for bidirectional control (same partition in both directions).
• Goals-as-agents equivalence: the terminal density qualifies as an Active-Inference agent in reverse time.
• Implementation takeaway: continuous-time Schrödinger-bridge solvers double as Active-Inference engines (within scope).
Downloads and citation
Zenodo record (archival DOI): 10.5281/zenodo.16096819
Suggested citation
Anderson, Thomas Orr. (2025). Goals as Reverse-Time Active-Inference Agents: A Schrödinger-Bridge Formulation for Bidirectional Control. Zenodo. https://doi.org/10.5281/zenodo.16096819
Rights
Copyright © 2025 Thomas Orr Anderson
Keywords
active inference, Schrödinger bridge, entropic optimal transport, stochastic control, Markov blanket, time symmetry, teleology, KL divergence