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Information geometry, Fisher information metric and gauge theory of belief space

Information geometry treats the parameters of a probability distribution as points on a statistical manifold, which carries a metric called the Fisher information metric.5 Friston, Wiese and Hobson tie this geometry to variational free energy, whose curvature is the Fisher metric.6 Gauge theory then describes how a system keeps its form against local disturbances.3 Together they are proposed as a formal account of belief in brains and other living systems.1

Earliest held
2016, Sengupta B, Tozzi A, Cooray…
Most discussed in
Sentience and the Origins of Consciousness…, 2020
In the library
68 passages in 4 works
Rewritten
2026-10-30
18501875190019251950197520002025Sengupta B, Tozzi A, Cooray GK, Douglas…, 20161Sengupta B, Tozzi A, Cooray GK, Douglas…, 20162Ramstead MJD, Badcock PB, Friston KJ, 20183Ramstead MJD, Badcock PB, Friston KJ, 20184Friston KJ, Wiese W, Hobson JA, 20205Friston KJ, Wiese W, Hobson JA, 20206Friston KJ, Wiese W, Hobson JA, 20207Fields C, Levin M, 20228
01

Statistical manifolds

Friston, Wiese and Hobson (2020) define a statistical manifold as a state space whose states are the parameters of a probability distribution.5 Each point on it corresponds to one distribution. For Gaussian densities, they give a two-dimensional example with mean and precision as coordinates. They state that any such manifold is equipped with a unique metric tensor, the Fisher information metric. The belief space or statistical manifold is here afforded by internal states. The authors cite earlier work for the metric.

02

Curvature as precision

Friston, Wiese and Hobson (2020) give the Fisher information metric several interpretations. First, it is the curvature of the variational free energy as one moves across the internal manifold.6 Second, this curvature equals the conditional precision, or the confidence placed in beliefs about external states.7 They add that precision plays a key role in predictive processing accounts of attentional selection and in interoceptive inference. They stress a seamless connection running from gauge theory, through information geometry, to variational inference.

03

Gauge theory of the brain

Sengupta and colleagues (2016) proposed variational free energy minimisation as a candidate neuronal gauge theory. Under it, every aspect of the brain, from neuronal activity to synaptic strengths, can be described as a process that minimises variational free energy.1 They name three ingredients: a system with symmetry, local forces, and gauge fields. In their reading, local forces are prediction errors that raise free energy and so activate gauge fields that explain them away.2 Ramstead, Badcock and Friston (2018) restate this: local forces break the symmetry, and a gauge field restores it by compensating for the perturbations.3

04

Nested scales

Sengupta and colleagues note that the gauge description can be applied at nested hierarchical levels. In predictive coding formulations, bottom-up messages are assumed to carry prediction error and top-down messages to carry predictions.2 Ramstead and colleagues argue for scale-free consistency. Free energy for a global Markov blanket, such as a species, must follow the same principles as its constituent blankets, such as phenotypes.4 Fields and Levin (2022) make a related point in terms of agency. Higher-level agents deform the energy landscape of actions for lower levels.8

SourcesEach quotation was checked word for word against the passage it opens.
  1. This principle says that every aspect of the brain, from neuronal activity to synaptic connection strengths, can be described as a process that minimises variational free energySengupta B, Tozzi A, Cooray GK, Douglas PK, Douglas PK…, 2016 · Towards a Neuronal Gauge Theory · open at passage 2
  2. From the perspective of a gauge theory, one can think of the local forces as prediction errors that increase variational free energy, thereby activating the gauge fields to explain away local forcesSengupta B, Tozzi A, Cooray GK, Douglas PK, Douglas PK…, 2016 · Towards a Neuronal Gauge Theory · open at passage 28
  3. This symmetry is broken by local forces, invoking a gauge field that restores the system's symmetry by compensating for these local perturbations.Ramstead MJD, Badcock PB, Friston KJ, 2018 · Answering Schrödinger's question: A free-energy formulation · open at passage 23
  4. the variational free energy associated with a global Markov blanket (e.g., a species) has to conform to the same principles as all of its constituent Markov blankets (e.g., phenotypes)Ramstead MJD, Badcock PB, Friston KJ, 2018 · Answering Schrödinger's question: A free-energy formulation · open at passage 23
  5. A statistical manifold is a special state space, in which the states represent the parameters of a probability distribution.Friston KJ, Wiese W, Hobson JA, 2020 · Sentience and the Origins of Consciousness: From Cartesian Duality to Markovian… · open at passage 24
  6. First, the Fisher information metric is simply the curvature of the variational free energy as one moves on the internal (statistical) manifold.Friston KJ, Wiese W, Hobson JA, 2020 · Sentience and the Origins of Consciousness: From Cartesian Duality to Markovian… · open at passage 89
  7. This is the same as the conditional precision or confidence placed in beliefs about external states.Friston KJ, Wiese W, Hobson JA, 2020 · Sentience and the Origins of Consciousness: From Cartesian Duality to Markovian… · open at passage 89
  8. Agents at higher levels (e.g., organs) deform the energy landscape of actions for the lower levels (e.g., cells or subcellular machinery)Fields C, Levin M, 2022 · Competency in Navigating Arbitrary Spaces as an Invariant for Analyzing… · open at passage 51
Linked ideas
The gauge and information-geometric readings are formulations of the free energy principle, where variational free energy supplies the curvature.
Friston and colleagues equate the Fisher metric with conditional precision, which links the geometry to attention and precision weighting.
Curvature or precision is said to play a key role in predictive processing accounts of attention and interoceptive inference.
Ramstead and colleagues place the gauge formulation within nested Markov blankets, each obeying the same free energy principles.
Sengupta and colleagues apply the gauge description at nested levels, with prediction errors passed up and predictions passed down.
Fields and Levin describe higher-level agents shaping the energy landscape of lower levels, a multiscale picture close to the nested gauge account.
Friston et al. propose 'Markovian monism' to repair Cartesian duality using physics and information theory, with beliefs as probability distributions, linking monism (A) to belief-space formalism (B); passage is truncated.
Where it is discussedPassages matching information geometry, Fisher information metric, statistical manifold, gauge theory
2020Sentience and the Origins of Consciousness: From Cartesian Duality to Markovian… · Friston KJ, Wiese W, Hobson JA45
2016Towards a Neuronal Gauge Theory · Sengupta B, Tozzi A, Cooray GK, Douglas…21
2018Answering Schrödinger's question: A free-energy formulation · Ramstead MJD, Badcock PB, Friston KJ1
2022Competency in Navigating Arbitrary Spaces as an Invariant for Analyzing… · Fields C, Levin M1