state-dependent-observation-noise-active-inference

name	state-dependent-observation-noise-active-inference
description	State-Dependent Observation Noise methodology that reintroduces epistemic value in Linear-Gaussian Active Inference models. This skill provides the mathematical framework and implementation guidance for restoring curiosity-driven behavior in Gaussian agents by introducing state-dependent observation noise covariance R(x). Use when working with active inference, Bayesian filtering, dual control theory, or neural dynamics models where epistemic drive has been lost in standard linear-Gaussian formulations.
license	Complete terms in LICENSE.txt

State-Dependent Observation Noise in Active Inference

Overview

This methodology addresses a fundamental limitation in linear-Gaussian active inference: the loss of epistemic drive (curiosity) under standard formulations. Recent work established that linear-Gaussian state-space models lose their epistemic incentive "under any circumstances" because the Expected Free Energy's epistemic term becomes constant, reducing the agent to a fixed Kalman filter.

Key insight: State-dependent observation noise covariance R(x) that varies with the state x (representing sensor accuracy degrading with range) restores epistemic value by making the posterior covariance and Kalman gain dependent on actions.

Mathematical Framework

Problem Statement

In standard linear-Gaussian active inference:

• Dynamics: x_{t+1} = Ax_t + Bu_t + w_t, w_t ~ N(0, Q)
• Observations: y_t = Cx_t + v_t, v_t ~ N(0, R)
• Issue: R is constant → epistemic term constant → no information-seeking behavior

Solution: State-Dependent Observation Noise

• Modified observations: y_t = Cx_t + v_t, v_t ~ N(0, R(x_t))
• Where R(x) represents sensor accuracy that degrades with state (e.g., range-dependent measurement noise)
• Agent uses standard first-order Gaussian filter with R evaluated at predicted mean: R(μ̂_t)

Key Results

1. Dual Effect Restoration: Actions now influence both state evolution AND future estimation quality
2. Non-Constant Epistemic Value: Under mild rank condition on C and non-degeneracy of R(x), epistemic value is no longer constant
3. Scalar Case: For scalar observations, reachable non-constancy alone suffices
4. Bar-Shalom-Tse Dual Effect: This represents the minimal constructive instance of dual control in maintained covariance

Implementation Guidelines

Detection of Incompatibility

The companion library cpomdp v0.4.2 automatically detects model incompatibility:

• Raises IncompatibleLinearizationError from model specification alone
• Provides executable witness that would refute both theorem and witness if any fixed filter reproduced the agent

Practical Applications

1. Robotics: Range-dependent sensor noise in LiDAR, cameras, or sonar
2. Neuroscience: Modeling attention mechanisms where sensory precision varies with stimulus properties
3. Finance: Volatility-dependent observation noise in market models
4. Control Systems: Signal-to-noise ratio varying with system state

Algorithm Steps

1. Define state-dependent observation covariance R(x)
2. Implement Gaussian filter using R(μ̂_t) where μ̂_t is predicted state mean
3. Compute Expected Free Energy with action-dependent posterior covariance
4. Optimize actions to balance extrinsic (utility) and intrinsic (information) objectives

Verification

The methodology includes an executable witness:

• Any attempt to reproduce the agent with a fixed linear-Gaussian filter would simultaneously refute both the theorem and the witness
• This provides precise, observation-side characterization of curiosity in Gaussian agents

References

• Primary Paper: Corva, D. (2026). "State-Dependent Observation Noise Reintroduces Epistemic Value in Linear-Gaussian Active Inference." arXiv:2607.20306 [q-bio.NC]
• Companion Software: cpomdp v0.4.2 (archived repository available)
• Theoretical Foundation: Bar-Shalom & Tse (1974) dual effect theory
• Active Inference: Friston et al. Expected Free Energy framework

Activation Keywords

• state-dependent observation noise
• active inference epistemic value
• linear-gaussian curiosity
• dual control bar-shalom-tse
• gaussian agent curiosity
• observation noise covariance
• expected free energy epistemic

Related Skills

• active-inference-framework
• kalman-filter-extensions
• dual-control-theory
• bayesian-filtering-neuroscience
• computational-neuroscience-models