Belief-Based Maximum Occupancy Principle and Active Inference

arXiv:2609.39342 · q-bio.NC, cs.AI · Submitted 2026-09-30 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: Today's paper: "Belief-Based Maximum Occupancy Principle and Active Inference".

Marcus: The paper introduces and compares three intrinsic motivation frameworks—Maximum Occupancy Principle (MOP), Active Inference, and Empowerment—by extending them to partially observable environments using belief-based inference.

Ines: First, who's behind it and why it matters.

Title and authors: Ines: So we're diving into the paper titled "Belief-Based Maximum Occupancy Principle and Active Inference," which sounds really dense, but essentially it looks at how intrinsic motivations like MOP and Active Inference handle things when you can't see everything directly. What do you make of that title?

Marcus: From a data science perspective, I think the title tells us they are combining two distinct theoretical approaches to decision-making—MOP and Active Inference—and applying them to belief-based inference, which means the agents are operating under uncertainty about what's actually going on.

Yuki: I find it interesting that they are looking at these intrinsic motivations because traditional reinforcement learning often relies purely on external rewards, whereas this paper suggests that internal objectives can drive behavior even when there's no clear payoff for a specific action.

Ines: Exactly, and the authors explain how MOP focuses on maximizing the entropy of future paths of states and actions without any external preferences or targets. This sounds like a way to build adaptability right into the core decision-making process.

Marcus: And then you have Active Inference, which aims to minimize expected free energy by balancing risk and ambiguity resolution, which is a very different kind of goal than just maximizing path entropy. It seems they're testing how these two principles play out against each other in complex settings.

Yuki: It reminds me a bit of how population genetics deals with how traits spread or persist across a population, where internal pressures—like selection or drift—shape the behavior without an external, immediate reward signal dictating every move.

The paper's summary: Ines: To summarize what they're saying in "Belief-Based Maximum Occupancy Principle and Active Inference," the core idea is extending these intrinsic motivation frameworks to environments where the agent doesn't have a perfect view of the situation, using belief-based inference to manage that uncertainty.

Marcus: Specifically, they show how MOP agents adjust their strategy based on their energy levels; for high energy, they explore broadly between food sources, but when energy is low, they become much more focused on survival.

Yuki: That connection between internal constraints like energy and the resulting behavioral strategy is something that resonates with my work in evolutionary biology; it suggests that physical limitations can directly modulate the type of adaptive behavior an organism displays.

Ines: Right, because Active Inference agents seem to prefer to stick near a single food source, and their decision-making is shaped by minimizing free energy through balancing risk and ambiguity. They don't just seek the nearest food; they try to resolve uncertainty about where that food might be while managing potential risks.

Marcus: And the mathematical setup involves defining an agent's state as a tuple of observable state and its belief, which is then updated using Bayes' rule over hidden variables to make those decisions tractable for computation. That’s a significant step in modeling partial observability.

Yuki: Modeling the agent’s belief as an evolving stochastic process over time is crucial because it mirrors how an organism might update its internal map of its environment based on noisy sensory inputs and past experiences.

The paper's improvements: Ines: The paper suggests several ways to improve these existing models, particularly focusing on how MOP agents can be better tuned when dealing with internal constraints like energy. They propose refining the value function to explicitly handle the trade-off between exploration and goal-directed survival based on that energy level.

Marcus: I think that's where they move from just describing behavior to actually designing an AI system that can adapt its risk profile dynamically, shifting between broad exploration and highly concentrated goal seeking as energy depletes.

Yuki: That idea of self-regulating the degree of stochasticity based on an internal physical state is compelling because it moves the behavior away from being dictated purely by external rewards or fixed parameters.

Ines: Furthermore, they suggest integrating information theory into the Active Inference framework by using Expected Free Energy to explicitly balance pragmatic drives—the risk aspect—with epistemic drives, which is a refinement over just minimizing free energy alone.

Marcus: That integration means the AI can prioritize actions that provide the most informative observations about hidden states rather than just randomly seeking out things, which is a much smarter way to gather data in an uncertain environment.

Yuki: From a systems view, this suggests that we need models where the agent's internal state isn't just static but actively shapes the information it seeks out; it’s about building structures that allow for context-dependent learning.

Conclusion: Ines: So, to wrap up our discussion on "Belief-Based Maximum Occupancy Principle and Active Inference," we see that these frameworks offer a unified way to approach adaptive behavior by comparing how maximizing path entropy versus minimizing expected free energy leads to very different strategies under uncertainty.

Marcus: The main implication for the field is that it gives us a solid, information-theoretic foundation for understanding why agents exhibit both spontaneous exploration and constrained survival behaviors in complex, partially observable settings.

Yuki: I think this work contributes by showing how intrinsic motivations can naturally generate these rich adaptive strategies without needing to program specific reward functions for every scenario.

Ines: And the suggested improvements point toward creating more robust AI systems that can manage internal constraints like energy levels to dynamically switch between broad exploration and focused goal-directed survival.

Marcus: I agree, and this belief state management approach via value iteration seems like a practical path forward for actually building these complex planners in real-world scenarios where we don't know everything upfront.

Yuki: It really shows that understanding the underlying information structure of the environment is as important as understanding the agent's immediate goals or rewards.

Manolis Mylonas, Rubén Moreno-Bote

Center for Brain and Cognition · Department of Engineering

q-bio.NC, cs.AI

Submitted: 2026-09-30

Updated: 2026-09-30

Comments: Accepted at the 7th International Workshop on Active Inference (IWAI 2026, Madrid). To appear in Springer CCIS proceedings

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 79/100

The gist: The paper introduces and compares three intrinsic motivation frameworks—Maximum Occupancy Principle (MOP), Active Inference, and Empowerment—by extending them to partially observable environments

Key concepts

Belief State (s, b)
The agent's state is defined as the tuple consisting of the observable state and its belief. The belief represents a probability distribution over hidden variables that are not directly seen by the agent. This combined state allows the framework to operate effectively in settings with noisy observations.
Maximum Occupancy Principle (MOP)
MOP agents maximize future action-state path entropy, preferring actions and states that support diverse possible future trajectories. This leads to adaptive exploration: high energy allows for broad action distributions, while low energy forces the agent to concentrate on reaching food sources for survival.
Active Inference (AI)
AI minimizes Expected Free Energy by balancing risk and ambiguity. Agents tend to stay near a single food source; increasing the parameter 'd' makes them commit to that source for longer survival, trading policy entropy for duration.

Terminology

Summary

The paper introduces and compares three intrinsic motivation frameworks—Maximum Occupancy Principle (MOP), Active Inference, and Empowerment—by extending them to partially observable environments using belief-based inference. This work matters because it provides a unified, information-theoretic perspective on adaptive behavior by comparing how different principles induce distinct strategies for exploration versus goal-directed survival under uncertainty.

The gist: MOP agents switch between goal-directed (food seeking) behavior and exploration between different food sources, depending on their energy available and their belief state, in contrast to Active Inference agents which mostly inhabit regions around a single food source.

Framework Extensions for Partial Observability

The research extends MOP to Partially Observable Markov Decision Processes (POMDPs) by introducing belief-based inference over hidden variables as part of the agent's state. The agent state is defined as the tuple (observable state, belief), denoted as (s, b). This allows the framework to operate in settings where agents do not directly observe the full system state but instead receive noisy observations, requiring a construction of a belief over hidden variables that evolves in time as a stochastic process.

The agent's belief is updated using Bayes' rule:

b′(h′) ∝ X x′, h q(ω′x′, h′) p(h′h, x) b(h).

This belief state (s, b) serves as the sufficient statistics in our problem, summarizing all available information for computing an optimal policy. The predictive model computes the next agent state distribution using the joint probability distribution over observations, future observable states, and hidden states:

P(s′, b′s, b, a) = X ω′ δ(b′ − b′(ω′, s, a)) P(s′, ω′s, b, a)

Maximum Occupancy Principle (MOP)

MOP proposes that agents act to maximize the entropy of future action-state paths: actions and states are preferred when they support a diverse set of possible future trajectories. This is formalized through a value function functional, where the optimal policy is derived by optimizing this value function over every agent's state:

V∗(s, b) = lnX a exp [βH(·, ·s, b, a) + γ X s′,b′ P(s′,b′s,b,a)V∗(s′,b′)]

The resulting optimal action-value function is computed via value iteration over the full belief-state space. The key finding is that MOP induces adaptive exploration strategies between and around the food sources that are strongly modulated by internal energetic constraints. Specifically, for high energy levels (E > 15), the agent exhibits increased policy entropy, corresponding to a broader distribution over actions, allowing for exploratory behavior. For low energy levels (E ≤ 15), the policy becomes significantly more concentrated, similar to the deterministic EFE and Empowerment agents, prioritizing reaching food sources to avoid starvation.

Active Inference (AI)

Active Inference aims to minimize the Expected Free Energy (EFE), which balances pragmatic drives (risk) with epistemic drives (ambiguity resolution). The per-action EFE is defined as:

G(s, b, a) = EP [ω′,s′,h′s,b,a] h Risk z ln P(s′s, b, a) − ln P(s′) Ambiguity z − ln q(ω′s′, h′) i Expected free energy of next action + γ EP (s′,b's,b,a) G(s′,b')]

The policy is computed as:

π(as, b) = σ − d G(s, b, a)

Active Inference agents tend to inhabit a single food source and remains in its vicinity, driven by the joint minimization of risk and ambiguity. The inverse temperature parameter 'd' controls the degree of stochasticity:

  1. For small d (e.g., d ≤ 1), the policy is stochastic, leading to high policy entropy, but survival is shortest (e.g., 539 steps at d = 0.5).

  2. As d increases (d ≥ 2), the agent commits to a single source and stays there, resulting in higher survival duration (e.g., 4883 steps at d = 10) but lower policy entropy, indicating a trade-off where survival is bought by abandoning exploration.

Empowerment

Empowerment is an intrinsic objective that encourages the agent to select actions that maximize its influence over future observations, quantified as the mutual information between an action sequence and the resulting future observation.

Improvements for AI systems

Based on the provided scientific paper, here are specific improvements that can be made to Artificial Intelligence systems by incorporating the concepts from Belief-Based Maximum Occupancy Principle (MOP) and Active Inference:

The following improvements focus on transitioning AI systems from purely reward-driven optimization toward more robust, adaptive, and information-theoretically grounded behavior in complex, partially observable environments.

  1. Refactor Policy Optimization for Goal-Directed Exploration vs. Survival Trade-off:

  2. Implement Belief State Management via Tractable Value Iteration:

  3. Integrate Information Theory into Risk/Ambiguity Balancing (Expected Free Energy):

  4. Develop Adaptive Exploration Strategies Modulated by Internal Constraints:

The improved AI system, incorporating these elements, can perform the following specific capabilities:

  1. Refactor Policy Optimization for Goal-Directed Exploration vs. Survival Trade-off:

The system will no longer solely maximize extrinsic rewards but will optimize a value function that balances two intrinsic pressures: maximizing path entropy (exploration/variability) and minimizing risk (survival/homeostasis).

  • It can intelligently switch between aggressive, broad exploration when energy is high.

  • It can transition to highly goal-directed, focused behavior (e.g., food seeking) when energy is low or survival becomes critical.

  1. Implement Belief State Management via Tractable Value Iteration:

The system will explicitly model uncertainty by maintaining a belief state (a probability distribution over hidden environmental variables like food source locations).

  • It can perform optimal decision-making under partial observability (POMDPs) without needing full environmental knowledge.

  • The use of Bellman reformulations and value iteration over the belief-state space allows for the computation of time-stationary policies offline, making planning feasible in complex, partially observable real-world scenarios.

  1. Integrate Information Theory into Risk/Ambiguity Balancing (Expected Free Energy):

The system will utilize a formulation based on Expected Free Energy (EFE) to select actions that minimize a cost balancing pragmatic drives (risk/deviation from prior expectations) and epistemic drives (ambiguity resolution).

  • It can resolve uncertainty about hidden environmental states by prioritizing actions that lead to observations most informative about those states.

  • This allows the AI to prioritize visiting areas where its current beliefs are highly uncertain, leading to targeted information-gathering behavior rather than random wandering.

  1. Develop Adaptive Exploration Strategies Modulated by Internal Constraints:

The system will learn how its own internal state (e.g., energy level, computational resources) modulates its level of stochasticity (policy entropy).

  • It can self-regulate its exploratory behavior: high energy allows for diverse actions and broad exploration; low energy forces the agent into conservative, goal-directed actions to ensure survival.

  • This creates a biologically plausible mechanism where internal physical constraints directly shape the AI's behavioral strategy, leading to adaptive control that is more robust than fixed reward schedules.

Sources

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