rSLDS-based Message Passing Techniques

• The rSLDS-based message passing approach is a Bayesian framework that utilizes message propagation in clone-structured models for efficient inference in ambiguous settings.
• It applies forward-backward and Viterbi recursions to update belief states and support active inference, enabling robust planning and context disambiguation.
• The method bridges cognitive neuroscience and AI by modeling hippocampal functions and sequential decision making, with prospects for online learning and hierarchical extensions.

rSLDS-based message passing refers to the class of belief-propagation techniques applied to Recurrent Switching Linear Dynamical Systems (rSLDS), a class of models that generalize Hidden Markov Models (HMMs) and Switching Linear Dynamical Systems (SLDS) by incorporating structured latent state spaces, temporal context, and in many cases, action-dependent transitions. Within the cognitive mapping and neuroscience modeling literature, the most advanced rSLDS architectures are exemplified by the Clone-Structured Cognitive Graph (CSCG) and related messaging algorithms. These approaches realize exact or approximate Bayesian inference and planning in latent state spaces with high contextual ambiguity, leveraging efficient message-passing through sparse, structured graphs.

1. Structural Definition and Generative Properties

The rSLDS framework, as instantiated in the CSCG, constructs a directed, action-augmented multigraph where each node represents a unique context-sensitive latent state or “clone.” For every observation symbol $o \in \mathcal{O}$, the model allocates multiple hidden states $s \in \mathcal{S}$ (clones), each encoding the context of its observation history. Edges between nodes are labeled by actions and parameterized by transition probabilities $B_{i a j} = p(s_{t+1}=j \mid s_t=i, a_t=a)$. The emission mapping $A_{oi} = p(o_t=o \mid s_t=i)$ is many-to-one: all clones of $o$ emit the same observation.

The full probabilistic joint is: $$p(o_{1:T}, s_{1:T}, a_{1:T-1} \mid \Theta) = p(s_1) \prod_{t=1}^{T} p(o_t \mid s_t) \prod_{t=1}^{T-1} p(a_t \mid s_t) p(s_{t+1} \mid s_t, a_t)$$ where in practice $a_t$ is typically set by the agent (Maele et al., 2023, Raju et al., 2022).

2. Message Passing Algorithms for Inference and Learning

Efficient inference in rSLDS-based models requires message passing algorithms that exploit sparse emission and structured transitions. Both marginal state estimation and Viterbi decoding are achieved via (generalized) forward–backward (Baum–Welch) recursions, adapted to action- and context-labeled transitions.

Forward–Backward (Baum–Welch) Recursions:

• Forwards: $\alpha_t(i) = p(o_{1:t}, s_t = i \mid \Theta)$
• Backwards: $\beta_t(i) = p(o_{t+1:T} \mid s_t = i, \Theta)$

Posterior marginals: $$\gamma_t(i) = p(s_t = i \mid o_{1:T}, a_{1:T-1}) = \frac{\alpha_t(i) \beta_t(i)}{\sum_j \alpha_t(j) \beta_t(j)}$$

Expected transition (pairwise) marginals: $s \in \mathcal{S}$0

Parameter updates (M-step) employ these marginals: $s \in \mathcal{S}$1

$s \in \mathcal{S}$2

(Maele et al., 2023, Raju et al., 2022).

Viterbi message passing identifies the MAP trajectory and is essential for clone pruning and hard assignment post-EM.

3. Planning via Active Message Passing

Planning within rSLDS/CSCG leverages message passing for both state estimation and trajectory optimization. In the context of active inference, the agent utilizes the learned transition (B), emission (A), preference (C), and prior (D) arrays to enumerate candidate policies $s \in \mathcal{S}$3, evaluating them via expected free energy (EFE):

$s \in \mathcal{S}$4

Policy distributions for control: $s \in \mathcal{S}$5

Belief propagation under candidate policies: $s \in \mathcal{S}$6

$s \in \mathcal{S}$7

(Maele et al., 2023).

This realizes a form of anticipatory forward message passing for multi-step lookahead.

4. Empirical Performance and Context Disambiguation

rSLDS-based message passing, as realized in CSCG agents, addresses environments with severe perceptual aliasing and ambiguous sensory cues. In paradigmatic experiments (Open Room, Ambiguous Maze, T-Maze), active inference-driven CSCG agents consistently outperform naive (greedy) variants and random baselines, especially in settings where context-sensitive disambiguation is critical (e.g., T-Maze with ambiguous cues).

Quantitatively, active inference leads to:

• Success rates of 100% across all scenarios when compared to lower rates for greedy agents in high-ambiguity tasks.
• Shorter or statistically equivalent path lengths in environments with ambiguity (mean path length: 5.0 vs 4.5 steps for T-Maze; difference not significant in Open Room). (Maele et al., 2023).

The ability to maintain and update posterior beliefs over clone states enables robust localization and rapid policy adaptation.

5. Neural and Computational Relevance

The rSLDS/CSCG framework provides a normative model for hippocampal function, capturing phenomena such as place fields, splitter cells, and context-dependent remapping. The unique message passing topology—split clones per observation with temporally contextual transitions—yields emergent representations mirroring observed neural coding diversity.

The same message-passing algorithms are interpreted as neural operations:

• State marginals correspond to Bayesian context beliefs.
• Backward messages support offline replay and preplay.
• Viterbi message passing underpins rapid replanning following environmental changes. (Raju et al., 2022).

Planning and schema transfer are naturally implemented as clamped message passing with priors on goal clones.

6. Constraints, Limitations, and Extensions

Known limitations of rSLDS-based message passing include:

• Offline learning procedures (EM) with large memory requirements for long sequences or high clone cardinalities.
• Exponential policy set growth in active inference lookahead, constraining planning horizon $s \in \mathcal{S}$8 in practice.
• Discrete state and emission structure, with current approaches requiring adaptation for continuous or high-dimensional sensory domains.

Proposed extensions involve:

• Online (recursive) parameter updates via Dirichlet processes.
• Hierarchical CSCGs supporting abstraction over temporal and spatial structure.
• Augmentation for structural causal modeling by encoding interventions through clones.
• Integration of neural or mixture-model-based emissions for richer observations (Maele et al., 2023, Raju et al., 2022).

7. Connections to Broader Paradigms

rSLDS-based message passing unifies classic HMM/SLDS inference with modern sequence learning paradigms under the Bayesian/active inference framework. It formalizes how cognitive maps and context-dependent planning arise from structured latent state inference and message propagation. This approach bridges cognitive neuroscience, sequential decision making, and structured probabilistic modeling, with empirical and theoretical justifications for observed hippocampal coding properties and planning flexibility.

The generality of the message-passing approach supports applications from grid/maze navigation to abstract hierarchical planning, providing a computational substrate for both biological and artificial agents operating in ambiguous and temporally extended settings (Raju et al., 2022).