Clone-Structured Causal Graph (CSCG)

• Clone-structured Causal Graph (CSCG) is a framework that uses latent clone states to resolve perceptual aliasing and capture nuanced sequential context from ambiguous observations.
• It employs a directed multi-graph architecture with multiple clones per observation, enabling higher-order contextual disambiguation and the emergence of cognitive maps.
• CSCG facilitates robust planning and offers mechanistic explanations for hippocampal phenomena through efficient inference and EM-based parameter optimization.

The Clone-structured Causal Graph (CSCG) is a class of structured, action-conditioned hidden Markov models engineered to capture the latent sequential organization of experience and to model the formation of context-specific cognitive maps from raw, aliased sensory and action sequences. CSCGs resolve the challenge of perceptual aliasing—where identical observations arise in different contexts—by assigning multiple latent “clone” states per observation and learning a contextually disambiguating multi-edge transition structure. The resultant latent graphs support the emergence of cognitive maps, robust planning, and the mechanistic explanation of hippocampal phenomena without relying on explicit spatial or Euclidean representations (Raju et al., 2022, Maele et al., 2023).

1. Formalization of the CSCG Model

Given a sequence of discrete observations $x_{1:N}$ and corresponding actions $a_{1:N-1}$, the CSCG introduces for each observation $x$ a set $C(x)$ of $M_x$ latent clone states $z$. Each clone deterministically emits its parent observation. The generative likelihood over an observation–action trajectory is: $$P(x_{1:N},a_{1:N-1}) = \sum_{z_1\in C(x_1)}\ldots\sum_{z_N\in C(x_N)} P(z_1) \prod_{n=1}^{N-1}P(z_{n+1}, a_n\mid z_n)$$ with action-conditional transitions

$$P(z_{n+1}, a_n \mid z_n) = T_{z_n, a_n, z_{n+1}}, \quad \sum_{a,z'} T_{z, a, z'} = 1\,\,\forall z$$

and deterministic emissions $P(x_n \mid z_n = z) = 1$ iff $z \in C(x_n)$. The initial clone prior $a_{1:N-1}$0 typically is uniform. To ensure transition normality (even for clones with no legal outbound transitions), a “dead” clone is introduced that self-loops and deterministically emits a special “dead” observation (Raju et al., 2022, Maele et al., 2023).

2. Graph Scaffolding and Contextual Disambiguation

The latent state space is a directed multi-graph $a_{1:N-1}$1 with nodes as clones $a_{1:N-1}$2 and edges $a_{1:N-1}$3 whenever $a_{1:N-1}$4 for some action $a_{1:N-1}$5. For each observation symbol $a_{1:N-1}$6, $a_{1:N-1}$7 disjoint clones are created, allowing the model to resolve identical observations presented in distinct sequential contexts. Higher-order dependencies across sequences are captured because a trajectory $a_{1:N-1}$8 encodes context history unavailable to first-order models. This architectural feature enables CSCG to “split” and “merge” sequential contexts, moving beyond the context-blindness of ordinary HMMs (Raju et al., 2022, Maele et al., 2023).

3. Inference, Learning, and Model Optimization

Inference in CSCG is performed via exact message-passing (forward filtering/backward smoothing) over the clone graph. At time $a_{1:N-1}$9, the posterior over possible clones $x$0 follows: $x$1 where $x$2 is the action-specific transition slice, $x$3 is the emission matrix (mapping clones to observations), and $x$4 is the one-hot encoding of $x$5.

Parameter learning uses expectation–maximization (EM), maximizing the joint likelihood over data to optimize the initial prior $x$6 and transition tensor $x$7. The E-step computes forward–backward posteriors: $x$8 The M-step updates are: $x$9

$C(x)$0

with $C(x)$1 a small pseudocount (e.g., $C(x)$2) for regularization.

After convergence, extraneous clones can be pruned by Viterbi decoding, retaining only those clones on the most likely paths, yielding a compact cognitive graph (Raju et al., 2022, Maele et al., 2023).

4. Clones, Perceptual Aliasing, and Cognitive Map Emergence

Perceptual aliasing—in which local sensory input is insufficient to specify state—poses a fundamental challenge to cognitive mapping. CSCG addresses this by assigning a sufficient number of clones per observation to accommodate all relevant sequential contexts. Learning splits the state space such that each clone within $C(x)$3 encodes a unique context, enabling specification even under extreme ambiguity. The resulting multi-graph acts as a cognitive map: the state activation vector $C(x)$4 after observing $C(x)$5 and past actions/observations delineates the agent’s location in latent (contextual) space (Maele et al., 2023).

Cognitive maps emerge without explicit access to spatial coordinates or representations. Instead, the correlation between clone activations and ground-truth positions can be revealed by binning posterior probabilities by true location, demonstrating emergent “place fields” analogous to biological place cells (Raju et al., 2022).

5. Planning, Active Inference, and Control in CSCG

CSCG facilitates planning and goal-seeking through principled message passing on the latent clone graph. Planning consists of clamping a start clone (inferred from sensory history) and a goal clone, then performing inference (forward/backward message passing) to discover the optimal action sequence mapping to the most probable path between clones. This approach supports replay-like transitive inference, extracting spatial (latent sequence) paths purely from transition structure (Raju et al., 2022).

Integration with the discrete Active Inference POMDP framework is accomplished by translating the learned CSCG transition and emission models to canonical POMDP matrices. The $C(x)$6-matrix encodes fixed clone–observation likelihoods, $C(x)$7-matrices encode action-specific latent transitions, the $C(x)$8-vector specifies log-preferences (goals), and the $C(x)$9-prior remains uniform. Policy selection minimizes Expected Free Energy $M_x$0 over action sequences $M_x$1, balancing epistemic value (uncertainty reduction) and pragmatic value (goal achievement) via softmax sampling: $M_x$2 This mechanism yields information-seeking and exploitation behaviors, with performance that matches or surpasses greedy baseline policies, particularly in ambiguous task settings (Maele et al., 2023).

6. Explanatory Power: Hippocampal Representation Phenomena

CSCG provides mechanistic explanations for a diverse array of hippocampal coding phenomena:

• Landmark-vector cells: Clones with contexts at fixed vector offsets from landmarks activate equivalently when the landmark moves, generating multi-component place fields as observed experimentally.
• Splitter cells: Identical observations (e.g., maze stem) are parsed into distinct clones encoding unique trial histories (e.g., left vs. right reward branches), resulting in context-dependent response splitting.
• Event-specific rate remapping: Clones track both spatial and event context (e.g., lap count), reproducing distinct activation profiles and remapping in response to shifts in reward contingencies.

These effects arise from the structure of the learned clone graph and the activation dynamics of clone posteriors $M_x$3, thereby offering a unified account for a dozen or more experimentally observed hippocampal phenomena (Raju et al., 2022).

7. Implementation, Hyperparameters, and Limitations

Typical hyperparameters for CSCG include the number of clones per observation (10–50 for visual tasks, exceeding the number of required context splits), EM stopping criterion (relative log-likelihood change $M_x$4 or $M_x$51000 iterations), and pseudocount $M_x$6 ($M_x$7) for transition smoothing. Observational sequences are typically pre-quantized to a closed set of symbols. Action sets are domain-dependent, e.g., {forward, turn left, turn right} for egocentric navigation or {north, south, east, west} for allocentric settings (Raju et al., 2022).

Identified limitations include the batch-offline nature of CSCG EM learning (no structure learning during active behavior), exponential scaling of policy evaluation with planning horizon $M_x$8, and basic goal/shaping heuristics in planning. The statistical and representation efficiency of CSCG is maximized when context splits are correctly matched to underlying task demands, but unnecessary excess of clones can incur computational overhead (Maele et al., 2023).

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CSCG offers a scalable, interpretable, and mechanistic framework for latent sequential representation, robust cognitive map formation, and hippocampal modeling, with wide implications for neuroscience, reinforcement learning, and biologically inspired AI (Raju et al., 2022, Maele et al., 2023).