Probabilistic Wisconsin Card Sorting Test

• The pWCST is a stochastic variant of the WCST that requires inferring latent sorting rules from probabilistic feedback rather than deterministic signals.
• It is modeled through frameworks such as Bayesian filtering, hidden-state inference, and adaptive representation learning to capture decision-making under uncertainty.
• Research in pWCST integrates computational modeling and neural circuit implementations to explore adaptive control, set-shifting, and evidence accumulation.

Searching arXiv for recent and directly relevant papers on the probabilistic Wisconsin Card Sorting Test and computational models of WCST/pWCST. The probabilistic Wisconsin Card Sorting Test (pWCST) is a stochastic variant of the Wisconsin Card Sorting Test in which latent sorting rules must be inferred under uncertain feedback rather than from deterministic correctness signals alone. In the standard WCST, a participant sorts stimuli according to one of three hidden rules—typically color, shape, or number—and must detect unannounced rule changes from binary feedback. In the pWCST, the same latent rule structure is preserved, but the feedback process is probabilistic, so single-trial outcomes are no longer sufficient for exact rule identification. This makes the pWCST a paradigmatic task for studying set-shifting, latent-state inference, adaptive control, evidence accumulation, and uncertainty-sensitive decision-making in humans and artificial agents. Contemporary arXiv work frames the task variously as a contextual bandit with hidden contexts, a Bayesian filtering problem over rule hypotheses, and a recurrent circuit implementation of approximate belief updating (Qin et al., 2022).

1. Task definition and formal structure

The pWCST inherits the canonical WCST rule space of three latent sorting dimensions, usually denoted $S=\{\text{color}, \text{shape}, \text{number}\}$. On each trial, a participant or model selects one of four response options, and feedback indicates whether the choice was “correct” or an “error.” What distinguishes the pWCST from the classical WCST is that feedback is stochastic: correct latent-rule application need not yield correct feedback with probability $1$, and incorrect application need not always be penalized. The task therefore separates immediate feedback from the underlying environmental state.

One formalization treats the task as a contextual bandit with changing hidden contexts. In that setting, the feedback process is written as

$y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$

where $x_t \in A \subset \mathbb{R}^d$ is the action, $y_t$ is the reward or feedback, $\theta_{\sigma(t)}$ is the task coefficient associated with the active context, and $\eta_t$ is $1$-sub-Gaussian noise. In the WCST-specific encoding, there are four fixed reference cards and each stimulus is represented by a matrix $A_t \in \mathbb{R}^{4\times 3}$, whose columns encode shape, number, and color matches via one-hot basis vectors. The active rule is represented by $B_\sigma \in \mathbb{R}^3$, and the effective task coefficient is

$1$0

so that the correct pile corresponds to a one-hot vector in $1$1. Deterministic WCST reward is then

$1$2

A pWCST is obtained by allowing this reward to be corrupted by stochastic feedback, either through $1$3 or through a Bernoulli observation model (Qin et al., 2022).

A second formalization treats the task as a hidden-state inference problem. Let $1$4 denote the dominant hidden rule at trial $1$5, $1$6 the selected rule or action, and $1$7 the feedback. In a block-structured pWCST, $1$8 is constant within a block and changes at block boundaries. A hazard indicator $1$9 encodes those boundaries, and the prior update is

$y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$0

with uniform baseline prior $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$1. Posterior beliefs follow Bayes’ rule: $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$2 This makes the pWCST a compact testbed for normative Bayesian filtering under latent-rule volatility (Su et al., 18 Aug 2025).

A third formulation, closer to classical cognitive modeling, defines the environment by a hidden-rule Markov process with hazard $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$3, noisy feedback with flip probability $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$4, and a choice policy over cards based on expected correctness under current beliefs. In that framework, the task becomes a partially observable sequential decision problem in which both uncertainty about the current rule and uncertainty about impending rule changes are behaviorally relevant (D'Alessandro et al., 2020).

2. Feedback stochasticity, switching regimes, and generative models

The defining property of the pWCST is the replacement of deterministic feedback by a probabilistic observation model. One formulation specifies a feedback noise parameter $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$5: for a correct action, correct feedback occurs with probability $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$6, whereas for an incorrect action, misleading positive feedback occurs with probability $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$7. This embeds the pWCST naturally in a linear bandit model with sub-Gaussian noise, and it implies that agents must integrate feedback across multiple trials rather than apply immediate win-stay/loss-shift logic (Qin et al., 2022).

Another formulation, used in recurrent-circuit simulations, defines a dominant rule within each block and lets feedback follow that dominant rule with probability $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$8, while the non-dominant rules account for the remaining $y_t = x_t^\top \theta_{\sigma(t)} + \eta_t,$9. The likelihood is then

$x_t \in A \subset \mathbb{R}^d$0

In those simulations, block lengths are $x_t \in A \subset \mathbb{R}^d$1–$x_t \in A \subset \mathbb{R}^d$2 trials, rule changes occur at block boundaries, and the first and last trials in each block are fixed to the dominant rule to mark switch timing in the analysis (Su et al., 18 Aug 2025).

A broader Bayesian-brain account introduces a stochastic rule-switch process explicitly. Let the true hidden rule be $x_t \in A \subset \mathbb{R}^d$3. Then a pWCST environment can be defined by

$x_t \in A \subset \mathbb{R}^d$4

where $x_t \in A \subset \mathbb{R}^d$5 is the hazard rate and $x_t \in A \subset \mathbb{R}^d$6 distributes switch probability across alternative rules. Feedback is then generated by a noisy correctness channel: $x_t \in A \subset \mathbb{R}^d$7 This formulation makes explicit the two independent sources of uncertainty in the pWCST: observation noise and environmental volatility (D'Alessandro et al., 2020).

These variants illustrate that the term “pWCST” does not denote a single canonical stochastic process. Some instantiations use blockwise stochastic feedback with scheduled switches; others use trialwise hazard-driven change-points. A plausible implication is that reported performance differences across studies may depend as much on the generative process as on the cognitive or algorithmic model under evaluation.

3. Normative inference and computational modeling

Because feedback is unreliable, pWCST performance is commonly analyzed as sequential inference over hidden rules. Under a normative Bayesian filter, belief updates combine a prior over rules, a change-point model, and a likelihood for observed feedback given the chosen action. The posterior

$x_t \in A \subset \mathbb{R}^d$8

is the central object: it quantifies the agent’s trial-by-trial confidence that the current rule is color, shape, or number. Decision policies can then choose the rule with maximal posterior mass or sample actions according to posterior-weighted expected correctness (Su et al., 18 Aug 2025).

A Bayesian-brain formulation augments this with internal dynamics over rule stability. The predictive prior over rules is computed by Chapman–Kolmogorov propagation through a stability matrix $x_t \in A \subset \mathbb{R}^d$9, whose diagonal entries reflect confidence that the current rule persists. These entries derive from state activations $y_t$0, updated by feedback using two psychologically interpretable parameters: flexibility $y_t$1, which controls disengagement from invalid rules after negative feedback, and information loss $y_t$2, which rescales evidence and slows convergence. The resulting framework separates three information-theoretic quantities: Shannon surprise,

$y_t$3

Bayesian surprise,

$y_t$4

and entropy,

$y_t$5

These quantities provide distinct measures of unexpectedness, belief revision, and uncertainty, respectively (D'Alessandro et al., 2020).

A different computational line models set-shifting using the Sequential Learning Model. There, each response card $y_t$6 has a binary match vector $y_t$7 over the three candidate rules, attention over rules is represented by a vector $y_t$8, and the choice policy is

$y_t$9

where $\theta_{\sigma(t)}$0 is a choice consistency parameter. Attention is updated after “correct” and “incorrect” feedback using separate reward sensitivity $\theta_{\sigma(t)}$1 and punishment sensitivity $\theta_{\sigma(t)}$2. Although this model was fitted in a deterministic WCST setting, it is directly relevant to pWCST because elevated $\theta_{\sigma(t)}$3 and high $\theta_{\sigma(t)}$4 imply strong, deterministic reactions to feedback and therefore predict sensitivity to misleading stochastic outcomes unless noise is explicitly represented (Li et al., 19 Jun 2025).

Taken together, these models locate the pWCST at the intersection of Bayesian filtering, latent-state learning, and controlled evidence integration. The main conceptual divide is between models that explicitly represent feedback noise and change-point structure and models that react to outcomes with high-gain deterministic updates. This distinction is central to interpreting both human and machine behavior in probabilistic environments.

4. Representation learning and adaptive control algorithms

Recent work also treats the pWCST as a benchmark for adaptive representation learning rather than only for latent-state inference. In the contextual-bandit formulation, tasks within a context are assumed to share a low-dimensional representation: $\theta_{\sigma(t)}$5 where $\theta_{\sigma(t)}$6 has orthonormal columns and $\theta_{\sigma(t)}$7. The practical implication is that learning a context-specific subspace $\theta_{\sigma(t)}$8 reduces the effective dimension from $\theta_{\sigma(t)}$9 to $\eta_t$0, which lowers exploration cost and accelerates adaptation after rule changes (Qin et al., 2022).

The algorithmic pipeline comprises five components. Representation Exploration (RE) uses explore-then-commit with exploration length $\eta_t$1, estimating $\eta_t$2 by least squares and then taking greedy actions. Representation Transfer (RT) assumes a current subspace estimate $\eta_t$3, restricts exploration to that subspace for $\eta_t$4 rounds, estimates $\eta_t$5, and reconstructs $\eta_t$6. Sequential Representation Learning (SeqRepL) alternates RE and RT over multiple tasks in the same context, constructing $\eta_t$7 via the top-$\eta_t$8 singular vectors of

$\eta_t$9

Outlier Detection (OD) probes the orthogonal complement of $1$0 and declares a potential context switch when the feedback vector falls outside the confidence set

$1$1

Adaptive Representation Learning (AdaRepL) combines these components by using OD to trigger RE and restarting SeqRepL after $1$2 consecutive outliers (Qin et al., 2022).

In pWCST terms, the hidden rule is treated as an unobserved context, and context change is inferred indirectly from deviations in reward statistics rather than by explicit Bayesian posterior tracking. This yields a change-point-like mechanism that is lightweight relative to full filtering. The reported interpretation is that each rule corresponds to a distinct low-dimensional subspace, so rapid set-shifting depends on learning and reusing those subspaces rather than relearning the full task from scratch.

This approach differs from classical pWCST models in a substantive way. Bayesian filters track posterior mass over discrete rules; AdaRepL tracks the geometry of task coefficients and detects regime shifts through orthogonal probing. A plausible implication is that the pWCST can support two distinct notions of adaptation: discrete rule inference and subspace transfer. The former emphasizes latent-state belief; the latter emphasizes reusable structure across related task instances.

5. Neural circuit implementations and approximate Bayesian belief updating

A biologically oriented account of the pWCST has been proposed using a recurrent divisive normalization circuit with self-excitation. The model contains $1$3 excitatory units $1$4 and one inhibitory pool $1$5, with dynamics

$1$6

Here, $1$7 is self-excitation, $1$8 is semi-saturation, $1$9 are positive excitatory-to-inhibitory weights, and $A_t \in \mathbb{R}^{4\times 3}$0 are external inputs. The normalized readout

$A_t \in \mathbb{R}^{4\times 3}$1

serves as a probability-like representation of rule belief (Su et al., 18 Aug 2025).

For WCST and pWCST simulations, $A_t \in \mathbb{R}^{4\times 3}$2 excitatory units encode the three rules. Divisive normalization enforces a sum-to-one constraint on the readout, while self-excitation maintains beliefs between observations. At the input-driven steady state,

$A_t \in \mathbb{R}^{4\times 3}$3

so the circuit maps normalized inputs to normalized rule beliefs. In the no-input regime, when $A_t \in \mathbb{R}^{4\times 3}$4, the system exhibits a continuous attractor, and the normalized proportions $A_t \in \mathbb{R}^{4\times 3}$5 remain constant. This is interpreted as maintenance of belief states during inter-trial delays (Su et al., 18 Aug 2025).

Feedback is converted into brief $A_t \in \mathbb{R}^{4\times 3}$6 ms input pulses. After a correct trial, the unit corresponding to the chosen rule receives $A_t \in \mathbb{R}^{4\times 3}$7 and other units receive $A_t \in \mathbb{R}^{4\times 3}$8. After an error, the chosen-rule unit receives $A_t \in \mathbb{R}^{4\times 3}$9 and the alternatives receive $B_\sigma \in \mathbb{R}^3$0. Beliefs are then updated by integrating the continuous-time dynamics through the pulse and the inter-trial interval. With parameters $B_\sigma \in \mathbb{R}^3$1 ms, $B_\sigma \in \mathbb{R}^3$2, $B_\sigma \in \mathbb{R}^3$3, and $B_\sigma \in \mathbb{R}^3$4 ms, the circuit tracks rule switches rapidly in deterministic WCST and maintains asymptotic accuracy near $B_\sigma \in \mathbb{R}^3$5 in the pWCST with dominant-rule probability $B_\sigma \in \mathbb{R}^3$6 across $B_\sigma \in \mathbb{R}^3$7 rule blocks (Su et al., 18 Aug 2025).

This circuit model is not a literal implementation of exact Bayes’ rule, and quantitative divergence from a normative HMM is not reported. Nevertheless, it demonstrates how normalized recurrent activity can approximate probabilistic belief updating while simultaneously maintaining working-memory-like persistence. This suggests a unification of noise suppression, memory retention, and latent-rule inference within a single cortical motif.

6. Empirical findings, behavioral signatures, and limitations

The empirical literature distinguishes clearly between deterministic WCST results and expected pWCST behavior. In deterministic set-shifting, LLMs have been shown to perform near an Expected Utility Maximization heuristic that switches the hypothesized rule after negative feedback and continues after positive feedback. In a $B_\sigma \in \mathbb{R}^3$8-trial WCST with rule changes after eight consecutive correct responses, GPT-4o, Claude, Gemini, and GPTo4m significantly outperformed humans on total correct matches, while DeepSeek matched human accuracy. LLMs other than DeepSeek also showed lower TRSET1, indicating faster initial rule discovery, and comparable FSET, indicating similar stability once a rule was established. At the same time, most LLMs exhibited more perseverative than non-perseverative errors, whereas humans showed the opposite pattern (Li et al., 19 Jun 2025).

These deterministic findings matter for pWCST because they reveal the interaction between sensitivity to outcomes and rule persistence. Hierarchical Bayesian fits of the Sequential Learning Model estimated higher reward sensitivity $B_\sigma \in \mathbb{R}^3$9, higher punishment sensitivity $1$00, and higher choice consistency $1$01 for most LLMs than for humans. The paper’s interpretation is that these models adopt an explicit, rule-centric and highly deterministic strategy. Under probabilistic feedback, such a strategy could produce either over-switching after misleading negative feedback or excessive perseverance if the model requires too much contradictory evidence before abandoning a rule. The study does not directly test pWCST behavior, so this remains an open empirical question (Li et al., 19 Jun 2025).

From the adaptive representation learning perspective, deterministic WCST experiments with rule switches every $1$02 rounds showed that tabular Q-learning and DQN performed barely better than random sorting, whereas AdaRepL adapted rapidly after switches and achieved substantially higher cumulative reward per trial. The interpretation offered is that standard RL baselines require too many samples and do not explicitly account for hidden context switching, while context-dependent representation learning yields faster adaptation. For pWCST, the same work argues that sub-Gaussian noise is already covered by the theoretical model, but empirical pWCST evaluations were not reported (Qin et al., 2022).

Several limitations recur across the literature. First, many studies use deterministic WCST and only extrapolate to pWCST theoretically. Second, outlier-based change detection is heuristic rather than a full probabilistic change-point detector. Third, parameter sensitivity is often underexplored: hyperparameters such as $1$03, $1$04, $1$05, and $1$06 in AdaRepL, or the input coding used in recurrent circuits, are not exhaustively analyzed. Fourth, some biologically inspired models are rate-based and do not model learning or synaptic plasticity. Fifth, the Bayesian-brain model identifies flexibility $1$07 well, but information loss $1$08 is harder to recover, especially at low values, indicating partial identifiability constraints (D'Alessandro et al., 2020).

The pWCST therefore occupies a distinctive methodological niche. It is simple enough to admit exact latent-state models, low-dimensional representation-learning approaches, and mechanistic neural circuits, yet difficult enough to expose differences between deterministic heuristics and uncertainty-aware adaptation. This suggests that the pWCST is best understood not merely as a noisy WCST, but as a canonical probe of how agents represent uncertainty, maintain task beliefs, detect volatility, and trade off robustness against flexibility under stochastic feedback.