causalstate

Causal state is a concept from computational mechanics describing the predictive structure of a stochastic process. A causal state is an equivalence class of past observations that yield the same conditional distribution over future observations. Formally, for a process Xt with finite alphabet, two histories x_{-∞:t} and x'_{-∞:t} are in the same causal state if P(X_{t:} | x_{-∞:t}) = P(X_{t:} | x'_{-∞:t}). The set of all such equivalence classes constitutes the causal state space.

The causal-state representation of a process is given by an epsilon-machine (ε-machine), a minimal, predictive, unifilar hidden Markov model whose states are the causal states. Transitions between causal states are triggered by observed symbols, with the next state determined by the current state and the observed symbol, and the emission probabilities defined by the conditional distributions over futures. This structure provides a compact, sufficient statistic for predicting future behavior and a transparent view of the process’s causal organization.

Two central quantities associated with causal states are the statistical complexity C_mu, the Shannon entropy of the stationary distribution over causal states, which measures the memory required for optimal prediction, and the entropy rate h_mu, the average uncertainty per symbol in the process.

Causal states can be inferred from data via causal-state reconstruction methods, such as CSSR (causal-state splitting reconstruction) or related CSR approaches, which cluster histories by their predictive distributions. The framework is used in time-series analysis, physics, and complex systems to quantify structure, build predictive models, and study the intrinsic information-processing capabilities of processes.