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It can be proved that when a set of states is used to build a prediction model, their statistical complexity is approximately equivalent to the size of the prediction model.
It can be proved that when a set of states is used to build a predictive model, their statistical complexity is approximately equivalent to the size of the prediction model.
Furthermore, to balance predictability and simplicity, computational mechanics defines [[causal equivalence]]. If <math>P\left ( \overrightarrow{s}|\overleftarrow{s}\right )=P\left ( \overrightarrow{s}|{\overleftarrow{s}}'\right )</math>, then <math>\overleftarrow{s}</math> and <math>{\overleftarrow{s}}'</math> are causally equivalent. This equivalence relation can divide all historical processes into equivalence classes and define them as [[causal states]]. All causal states of the historical process <math>\overleftarrow{s}</math> can be characterized by a mapping <math>\epsilon \left ( \overleftarrow{s} \right )</math>. Here, <math>\epsilon: \overleftarrow{\mathcal{S}}\rightarrow 2^{\overleftarrow{\mathcal{S}}}</math> is a function that maps the historical process <math>\overleftarrow{s}</math> to the causal state <math>\epsilon(\overleftarrow{s})\in 2^{\overleftarrow{\mathcal{S}}}</math>.
Furthermore, to balance predictability and simplicity, computational mechanics defines [[causal equivalence]]. If <math>P\left ( \overrightarrow{s}|\overleftarrow{s}\right )=P\left ( \overrightarrow{s}|{\overleftarrow{s}}'\right )</math>, then <math>\overleftarrow{s}</math> and <math>{\overleftarrow{s}}'</math> are causally equivalent. This equivalence relation can divide all historical processes into equivalence classes and define them as [[causal states]]. All causal states of the historical process <math>\overleftarrow{s}</math> can be characterized by a mapping <math>\epsilon \left ( \overleftarrow{s} \right )</math>. Here, <math>\epsilon: \overleftarrow{\mathcal{S}}\rightarrow 2^{\overleftarrow{\mathcal{S}}}</math> is a function that maps the historical process <math>\overleftarrow{s}</math> to the causal state <math>\epsilon(\overleftarrow{s})\in 2^{\overleftarrow{\mathcal{S}}}</math>.