mathematics//dynamical systems//sufficient state
A **sufficient state** keeps all the information from the past that the model needs to predict the future, **given the future inputs**. It does not have to reconstruct the past; it has to distinguish the futures that can still differ.
A sufficient state keeps all the information from the past that the model needs to predict the future, given the future inputs. It does not have to reconstruct the past; it has to distinguish the futures that can still differ.
In a well-posed deterministic model, knowing x(t0)x(t_0)x(t0) and the later input selects the trajectory. In a stochastic model it determines a distribution of possible evolutions, not necessarily a certain outcome.
The present has to carry the right memory. In x˙=u−λx\dot x=u-\lambda xx˙=u−λx, two different histories that produce the same x(t0)x(t_0)x(t0) share the same future under the same later input: the relevant past has been compressed into one number. But with
x˙1=u−λ1x1,x˙2=u−λ2x2,y=x1+x2,\dot x_1=u-\lambda_1x_1,\qquad \dot x_2=u-\lambda_2x_2,\qquad y=x_1+x_2,x˙1=u−λ1x1,x˙2=u−λ2x2,y=x1+x2,
and λ1≠λ2\lambda_1\ne\lambda_2λ1=λ2, knowing only yyy is usually not enough. The same total can hold different proportions of fast and slow memory, and the instantaneous output does not reveal them. Mechanics is analogous: position alone cannot continue a second-order motion; position plus velocity can.
Markov property. For a discrete process without explicit inputs,
Pr(Sk+1∣Sk,Sk−1,…,S0)=Pr(Sk+1∣Sk).\Pr(S_{k+1}\mid S_k,S_{k-1},\ldots,S_0)=\Pr(S_{k+1}\mid S_k).Pr(Sk+1∣Sk,Sk−1,…,S0)=Pr(Sk+1∣Sk).
Given the current state, consulting earlier states does not improve the prediction inside the model. This does not mean physical absence of memory: the relevant memory is already in the state. It does not mean independence between successive states either, nor is it a synonym of Bayesian inference. A finite Markov chain is a particular case; a deterministic equation with a sufficient state shares the idea of sufficiency without being a random walk over a finite list of nodes.
Sufficient does not mean minimal, observed or controllable. A state can be sufficient and redundant, sufficient and partly hidden, sufficient and contain modes no input can act on. Input-output minimality is a separate question (controllability and observability).
With a pure delay, the state may need a segment of history rather than a finite vector. Enlarging the state represents many memories, but does not guarantee compressing them exactly into a few variables. See delay and lag.