MARKOVIAN SYSTEM
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 2014 ▶ |
| Object type | General information, Methodology or model |
G. PASK states: “The probability that a Markovian system will occupy any of its states at t+1, depends only upon its state at t, and probabilistic transformation P made up of fixed transition probabilities P$_{ij}$” (1961, p.42).
Thus, its behavior is random only in some sense: It is impossible to predict with total certainty the next states of the system, but the matrix of the fixed probabilities of transition between the different states gives us three important informations:
- the number of possible states is finite, however enormous it may be. In other words, the system is more or less constrained in AHSBY's sense.
- As the transition probabilities are fixed, some states are more probable and frequent than other ones. Transition probabilities are constraints, but they merely restrict, but do not suppress randomness.
- When the possible states are numerous, most specific transition probabilities become very small and nearly equi-probable. This consequently reduces predictability.
In PASK's words: “…an observer could perfectly well appreciate much longer-term dependencies” (p.43). The observer “could”, but generally does not because the Markovian system has no precise memory of its past, but merely a general statistical one, which does not permit any secure prediction.
A population's behavior is generally markovian, as can be seen in some examples given by ASHBY (1956).
If, as observed by ASHBY some specific state of the system has zero probabilities of transition unto any other state, the system becomes stuck in this state. Such a situation , most of the time overlooked, generally implies its destruction.
PASK states moreover: “A distribution P$_{i}$(t) is the state of the Markovian system and a sequence of distributions is a behavior of the Markovian system, conditional upon the chosen initial state” (p.44).
And still: “By analogy with a state determined system any Markovian system reaches statistical equilibrium” and “In equilibrium…, regarded as an information source, it has a measurable variety. For n states, the maximum variety is Log$_{2}$n, the variety of the reference frame, without any statistical constraints” (p.45).
Another significant property of the Markovian system is that it has no “memory”. In PASK's words: “Memory or the ability to learn is not a property of the system, but of the relation between the system and an observer” (p.46). Indeed, it is the observer who construct the statistical matrix, after repeated observations. Even so, more observations could lead to a still different matrix.
A Markovian system is metastable: within its global stability, it undergoes stochastic jumps from state to state.
Under some constraints, Markovian systems become ergodic. Conversely, as observed by PASK, some ergodic systems include a “trapping” state, thus named because the system must end up in this single state (p.123).
A very important comment of PASK is that: “A system which is Markovian when observed in n states may not be Markovian if the observer combines some of its states and inspect a less detailed image” (p.125).
Thus, the 'lumping' of states (intentional or not), may very well deeply modify our understanding of the system.