Sion was hinted at in S.2/10, and we can now explore the possibil-
Ity of an operand having more than one transform. Some
Supplementary restriction, however, is required, so as to keep the
Possibilities within bounds and subject to some law. It must not
Become completely chaotic. A case that has been found to have
Many applications is that in which each operand state, instead of
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A N I N T R O D UC T I O N T O C Y B E R NE T I C S
I N CESSA N T TR AN SMI SSIO N
Being transformed to a particular new state, may go to some one
Of the possible states, the selection of the particular state being
Made by some method or process that gives each state a constant
Probability of being the transform. It is the unchangingness of the
Probability that provides the law or orderliness on which definite
Statements can be based.
Such a transformation would be the following: x' = x + a, where
The value of a is found by spinning a coin and using the rule Head:
a = 1; Tail: a = 0. Thus, if the initial value of x is 4, and the coin
Gives the sequence T T H H H T H T T H, the trajectory will be 4,
If the coin gives H T H H T T T H T T, the
Trajectory will be 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8. Thus the transforma-
Tion and the initial state are not sufficient to define a unique tra-
Jectory, as was the case in S.2/17; they define only a set of
Trajectories. The definition given here is supplemented by instruc-
Tions from the coin (compare S.4/19), so that a single trajectory is
Arrived at.
The transformation could be represented (uniformly with the
Previously used representations) as:
3
1/2
1/2
1/2
Ing Head, then the probability of its going to state 5 is 1/2 and so
Would be its probability of staying at 4.
↓ … 3
4
…
0
1/2
1/2
0
…
5
…
0
0
1/2
1/2
…
6 …
…
0
0
0
1/2
…
…
…
…
…
…
…
…
3
4
5
6
…
…
…
…
…
…
…
…
1/2
1/2
0
0
…
4
1/2
|
|
|
1/2
5
1/2
Etc.
3
4
4
5
5
6
Where the 1/2 means that from state 3 the system will change
With probability 1/2 to state 3,
And ,,,,,, ,, ,, 4.
Such a transformation, and especially the set of trajectories that it
May produce, is called “stochastic”, to distinguish it from the sin-
Gle-valued and determinate.
Such a representation soon becomes unmanageable if many
Transitions are possible from each state. A more convenient, and
Fundamentally suitable, method is that by matrix, similar to that
Of S.2/10. A matrix is constructed by writing the possible oper-
Ands in a row across the top, and the possible transforms in a col-
Umn down the left side; then, at the intersection of column i with
Row j, is put the probability that the system, if at state i, will go to
State j.
As example, consider the transformation just described. If the
System was at state 4, and if the coin has a probability 1/2 of giv-
162
All other transitions have zero probability. So the matrix can be
Constructed, cell by cell.
This is the matrix of transition probabilities. (The reader
Should be warned that the transposed form, with rows and col-
Umns interchanged, is more common in the literature; but the form
Given has substantial advantages, e.g. Ex. 12/8/4, besides being
Uniform with the notations used throughout this book.)
We should, at this point, be perfectly clear as to what we mean
By “probability”. (See also S.7/4.) Not only must we be clear
About the meaning, but the meaning must itself be stated in the
Form of a practical, operational test. (Subjective feelings of
“degree of confidence” are here unusable.) Thus if two observers
Differ about whether something has a “constant probability”, by
What test can they resolve this difference ?
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Probabilities are frequencies. “A ‘probable’ event is a frequent
Event.” (Fisher.) Rain is “probable” at Manchester because it is
Frequent at Manchester, and ten Reds in succession at a roulette
Wheel is “improbable” because it is infrequent. (The wise reader
Will hold tight to this definition, refusing to be drawn into such
Merely speculative questions as to what numerical value shall be
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