INCESSANT TRANSMISSION 9/4 



of the type we have considered throughout the book till now. The 

 single-valued, determinate, transformation is thus simply a special, 

 extreme, case of the stochastic. It is the stochastic in which all the 

 probabilities have become or 1 . This essential unity should not be 

 obscured by the fact that it is convenient to talk sometimes of the 

 determinate type and sometimes of the types in which the important 

 aspect is the fractionality of the probabilities. Throughout Part III 

 the essential unity of the two types will play an important part in 

 giving unity to the various types of regulation. 



The word "stochastic" can be used in two senses. It can be used 

 to mean "all types (with constant matrix of transition probabilities), 

 the determinate included as a special case", or it can mean "all 

 types other than the determinate". Both meanings can be used; but 

 as they are incompatible, care must be taken that the context shows 

 which is implied. 



THE MARKOV CHAIN 



9/4. After eight chapters, we now know something about how a 

 system changes if its transitions correspond to those of a single- 

 valued transformation. What about the behaviour of a system 

 whose transitions correspond to those of a stochastic transformation? 

 What would such a system look like if we met one actually working? 

 Suppose an insect lives in and about a shallow pool — sometimes 

 in the water (W), sometimes under pebbles (P), and sometimes on the 

 bank (B). Suppose that, over each unit interval of time, there is a 

 constant probability that, being under a pebble, it will go up on the 

 bank; and similarly for the other possible transitions. (We can 

 assume, if we please, that its actual behaviour at any instant is 

 determined by minor details and events in its environment.) Thus a 

 protocol of its positions might read : 



WBWBWPWBWBWBWPWBBWBWPWBWPW 

 BWBWBBWBWBWBWPPWPWBWBBBW 



Suppose, for definiteness, that the transition probabihties are 



\ 



B W P 



These probabilities would be found (S.9/2) by observing its 

 behaviour over long stretches of time, by finding the frequency of, 

 say, B-^ W, and then finding the relative frequencies, which are 



165 



