Main trends of research in the social and human ... - unesdoc - Unesco

546 Raymond Boudon
By substracting P.(t) from both sides, dividing by At and making At tend
towards zero, we get the differential equation
-= dPx(t)
-kxPx (t) + k(x-~)P,-, (1)
dt
which is resolved :
Px(t) = C::? e--Kx~t(l-e--Kf)x-xo
This equation, the expression of the ‘Yule-Ferry’ law, is obviously more
complicated than the corresponding deterministic equation. Its advantage is
that it describes the probability of finding x converts at time t, instead of asserting,
like the corresponding deterministic model, that x converts wil be found
at time t. In this specific case, assuming that we observe the process described a
great many times with attention to the number of converts we can expect to find
on average at time t, we shall find that the stochastic model gives average
values of Px(t), whose expression is simply the expression yielded by the deterministic
model. In other words, average of
~,(t)
03
= C XP,(~) = xoeKt
XFO
Here then we have a specimen of the cases where the deterministicmodel can
be regarded as expressing the states of highest probability of the corresponding
stochastic process. But sometimes the average values of Px(t) do not obey the
same law as the corresponding deterministic model. Hence the value of the
theory of stochastic processes : when the mechanisms representing the passage
of the system from one state to another are properly expressed, it is possible
firstly to deduce from them the degree of probability of each possible state of
the system at any moment, and secondly to observe that the sequence of most
probable states of the system does not match the answer given by any simple
deterministic model. Further, it is unquestionably easier to formulate reason-
able hypotheses on the mechanisms of the changes of state than on the overall
changes.
Taken together with the observed fact that several disciplines are tending to
abandon deterministic models in favour of stochastic models, the above con-
siderations seem to indicate that one of the essential lines of research in the
field of mathematical methodology should be on wider applications of the
theory of stochastic processes to human phenomena.
Already most of the social sciences can instance some few applications of the
theory of stochastic processes.
In psychology, learning theory makes extensive use of Markov chain theory
while applications of the Poisson processes are to be found in psychometrics.
In anthropology, stochastic processes are used in the models for kinship
structure analysis propounded by Harrison Whyte. In linguistics, the use of
the Markov chain theory dates from the work of Markov himself, as this
theory was first formulated in connexion with an analysis of the successions of
consonants and vowels in Eugene Onegin. Also worth noting in this regard
are Simon’s analyses of Zipf‘s law with the help of a stochastic-type formaliza-