9.4 Likelihood

111

random walk; cf. Chap. 11). The probability distribution of the net displacement

after nn steps is the binomial function. The central limit theorem guarantees that

this distribution is Gaussian as n right arrow normal infinityn →∞, a universal property of random additive

processes.

Although their formalism is less familiar, random multiplicative processes (RMP)

are not less common in nature. An example is rock fragmentation. From an initial

value x 0x0, the size of a rock undergoing fragmentation evolves as x 0 right arrow x 1 right arrow x 2 right arrow midline horizontal ellipsis right arrow x Subscript upper Nx0 →x1 →x2 →

· · · →xN. If the size reduction factor

r Subscript n Baseline equals StartFraction x Subscript n Baseline Over x Subscript n minus 1 Baseline EndFractionrn =

xn

xn−1

(9.47)

is less than 1, we have

x Subscript upper N Baseline equals x 0 product Underscript k equals 1 Overscript upper N Endscripts r Subscript k Baseline periodxN = x0

N

|

k=1

rk .

(9.48)

Extreme events, although exponentially rare, are exponentially different. Hence, the

average is dominated by rare events. This is quite different from the more intuitively

acceptable random additive process. If the phenomenon is of that type, the more

measurements one can take, the better the estimate of its value. However, if the

phenomenon is an RMP, as one increases the number of measurements, the estimate

of the mean will fluctuate more and more, before ultimately converging to a stable

value. Since multiplication is equivalent to adding logarithms, it is not surprising that

the distribution of the result of an RMP is lognormal (i.e., ln p equals sigma summation ln p Subscript iln p = ^E ln pi), and the

average value (expectation) of pp is

p overbar equals sigma summation Underscript n equals 0 Overscript upper N Endscripts left parenthesis upper N n right parenthesis p Superscript n Baseline q Superscript upper N minus n Baseline period ¯p =

N

E

n=0

(Nn)pⁿq ^N−n .

(9.49)

9.4

Likelihood

The search for regularities in nature has already been mentioned as the goal of

scientific work. Often, these regularities are framed in terms of hypotheses. ¹⁷ With

hypotheses (which may eventually become theories), laws and relations acquire more

than immediate validity and relevance (cf. unconditional information, Sect. 6.1.1).

17 Strictly speaking, one should instead refer to propositions. A hypothesis is an asserted proposition,

whereas at the beginning of an investigation it would be better to start with considered propositions,

to avoid prematurely asserting what one wishes to find out. Unfortunately, the use of the term

“hypothesis” seems to have become so well established that we may risk confusion if we avoid

using the word.