114

9

Probability and Likelihood

and

upper P left brace upper R 1 and upper R 2 vertical bar upper H right brace equals upper P left brace upper R 1 vertical bar upper H right brace upper P left brace upper R 2 vertical bar upper H right brace periodP{R1 and R2|H} = P{R1|H}P{R2|H} .

(9.52)

The method of likelihood reposes on the definitions of likelihood per se and of the

likelihood ratio.

Example. The problem is to determine the probability that a baby will be a boy. We

take a binomial model (cf. Sect. 9.2.3) for the occurrence of boys and girls in a family

of two children; we have two sets of data—upper R 1R1: one boy and one girl; and upper R 2R2: two

boys—and two hypotheses—upper H 1H1: the probabilitypp of a birth being male born equals

one fourth 1

4; and upper H 2H2: p equals one halfp = 1

2. Hence,

StartLayout 1st Row StartLayout 1st Row 1st Column upper P left brace upper R vertical bar upper H right brace 2nd Column upper R 1 3rd Column upper R 2 2nd Row 1st Column upper H 1 2nd Column 2 p left parenthesis 1 minus p right parenthesis equals three eighths 3rd Column p squared equals one sixteenth 3rd Row 1st Column Blank 4th Row 1st Column upper H 2 2nd Column 2 p left parenthesis 1 minus p right parenthesis equals one half 3rd Column p squared equals one fourth EndLayout period EndLayout

P{R|H}

R1

R2

H1

2p(1 −p) = 3

8 p2 =

1

16

H2

2p(1 −p) = 1

2 p2 = 1

4

.

By inspection,upper P left brace upper R vertical bar upper H right braceP{R|H} forupper H 2H2 exceeds that forupper H 1H1 for both sets of data, from which

we may infer that upper H 2H2 is better supported by the data.

The concept of likelihood ratio can easily be extended to continuous distributions;

that is,upper P left brace upper R vertical bar upper H right braceP{R|H} becomes a probability density. The likelihood ratio is computed for

the distribution with respect to one value chosen arbitrarily and the maximum is

sought. Usually it is better to work in logarithms, and the support German upper SS is defined as the

logarithm of the likelihood, namely

German upper S left parenthesis p right parenthesis equals log upper L left parenthesis p right parenthesis periodS(p) = log L(p) .

(9.53)

The curvature of German upper S left parenthesis p right parenthesisS(p) at its maximum has been called the information, and its

reciprocal is a natural measure of the uncertainty aboutpp (i.e., the width of the peak

is inversely related to the degree of certainty of the estimation).

The method of maximum likelihood provides the ability to deliver a conclusion

compatible with the given evidence.

References

Feller W (1967) An introduction to probability theory and its applications, vol 1, 3rd edn. Wiley,

New York

Mood AM (1940) The distribution theory of runs. Ann Math Statist 11:367–392

Planck M (1932) The concept of causality. Proc Phys Soc 44:529–539

Sommerhoff G (1950) Analytical biology. Oxford University Press, London

von Mises R (1931) Wahrscheinlichkeitsrechnung. Deuticke, Leipzig