Physics and Engineering of Radiation Detection

534 Chapter 9. Essential Statistics for Data Analysis

If we now takeNmeasurements, the joint probabilities associated with a particular
result of each of these hypotheses will be given by

dp 1 =

∏N

i=1

f 1 (xi)dxi and

dp 2 =

∏N

i=1

f 2 (xi)dxi. (9.3.17)

However we are not interested in these individual probabilities since our aim is
to judge the first hypothesis against the other. In other words we want to know
the odds that hypothesis 1 is true against hypothesis 2 (or vice versa). This can be
quantified by using the so calledlikelihood ratiodefined as

R=

∏N

i=1

f 1 (xi)

f 2 (xi)

(9.3.18)

This ratio tells us how much faith we should put into a hypothesis against the
other but does not in any way rule out the possibility of other hypotheses being more
correct than these two. This is especially true for situations where a large number
of hypotheses can be associated with the experiment. Let us suppose that we have
an infinite number of hypotheses, which can be represented by a continuous variable
hof the normalized probability density functionf(h;x). The joint probability that
a particular hypothesis is true can then be obtained by taking the product of all
the individual distributionsf(h;xi) associated with each of the experimental results
x 1 ,x 2 , ...., xN. Thisiscalledthelikelihood functionand is represented by

L(h)=

∏N

i=1

f(h;xi). (9.3.19)

The likelihood functionL(h) is a distribution function ofhand can assume any
shape depending on the probability density functions from which it has been derived.
If we plot this function with respect toh, the most probable value ofh(generally
represented byh∗) will be the value at whichL(h) is maximum (see Fig.9.3.2).
However in terms of computing this mathematically, since most of the probability
density functions are exponential in nature therefore generally the natural logarithm
of this function is used used instead. This function defined as

l(h)=ln(L(h)), (9.3.20)

is commonly known aslog-likelihood function. h∗ can then be found by simply
equating the derivative of this function with respect tohto zero, that is

∂l(h)

∂h

= 0 (9.3.21)

Although the most probable value as obtained from the expression above is very
useful, still it alone is not a faithful representation of the function since it does not