Usually it is a matter of taste what physical quantity is
chosen as . For example, in
a lifetime experiment some workers would solve for the lifetime,
*, while others would solve for
*, where
=
1/. Some workers prefer to use
momentum, and
others energy, etc. Consider the case of two related physical
parameters and
. The maximum-likelihood
solution for is
obtained from the equation
ð*w* / ð = 0. The
maximum-likelihood
solution for is obtained
from
ð*w* / ð =
0. But then we have

Thus the condition for the maximum-likelihood solution is unique and independent of the arbitrariness involved in choice of physical parameter. A lifetime result * would be related to the solution * by * = 1/*.

The basic shortcoming of the maximum-likelihood method is
what to do about the prior probability of
. If the prior
probability of is
*G*() and the
likelihood function obtained for the experiment alone is
(), then the joint likelihood
function is

give the maximum-likelihood solution. In the absence of any
prior knowledge the term on the right-hand side is zero. In
other words, the standard procedure in the absence of any prior
information is to use a prior distribution in which all values
of are equally
probable. Strictly speaking, it is impossible
to know a "true" *G*(),
because it in turn must depend on its
own prior probability. However, the above equation is useful
when *G*() is the
combined likelihood function of all previous experiments and
() is the likelihood function of
the experiment under consideration.

There is a class of problems in which one wishes to determine
an unknown distribution in
,
*G*(), rather than a single
value . For example, one may
wish to determine the momentum
distribution of cosmic ray muons. Here one observes

where (; *x*) is known from the
nature of the experiment and
G() is the function to be
determined. This type of problem is discussed in Reference 5.