### 10. UNIQUENESS OF MAXIMUM-LIKELIHOOD SOLUTION

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 ðw / ð = 0. The maximum-likelihood solution for is obtained from ðw / 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.