652
L. Lyons
density by multiplying the likelihood by the prior. The Bayesian approach thus
obeys the likelihood principle, which states that the only way the experimental data
affects inference is via the likelihood function. In contrast, the Neyman construction
requires not only the likelihood for the actual data, but also for all possible data that
might have been observed.
Because the likelihood is not a probability density, it does not transform like one.
Thus the value of the likelihood for a parameter μ 0 is identical to that for λ 0 = 1/μ 0 .
This means that ratios of likelihoods (or differences in their logarithms) are useful to
consider, but that the integration of tails of likelihoods is not a recognised statistical
procedure.
A longer account of the Bayesian and frequentist approaches can be found in
ref. [28]. Reference [29] provides a very readable account for a Poisson counting
experiment.
15.2 Likelihood Issues
In this section, we discuss some potential misunderstandings of likelihoods.
15.2.1 (lnL) = 0.5 Rule
In the maximum likelihood approach to parameter determination, the best value
λ 0 of a parameter is determined by finding where the likelihood maximises; and
its uncertainty is estimated by finding how much the parameter must be changed 6
in order for the logarithm of the likelihood to decrease by 0.5 as compared with the
maximum. 7 From a frequentist viewpoint, this should ideally result in the parameter
range having 68% coverage. That is, in repeated use of this procedure to estimate
the parameter, 68% of the intervals should contain the true value of the parameter,
whatever its true value happens to be.
If the measurement is distributed about the true value as a Gaussian with constant
width, the likelihood approach will yield exact coverage, but in general this is not so.
For example, Garwood[31] and Heinrich[32] have investigated the properties of the
likelihood approach (and other methods too) to estimate μ, the mean of a Poisson,
when n obs events are observed. Because n obs is a discrete variable, the coverage is
6 If there are more than just one parameter, the likelihood must of course be remaximised with
respect to all the other parameters when looking for the = 0.5 points. Alternatively, a
region in multi-parameter space can be selected by finding the contour at which (lnL) decreases
from its maximum by an amount which depends on the number of parameters.
7 This (like several other methods) can give rise to asymmetric uncertainties. Techniques for dealing
with this have been discussed by Barlow[30].
L. Lyons
density by multiplying the likelihood by the prior. The Bayesian approach thus
obeys the likelihood principle, which states that the only way the experimental data
affects inference is via the likelihood function. In contrast, the Neyman construction
requires not only the likelihood for the actual data, but also for all possible data that
might have been observed.
Because the likelihood is not a probability density, it does not transform like one.
Thus the value of the likelihood for a parameter μ 0 is identical to that for λ 0 = 1/μ 0 .
This means that ratios of likelihoods (or differences in their logarithms) are useful to
consider, but that the integration of tails of likelihoods is not a recognised statistical
procedure.
A longer account of the Bayesian and frequentist approaches can be found in
ref. [28]. Reference [29] provides a very readable account for a Poisson counting
experiment.
15.2 Likelihood Issues
In this section, we discuss some potential misunderstandings of likelihoods.
15.2.1 (lnL) = 0.5 Rule
In the maximum likelihood approach to parameter determination, the best value
λ 0 of a parameter is determined by finding where the likelihood maximises; and
its uncertainty is estimated by finding how much the parameter must be changed 6
in order for the logarithm of the likelihood to decrease by 0.5 as compared with the
maximum. 7 From a frequentist viewpoint, this should ideally result in the parameter
range having 68% coverage. That is, in repeated use of this procedure to estimate
the parameter, 68% of the intervals should contain the true value of the parameter,
whatever its true value happens to be.
If the measurement is distributed about the true value as a Gaussian with constant
width, the likelihood approach will yield exact coverage, but in general this is not so.
For example, Garwood[31] and Heinrich[32] have investigated the properties of the
likelihood approach (and other methods too) to estimate μ, the mean of a Poisson,
when n obs events are observed. Because n obs is a discrete variable, the coverage is
6 If there are more than just one parameter, the likelihood must of course be remaximised with
respect to all the other parameters when looking for the = 0.5 points. Alternatively, a
region in multi-parameter space can be selected by finding the contour at which (lnL) decreases
from its maximum by an amount which depends on the number of parameters.
7 This (like several other methods) can give rise to asymmetric uncertainties. Techniques for dealing
with this have been discussed by Barlow[30].