Another approach for evaluating significance
is to directly compare the likelihoods for two competing
hypotheses [143,146].
Let the first hypothesis, 
, be that the data are due
entirely to background, with its cross section calculated from the
counting experiment. Let the second hypothesis, 
, be that there
is also top in the data, with a mass and cross section given by the
maximum likelihood fit. Then, by Bayes' theorem, the likelihood ratio
is
Note that
and
The individual likelihoods are then
The factor 
is given
by equation (7.28).
The corresponding factor for , 
, is
the same
thing with 
set to zero:
The priors for the parameters are taken to be gaussians.
The result of the background calculation is used for 
, while
and 
are taken from the result of the maximum
likelihood estimation. For definiteness, the priors
and 
are taken to be equal
( = = 1 / 2).
The results of this calculation are 
for the
loose cuts, and 
for the standard cuts.
For R this small, 
R, so the interpretation is
that the data are very unlikely to be due entirely to background.
As a check, the calculation was redone using the loose cut parameters
on a sample consisting of 12 (
) background events.
The result was R = 1.03, or 50%.
These results complement the confidence limit results of the previous section. Both indicate that the data are very unlikely to be entirely due to the known backgrounds. The confidence limit result, however, does not say anything about the plausibility of the top hypothesis; the likelihood ratio test shows that the top hypothesis is indeed much more plausible than the background-only hypothesis.