Probability is a way of comparing observed data or results for a given hypothesis. Likelihood on the other hand is a way of comparing hypotheses for a given set of data or results. The two are related, differing only by an arbitrary constant, with the likelihood of a hypothesis given the data or results being proportional to the probability of the data given the hypothesis (Edwards 1972). When hypotheses are compared, this is conveniently done in the form of likelihood ratios or relative likelihoods, L, ratios of any individual likelihood to the maximum, scaled from 0 to 1. With the maximum likelihood scaled to 1, as it is in the following calculations, the likelihood ratios are just the individual likelihoods, which are in turn upper limits of the individual probabilities of the data for each hypothesis.
Likelihood estimation involves comparison of the relative likelihoods of different hypotheses concerning t1 and ETD (Fig. 1), where the likelihood of a particular hypothesis is proportional to the probability of the observed results, n samples in OTD, for that hypothesis: k ˇ P(OTD, n|ETD), with k being an arbitrary constant. Given the geometric model shown here and the observed results, n samples falling in OTD, the hypothesis about t1 and ETD that has maximum likelihood is the hypothesis that t1 = t2 and ETD = OTD. No matter what the size of n, ETD = OTD has maximum likelihood because the exponentiated quotient (OTD/ETD)n = (OTD/OTD)n = 1n = 1, and by definition no probability of observed data can be greater than 1. Maximum likelihood, by convention, has an associated likelihood ratio L of k ˇ P divided by itself: L = (k ˇ P)/(k ˇ P) = 1, and competing likelihood ratios are necessarily smaller, lying in the range 0 to 1. Note that while P is greater than or equal to 1, there is always a likelihood ratio L = 1 and L is consequently an upper bound for P.
It is useful to distinguish sets of hypotheses that satisfy some minimal likelihood criterion, and we are here interested in all hypotheses for which likelihoods exceed the critical likelihood l = 0.5 and all hypotheses for which likelihoods exceed l = 0.05, that is, hypotheses with at least 1 in 2 chances of occurrence (for which the betting odds are an even "50-50") or at least 1 in 20 chances of occurrence (for which the odds are "5-95"). These lambdas are upper limits for ordinary levels of significance a = 0.5 and a = 0.05 and hence define conservatively narrow 50% and 95% confidence limits for t2 and OTD in terms of an hypothesized origination time t1 and ETD. In the uniform case, as before,
l = (OTD/ETD)n-1 (4)
and
ETD = OTD / (n-1)Öl (5)
We are interested to know how large we can
make ETD and still expect all samples from ETD to fall in OTD in
1 out of 2 or in 1 out of 20 trials—in other words, how
large can ETD be and still yield OTD some small, but still
reasonable, proportion of the time?
The hypotheses being compared are
constructed to differ by some number of arbitrarily small
increments i added to OTD, and there is in theory no
limit on the fineness of the increment nor on the number of
incremented hypotheses that can be compared. Increments of i
are added to OTD until the sum ETD satisfies equations 4 and 5.
OTD, n, and l are, of course, known or specified in
advance. Relationship of ETD to OTD is not sensitive to n
nor to l if n is in the range considered
here (Fig. 2
and Fig. 3).
Increments of i added to OTD correspond to addition of
some amount of time to t2. The time involved
is proportional to i in the uniform case, but not in the
nonuniform models considered here.