This is my current model of your problem:
You have a set S of start sites, each of which we can make propositions about. Each one of them has some position on the genome.
You're interested in looking at each of the start sites and assessing some property- "does this start site overlap the previous gene's stop site?" If that's true for the particular start site s, we say Q(s)=1; otherwise, Q(s)=0 (using 0 and 1 as synonymous with true and false). This is unknown, so we refer to our uncertainty as P(Q(s)), which might starts off as 1/S for all s, or might vary with the start site. Knowing P(Q(i)) doesn't tell us anything about P(Q(j)).
When we do an experiment, we get back an observation about s; suppose it signals either "heads" or "tails," which I'll shorten to H or T. We can calculate P(O(s)=H|Q(s)=1) and P(O(s)=H|Q(s)=0), and from that we can calculate the likelihood ratio used to update Q(s). Note that the likelihood ratio is dependent only on the probabilities of H, and thus is totally independent of the prior probability on Q(s).
We can do the experiments in batches- on, say, four sites at once. It will give a H or T reading for each start site, and the probabilities may depend on the number of sites measured at once. Thus, the likelihood ratio will be different based on the batch size- suppose we call a "heads" result when there are 2 sites tested H2.
Thus, we want to figure out, say, P(Q(s)|&, H2,T4,H4,H4). "&" stands for "all background knowledge," which will basically be our prior, and assuming multiple experiments are independent conditioned on Q(s), then we can just multiply the odds contributed by the prior and each of the tests to get one final estimate for Q(s). Assuming we started off with 1:20, H4 contributes 2:1, T4 contributes 1:2, and H2 contributes 4:1, we end up with 1*4*1*2*2:20*1*2*1*1=2:5, and so Q(s)=2/7.
(Rewritten entirely after seeing pragmatist's answer.)
In this post, helpful people including DanielLC gave me the multiply-odds-ratios method for combining probability estimates given by independent experts with a constant prior, with many comments about what to do when they aren't independent. (DanielLC's method turns out to be identical to summing up the bits of information for and against the hypothesis, which is what I'd expected to be correct.)
I ran into problems applying this, because sometimes the prior isn't constant across samples. Right now I'm combining different sources of information to choose the correct transcription start site for a gene. These bacterial genes typically have from 1 to 20 possible start sites. The prior is 1 / (number of possible sites).
Suppose I want to figure out the correct likelihood multiplier for the information that a start site overlaps the stop of the previous gene, which I will call property Q. Assume this multiplier, lm, is constant, regardless of the prior. This is reasonable, since we always factor out the prior. Some function of the prior gives me the posterior probability that a site s is the correct start (Q(s) is true), given that O(s). That's P(Q(s) | prior=1/numStarts, O(s)).
Suppose I look just at those cases where numStarts = 4, I find that P(Q(s) | numStarts=4, O(s)) = .9.
9:1 / 1:3 = 27:1
Or I can look at the cases where numStarts=2, and find that in these cases, P(Q(s) | numStarts=2, O(s)) = .95:
19:1 / 1:1 = 19:1
I want to take one pass through the data and come up with a single likelihood multiplier, rather than binning all the data into different groups by numStarts. I think I can just compute it as
(sum of numerator : sum of denominator) over all cases s_i where O(s_i) is true, where
numerator = (numStarts_i-1) * Q(s_i)
denominator = (1-Q(s_i))
Is this correct?