How to model the probability of a truth claim given an arrangement of eyewitness accounts supporting specific instances of that claim? Note: the title is adapted from my Philosophy.SE question Epistemic value of multiple eyewitness accounts: single event vs. multiple events given a fixed number of eyewitnesses?. See meta discussion here.

Intuitively speaking, multiple independent eyewitness accounts of a single event are more convincing than a single eyewitness account. For example, multiple independent eyewitness accounts of a loud explosion in a remote area (e.g. from different locations and viewpoints) are more convincing/reliable than a single account (e.g. maybe the single witness hallucinated the explosion).
A bit more formally, if we define X as some truth claim about some event, process or phenomenon in the real world, we could say that:
P(X is true | multiple eyewitness accounts) > P(X is true | a single eyewitness account)
However, what happens if we keep the number of eyewitness accounts constant and only change the number of events?
For example, let X = "alien abductions are real", and let Ei be a concrete example of an (alleged) alien abduction. X is a general claim, Ei is a claim about a very specific instance of X. It is clear that Ei entails X. Thus, which of the following probabilities is greater than the others?

*

*P(E1 is true | N eyewitness accounts for E1)

*P(E1 is true or E2 is true or ... or EN is true | one eyewitness account for Ei, for each i in {1, ..., N})

*P(E1 is true or E2 is true or ... or EN/2  is true | two eyewitness accounts for Ei, for each i in {1, ..., N/2})

*P(E1 is true  or E2 is true or ... or EN/3 is true | three eyewitness accounts for Ei, for each i in {1, ..., N/3})

*Etc.

In other words, given N eyewitness accounts, what is the optimal distribution of eyewitnesses over specific alleged instances of X that maximizes the probability of X being true? What should be more convincing, 1000 eyewitness accounts for E1, 500 eyewitness accounts for E1 + 500 eyewitness accounts for E2, etc.?
Notice that I used eyewitness reports of abductions by aliens as an illustrative example, but the reasoning can be extended to other rare events, such as reports of miracles, paranormal phenomena, angelic encounters, Bigfoot sightings, testimonials from whistleblowers (conspiracy theories), etc.

You use an expression like 'P(E1 is true | N eyewitness accounts for E1)', but I assume that the question is not whether E1 is true but instead whether X is true.
Instead you might consider the probability that $X$ is true given the number of eyewitnesses $N_1, N_2, etc.$
$$P(\text{$X$ true} | N_1, N_2, \dots, N_k)$$
This can be expressed with Bayes' rule
$$P(\text{$X$ true} | N_1, N_2, \dots, N_k) = \frac{P(N_1, N_2, \dots, N_k |  \text{$X$ true})}{P(N_1, N_2, \dots, N_k ) } P(\text{$X$ true})$$
This expresses a computation of posterior probability or believe (given the eyewitnesses) in terms of the prior probability or believe (without the eye witnesses).
The change of $P(\text{$X$ true})$ to $P(\text{$X$ true} | N_1, N_2, \dots, N_k)$ depends on the probability of the eyewitnesses given that $X$ is true and also given that $X$ is false.
You can also express it as a ratio with the probability $P(\text{$X$ false})$
$$\frac{P(\text{$X$ true} | N_1, N_2, \dots, N_k) }{P(\text{$X$ false} | N_1, N_2, \dots, N_k) } = \frac{P(\text{$X$ true})}{ P(\text{$X$ false})}   \cdot \frac{P(N_1, N_2, \dots, N_k |  \text{$X$ true})}{P(N_1, N_2, \dots, N_k |  \text{$X$ false})} $$
The odds for $X$ being false or true change depending on whether the eyewitnesses are more or less probable given $X$ being false or true.
The problem in cases of 'esoteric theories' like alien abductions is

*

*that the previous odds are very low. "extraordinary claims require extraordinary evidence"


*and the observations like eyewitnesses do not change it much.

*

*When alien abductions are true, then it is very probable to have eye witnesses for it.

*But, when alien abductions are false then it is also very probable to have eye witnesses for it.

(Because it is not true does not mean that there won't be eyewitnesses that in some way believe they saw or experienced abduction)


*Also problematic is that these mathematical formulations don't capture the entire situation very well. Asside from the number of eyewitnesses it is also important what the quality of the eyewitnesses is.
How did people experience and witness the events? For instance during the broadcast of war of the world's in 1938 a lot of people thought that aliens were landing on earth. But, they all saw (listened to) the same event where the event itself was fake. The number of witnesses might not be very useful if the event is not useful. It also matters whether some event is contrary to other previous believes (for instance do the spaceships and aliens disobey known physical laws or not).
But, possibly for some less extreme theories it might be possible to fill in some of the terms. An example could be finding the landing place of a crashing airplane that was undetected by the radar. Then the theories to consider are not whether the plane crashed or not, but more like how it had been flying. In this case observations along with triangulation may help to reduce the distribution of theories and pinpoint the crash site.
For events like meteors and earthquakes these eyewitness accounts are gathered. (I am not sure how useful it is for meteors since most often a camera is capturing it now. In the case of earthquakes it can help to learn about the spread and impact of an earthquake)
An example:

