I asked this question on Philosophy Stack Exchange: Epistemic value of multiple eyewitness accounts: single event vs. multiple events given a fixed number of eyewitnesses?
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.?
Note: 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.
I received the following suggestions in the comments:
Interesting question, but probably would get better answers on math.stackexchange.com or statistics.stackexchange.com ! If a witness claims they have seen event E_1, they might be lying or hallucinating, or someone else might have staged E_1. So you have to account for P(E_1 | eyewitness account), but also for P(X | E_1), which is the probability that aliens exist knowing that a possibly-staged alien abduction event happened.
You may get better answers on Cross Validated. Under some simplifying assumptions (like equal credibility of reports, independent events, etc.), this is essentially a mathematical question about calculating Bayesian posteriors. The formula for multiple reports of a single event is standard, but I am not sure how they are calculated for derivative statements implied by multiple reported events. You may want to use a less provocative example than alien abductions there, so it doesn't become a distraction.
Question: How can I adapt this question to fit CV on-topic standards?
