<p>I asked this question on Philosophy Stack Exchange: <a href="https://philosophy.stackexchange.com/q/93007/48437">Epistemic value of multiple eyewitness accounts: single event vs. multiple events given a fixed number of eyewitnesses?</a></p>
<blockquote>
<p>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).</p>
<p>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>
<p>P(X is true | multiple eyewitness accounts) &gt; P(X is true | a single
eyewitness account)</p>
<p>However, what happens if we keep the number of eyewitness accounts
constant and only change the number of events?</p>
<p>For example, let X = &quot;alien abductions are real&quot;, and let
E<sub>i</sub> be a concrete example of an (alleged) alien abduction. X
is a general claim, E<sub>i</sub> is a claim about a very specific
instance of X. It is clear that E<sub>i</sub> entails X. Thus, which
of the following probabilities is greater than the others?</p>
<ul>
<li>P(E<sub>1</sub> is true | N eyewitness accounts for E<sub>1</sub>)</li>
<li>P(E<sub>1</sub> is true or E<sub>2</sub> is true or ... or E<sub>N</sub> is true | one eyewitness account for E<sub>i</sub>, for
each i in {1, ..., N})</li>
<li>P(E<sub>1</sub> is true or E<sub>2</sub> is true or ... or E<sub>N/2</sub>  is true | two eyewitness accounts for E<sub>i</sub>,
for each i in {1, ..., N/2})</li>
<li>P(E<sub>1</sub> is true  or E<sub>2</sub> is true or ... or E<sub>N/3</sub> is true | three eyewitness accounts for E<sub>i</sub>,
for each i in {1, ..., N/3})</li>
<li>Etc.</li>
</ul>
<p>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 E<sub>1</sub>, 500 eyewitness
accounts for E<sub>1</sub> + 500 eyewitness accounts for
E<sub>2</sub>, etc.?</p>
<p><strong>Note</strong>: 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.</p>
</blockquote>
<p>I received the following suggestions in the comments:</p>
<blockquote>
<p><strong>Interesting question, but probably would get better answers on math.stackexchange.com or statistics.stackexchange.com !</strong> 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.</p>
</blockquote>
<blockquote>
<p><strong>You may get better answers on Cross Validated</strong>. 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.</p>
</blockquote>
<p><strong>Question</strong>: How can I adapt this question to fit CV on-topic standards?</p>
