well taking your lottery example. if person 1 is lying and person 2 is lying, then they seem equal, if you don't know whether they are lying or not.

Is "Person 2"'s claim have a slightly higher probability of being correct due to the number of eye witnesses they claim are available? I'm just talking about the claim on its own, without doing further investigation like questioning the witnesses or any other analysis. — coolguy8472

According to a Bayesian view of evidence, the answer is yes. Though this of course does not alter the likelihood of winning the lottery in and of itself. An alternative interpretation is to say person 2 theory has higher predictive power (it predicts the statements of 10 people) and is easier to falsify (you can ask any of the 10).

More observations give more evidence, as a general rule. Say we have n of them. However, if you repeat the same observation n times - like copy pasting a row of a spreadsheet - you have no more information than a single observation. The number of independent observations is between 1 and the number of reports n. In anecdotal examples the true number of independent reports usually is somewhere close to 1 or somewhere close to the number of reports.

The scenarios when it's somewhere close to the number of reports are when you have something like a controlled randomised double-blind experiment. The scenarios where it's close to 1 are when each subject has either been conjured into existence or is likely to have exactly the same reaction to the treatment/question due to unobserved factors; correlating their responses/beliefs strongly.

I leave it up to you whether the anecdotal evidence in the bible resembles more strongly people passing along and writing down stories with a selection mechanism that makes only strong believers write stories or a controlled and randomised double blind experiment.

First, the only way we could establish that the number of witnesses testifying to something implies that it has a greater probability of being the case would be if we had a large set of data showing, for multiple scenarios, that there is some correlation to how many witnesses there are relative to whether something turned out to be the case, where the latter was checked via independent means.

In other words, we'd need actual frequentist data to plausibly support a probability claim, in my opinion.

Even with the frequentist data, however, there would still be a number of problems to overcome. That's because there are so many different variables that can come into play. Making a probability claim on this sort of frequentist data implies that we're parsing the witnesses as ideal--no sort of bias, no sort of hidden agenda, no perceptual problems, ideally intelligent and rational, etc., and it also implies that we're assuming they have a more or less ideal access to information. Otherwise there would be no way to establish that the correlation is implicational, and that's what you'd be looking for here.

Person 1 Claim: "I won the lottery, my friend saw the ticket and can confirm"
Person 2 Claim: "I won the lottery, 10 people were saw the ticket and can confirm" — coolguy8472

Not to be too nit-picky, but these statements are equivalent. In both cases, the only information provided is your claim. If adequate documentation from the other observers is provided, which it hasn't been here, that will probably change.

So, do you have independent confirmation of the biblical observations by the people who observed, or only claims by the writer?

It's easier to get one or two people to lie and generally share a similar story, but when one gets to hundreds of people claiming they saw the same thing (considering the various perspectives of course) this in and of itself would be a miracle. Then when you add to that, people willing to die because of what they saw and therefore believed, that also is something that normally doesn't happen. If one saw something and knew it to be fake and a lie but chose to suffer and die for it anyway, then that is very rare.

First, the only way we could establish that the number of witnesses testifying to something implies that it has a greater probability of being the case would be if we had a large set of data showing, for multiple scenarios, that there is some correlation to how many witnesses there are relative to whether something turned out to be the case, where the latter was checked via independent means. — Terrapin Station

While @fdrake has already provided a fairly in-depth post on the value of multiple accounts, for most everyday examples it seems fairly self-evident that multiple witnesses increase the probability of the event having occurred. It's unlikely that multiple people hallucinate similar observations.

Even with the frequentist data, however, there would still be a number of problems to overcome. That's because there are so many different variables that can come into play. Making a probability claim on this sort of frequentist data implies that we're parsing the witnesses as ideal--no sort of bias, no sort of hidden agenda, no perceptual problems, ideally intelligent and rational, etc., and it also implies that we're assuming they have a more or less ideal access to information. Otherwise there would be no way to establish that the correlation is implicational, and that's what you'd be looking for here. — Terrapin Station

The witnesses need not be ideal. It's sufficient that every individual witness account has a non-zero probability of relating the true event.

Person 1 Claim: "I won the lottery, my friend saw the ticket and can confirm"
Person 2 Claim: "I won the lottery, 10 people were saw the ticket and can confirm" — coolguy8472

There no difference, because in both cases it is just a claim that YOU are making. If it's a lie, it's just a somewhat bigger lie to claim 10 people have confirmed.

On the other hand, if 10 people actually tell me they saw your winning ticket, that increases the epistemic probability to me that you actually won.

What got me thinking about it was accessing biblical accounts of miracles that claim many eyewitnesses and wondering if the claim of more eyewitnesses adds any credibility to the claim or not. — coolguy8472

There is the claim of the miracle and the claim of eyewitnesses. Cut to the chase. What reason can you give to affirm that miracles occur? Please note I did not say "believe"; belief is a separate subject.

Your question assumes the account is accurate. "Accurate" needs a good deal of explication, here, that I hope is unnecessary. But given 150 years of increasing clarity about the writing and compilation of the Bible, I argue that acceptance must occur under the aegis of belief - not subject to argument or reason. Which itself is not at all unreasonable, because plowed up to the bedrock, everything we hold to be so is in some way based in belief - a/k/a presuppositions.

Beliefs, in turn, are not held for the purpose of being true, but for being efficacious.The original Christian thinkers, the Patristic fathers, were not confused on this point, nor thinkers after them. But a lot of people are, and waste tremendous time and energy on questions of the truth of the beliefs, not realizing that the question is a nonsense question - with respect to the thing questioned.

There no difference, because in both cases it is just a claim that YOU are making. If it's a lie, it's just a somewhat bigger lie to claim 10 people have confirmed.

On the other hand, if 10 people actually tell me they saw your winning ticket, that increases the epistemic probability to me that you actually won. — Relativist

Good catch.

Which would also be the answer to:

That was my non-supernatural example. What got me thinking about it was accessing biblical accounts of miracles that claim many eyewitnesses and wondering if the claim of more eyewitnesses adds any credibility to the claim or not. — coolguy8472

Just cause Matthew says so and so many people saw miracle X, doesn't mean they did.

Additionally, the probability nears zero when the allegedly witnessed X (miracle or lottery ticket) is logically or practically impossible. Like, say, you claim to have won the lottery and people claim to have seen you, but you didn't actually play. Or you lost the ticket. Or there was no lottery.

Jesus returning sight to the blind or walking on water, or Moses dividing the Red Sea... those are pretty impossible things, and so eyewitness claims are not as convincing.

Just cause Matthew says so and so many people saw miracle X, doesn't mean they did. — NKBJ

That's even worse, because the author of Matthew was not even an eyewitness. He's just passing along hearsay.

While fdrake has already provided a fairly in-depth post on the value of multiple accounts, for most everyday examples it seems fairly self-evident that multiple witnesses increase the probability of the event having occurred. It's unlikely that multiple people hallucinate similar observations. — Echarmion

Basically you're restating the common belief that witnesses matter re probability of something being the case. I'm aware of the belief. I addressed. You didn't address anything I said. You're just restating the status quo.

Basically you're restating the common belief that witnesses matter re probability of something being the case. I'm aware of the belief. I addressed. You didn't address anything I said. You're just restating the status quo. — Terrapin Station

Your argument is that we'd need a large set of data. I say we already have a large set of data for everyday occurrences. Our knowledge of current events essentially relies on witness reports.

I also don't see how you can deny, in principle, that a witness report of an event is more likely in a world where that event happened.

Your argument is that we'd need a large set of data. — Echarmion

That was the beginning of the sentence. The rest was:

showing, for multiple scenarios, that there is some correlation to how many witnesses there are relative to whether something turned out to be the case, where the latter was checked via independent means. — Terrapin Station

And I said that this was just the start of what we'd need to do.

In a situation where Person 1 and Person 2 are being truthful it seems to me that Person 2 is more likely to be an actual lottery winner because of corroborating evidence for the same reason why 10 eye witnesses in the trial describing the same thing is stronger evidence than just a single eye witness.

Stated differently for that reason I can buy that P(Person 1 won the lottery | Person 1 is being truthful) < P(Person 2 won the lottery | Person 2 is being truthful). That part make the case that claiming more people observed it helps the probability of the claim.

But there are multiple scenarios to consider besides that one I put in bold

P(Person 1 won the lottery | Person 1 is being truthful) + P(Person 1 won the lottery | Person 1 is not being truthful) + P(Person 1 did not win the lottery | Person 1 is being truthful) + P(Person 1 did not win the lottery | Person 1 is not being truthful) = 1

P(Person 2 won the lottery | Person 2 is being truthful) + P(Person 2 won the lottery | Person 2 is not being truthful) + P(Person 2 did not win the lottery | Person 2 is being truthful) + P(Person 2 did not win the lottery | Person 2 is not being truthful) = 1

conversely someone is more likely to have actually won the lottery and not be mistaken if more people look at the ticket can confirm it.

Stated differently for that reason I can buy that P(Person 2 did not win the lottery | Person 2 is being truthful) < P(Person 1 did not win the lottery | Person 1 is being truthful).

P(Person 1 won the lottery | Person 1 is not being truthful) and P(Person 2 won the lottery | Person 2 is not being truthful) seem negligible or about the same low value.

The main thing it looks like that determines how P(Person 1 won the lottery) compares with P(Person 2 won the lottery) is how P(Person 1 did not win the lottery | Person 1 is not being truthful) compares with P(Person 2 did not win the lottery | Person 2 is not being truthful). But it seems to me that if someone were trying to be dishonest, they would choose to be as convincing as possible making P(Person 2 did not win the lottery | Person 2 is not being truthful) > P(Person 1 did not win the lottery | Person 1 is not being truthful) for the same reason why someone who bluffs in poker might bet more if it's their goal to deceive others.

If Person 2 is more likely to lie big when they are lying and Person 2 is also more likely to have more impressive evidence when being honest, I don't know if that helps if it just leads back to determining whether likelihood of comparative probabilities of Person 1 and Person 2 being honest versus deceptive.

Is "Person 2"'s claim have a slightly higher probability of being correct due to the number of eye witnesses they claim are available? — coolguy8472

Not in the least. The sole arbiter is the issuing authority of the lottery, whether a country or a state or a church group. No number of non-arbiters, no matter how large, can confirm a win. If a thousand people see your winning ticket, the lottery authority can always claim machine error. Here is a real life case. https://www.npr.org/sections/thetwo-way/2017/12/28/574070736/how-the-glitch-stole-christmas-s-c-lottery-says-error-caused-winning-tickets

And now you're the one just repeating what you already said without engaging with the substance of my reply. Perhaps I am fundamentally misunderstanding you. Could you rephrase the argument that you think I am not addressing?

conversely someone is more likely to have actually won the lottery and not be mistaken if more people look at the ticket can confirm it. — coolguy8472

Unless of course the ticket is fake or otherwise invalid. No amount of witnesses will modify that probability.

But it seems to me that if someone were trying to be dishonest, they would choose to be as convincing as possible making P(Person 2 did not win the lottery | Person 2 is not being truthful) > P(Person 1 did not win the lottery | Person 1 is not being truthful) for the same reason why someone who bluffs in poker might bet more if it's their goal to deceive others. — coolguy8472

But it's harder to find 10 people willing to lie for you than it's to find 2, so even if they were willing to forge more evidence, the evidence still increases the probability of them being truthful. You can always construct reasons to not consider any single piece of evidence convincing, but it's still evidence and you still need to take it into account.

I think you’ve misused “probability” here? The numbers are irrelevant in the manner you’ve presented the problem. The of ONE person being “correct” would rise for sure.

To expand a little, let us assume that some people are “duped” by what they see and some are not. It would seem to me, psychologically speaking, that there is a threshold where once a certain proportion believe something to be true the causal bystander will just agree out of the psychological need to “fit in” - this kind of experiment can be easily demostrated where a group of people repeatedly and purposefully give the wrong answers and the one left in the dark starts to agree with them against their better judgement.

Not in the least. The sole arbiter is the issuing authority of the lottery, whether a country or a state or a church group. No number of non-arbiters, no matter how large, can confirm a win. If a thousand people see your winning ticket, the lottery authority can always claim machine error. Here is a real life case. https://www.npr.org/sections/thetwo-way/2017/12/28/574070736/how-the-glitch-stole-christmas-s-c-lottery-says-error-caused-winning-tickets — fishfry

Unless of course the ticket is fake or otherwise invalid. No amount of witnesses will modify that probability. — Echarmion

Yeah I've considered that when determining "P(Person 1 won the lottery | Person 1 is being truthful) < P(Person 2 won the lottery | Person 2 is being truthful)". I would agree P(Person 1 did not win the lottery | Person 1 is being truthful and the ticket is invalid or fake) >= P(Person 2 did not win the lottery | Person 2 is being truthful and the ticket is invalid or fake). But I was thinking there's more scenarios of individual people thinking they won the lottery and are just mistaken due to human error within P(Person 1 won the lottery | Person 1 is being truthful). In cases of human error I consider more people making the same verification a way to minimize that.

But it's harder to find 10 people willing to lie for you than it's to find 2, so even if they were willing to forge more evidence, the evidence still increases the probability of them being truthful. You can always construct reasons to not consider any single piece of evidence convincing, but it's still evidence and you still need to take it into account. — Echarmion

Except we don't know if the 10 people exists when considering the probability. They could just being saying there are 10 people that can verify and are making it up. That's the part I'm tripped up on the most: determining the likelihood that someone is being untruthful then the probability that they would make a claim like "1 other person can verify" versus "10 other people can verify" if their goal is to be as convincing as possible.

Except we don't know if the 10 people exists when considering the probability. They could just being saying there are 10 people that can verify and are making it up. That's the part I'm tripped up on the most: determining the likelihood that someone is being untruthful then the probability that they would make a claim like "1 other person can verify" versus "10 other people can verify" if their goal is to be as convincing as possible. — coolguy8472

Oh, the scenario was supposed to be just a claim? Well in that case the answer is that a statement alleging more witnesses is less likely to be true, by virtue of alleging extra facts. For a reasonable number of witnesses, the probability of the statements is roughly identical and only depends on the likelihood the person is lying in the first place.

Oh, the scenario was supposed to be just a claim? Well in that case the answer is that a statement alleging more witnesses is less likely to be true, by virtue of alleging extra facts. For a reasonable number of witnesses, the probability of the statements is roughly identical and only depends on the likelihood the person is lying in the first place. — Echarmion

I would have thought the more witnesses with consistent answers adds credibility. Assuming honesty and the existence of the witnesses in order for them to be mistaken every witness has to be wrong. The likelihood of all witnesses being wrong approaches 0 with the more witnesses you have.

We can see this in real life all the time when rumors and accusations spread. Like if someone just claims they were assaulted versus someone claims they were assaulted with many eye witnesses according to them. Or if someone makes a claim about the government versus someone a claim about the government that was corroborated by many anonymous sources according to them. Do people correctly apply more likelihood of the event being true when introducing more facts like that? Whether the person expects to be fact checked, how disprovable the facts are, and how intelligent the person is all pay a factor too.

as already said, lots of variables involved

Even with the frequentist data, however, there would still be a number of problems to overcome. That's because there are so many different variables that can come into play. Making a probability claim on this sort of frequentist data implies that we're parsing the witnesses as ideal--no sort of bias, no sort of hidden agenda, no perceptual problems, ideally intelligent and rational, etc., and it also implies that we're assuming they have a more or less ideal access to information. Otherwise there would be no way to establish that the correlation is implicational, and that's what you'd be looking for here. — Terrapin Station

Because we don't know a lot of the facts it makes it difficult to make a probability judgement. But maybe there's some kind of ambient probability like a weighted average of people who would tell a big lie that has more to attack versus a small lie that's harder to verify.

I would have thought the more witnesses with consistent answers adds credibility. Assuming honesty and the existence of the witnesses in order for them to be mistaken every witness has to be wrong. The likelihood of all witnesses being wrong approaches 0 with the more witnesses you have. — coolguy8472

But you just said that we are dealing with merely the claim of witnesses, not actual witness testimony. Only actual witnesses add credibility.

Do people correctly apply more likelihood of the event being true when introducing more facts like that? Whether the person expects to be fact checked, how disprovable the facts are, and how intelligent the person is all pay a factor too. — coolguy8472

People are generally bad at intuitively assigning correct probabilities. There is a tendency to evaluate how vivid and plastic a story is when determining whether it's likely. This is, however, a mistake. A naked claim is more likely than one with added details (such as alleged additional witnesses) because every detail is also an additional claim.

A naked claim is more likely than one with added details (such as alleged additional witnesses) because every detail is also an additional claim. — Echarmion

In the P("I own a car") > P("I own a red car") sense yeah.

More detail can increase the likelihood too like:

P("I own a red car given that I own something that's red, it makes noise, and has lights on it") > P("I own a red car")

But the original scenario is different than that example because we're dealing with claims and not "givens". But I'm thinking often times we can see that a statement is more likely to be true when it's claimed versus when it's not claimed if we can determine that it's more likely to not be fabricated. Maybe an example of that would be if I forgot what day of the week it was and asked someone then they told me "Wednesday", then that should raise the probability of it being "Wednesday" from 1 in 7 to something pretty close to 100% even though all that's changed is the introduction of someone else claiming it's Wednesday.

In the P("I own a car") > P("I own a red car") sense yeah.

More detail can increase the likelihood too like:

P("I own a red car given that I own something that's red, it makes noise, and has lights on it") > P("I own a red car")

But the original scenario is different than that example because we're dealing with claims and not "givens". — coolguy8472

Yes, the difference is between P(X and Y) and P(X, given Y). When we're looking at the content of a claim, we have P(X and Y). When we are looking at a claim within a specific situation, we are additionally dealing with P(X, given Y). The resolution of these will also depend on whether the probabilities are independent (as in your first example) or not (as in your second one).

But the original scenario is different than that example because we're dealing with claims and not "givens". But I'm thinking often times we can see that a statement is more likely to be true when it's claimed versus when it's not claimed if we can determine that it's more likely to not be fabricated. Maybe an example of that would be if I forgot what day of the week it was and asked someone then they told me "Wednesday", then that should raise the probability of it being "Wednesday" from 1 in 7 to something pretty close to 100% even though all that's changed is the introduction of someone else claiming it's Wednesday. — coolguy8472

When we evaluate the likelihood of a person being truthful, we need to evaluate both the content of their claim and the fact that they make the claim given what we know about the person and the situation. Given that there are no ordinary reasons why a random person should lie to us about the current date, their claim has a high likelihood of being true. This is a case of P(X, given Y). Mathematically, we take the chance that it's Wednesday (i.e. P(X), 1/7) and modify it with the chance a random person would lie (or be mistaken etc.) about the date (i.e. P(Y, given ~X), say 1/10). The prior likelihood was 1/7, the new likelihood is roughly 2/3 (1-(6/7 * 1/10)).

Firstly, in my opinion, Bayesian probability should be interpreted as being reducible to frequentist statistics. For ultimately the empirical distribution in front of us is all there is. A so-called 'prior' is what happens when an observed empirical distribution, say f(x,y), is mathematically represented as a product g(x)h(x,y). From this product it is clear that the so-called 'prior' g(x) is nothing more than a factor of the observed distribution f, and that g does not possess meaning that is independent of the "likelihood" h (and vice versa).

Yet in scientific practice Bayes' rule is usually used for prescriptive induction; it is often the case that g is derived from a different data-set from that used to derive h, such that the product of g and h constructs an unseen joint-distribution that is used to make novel inferences. As with all induction, no statistical justification for this can be given and Bayesian statisticians should remain silent.

Of course, g and h are rarely known explicitly and are more naturally represented in terms of computer programs representing our physical knowledge and assumptions from which we can simulate a distribution of pseudo-data for comparison against new real-world data.

But none of that should detract from the fact that g together with h are synonymous with empirical knowledge + empirical assumptions ; for whatever we are ignorant about can play no role in our predictions or calculations.

Returning to your question, it is under-determined without reference to a distribution correlating independent witness reports to the identity of lottery winners. Of course, we might say that we know this intuitively and are prepared to make an induction, but this further serves to illustrate why Bayesian statistics is pretty useless as a formalism for directly expressing prescriptive induction.

I tried to research the problem a bit, it's considered double hearsay in court and generally not admissible as evidence.

Person 1 Claim: "I won the lottery, my friend saw the ticket and can confirm"
Person 2 Claim: "I won the lottery, 10 people saw the ticket and can confirm"

My guess is that that in a lottery where the odds are 1 in a billion:
P(Person 1 won the lottery given they claimed "I won the lottery, my friend saw the ticket and can confirm") = 1%
P(Person 2 won the lottery given they claimed "I won the lottery, my friend saw the ticket and can confirm") = 1.01%

It depends on the setting, odds of invalid tickets, odds of being mistaken, odds that they were joking, etc... I didn't specify what would be random variables.

In terms of christian apologetics, making an unconfirmable claim that many people witnessed a miracle versus not making that claim I would say probably slightly increases the likelihood of the claim being true but the probability amount that increases from them making that claim is so minuscule it's really not worth mentioning as evidence that the claim is true. But people cite it as an argument that miracles occurred and seems to have persuasive power to some so that leads me to think double and tripple hearsay carry some slight amount of weight.

This is just like saying if 1,000,000 people each try to eat a fully grown elephant in 2 seconds the probability of someone doing so is greater than if 5 people try. Wrong! The probably is always 0.

When it comes to the lottery the chance of winning, or guessing that someone will win, is the same for everyone. Guesswork doesn’t change this, it only a\narrows the margin down that SOMEONE will guess correctly.

Witnessed experiences (illusionary of otherwise) are not in the same ball park.

My guess is that that in a lottery where the odds are 1 in a billion:
P(Person 1 won the lottery given they claimed "I won the lottery, my friend saw the ticket and can confirm") = 1%
P(Person 2 won the lottery given they claimed "I won the lottery, my friend saw the ticket and can confirm") = 1.01% — coolguy8472

Unless this really is a completely random guess, can you give any reason for your numbers, or for the difference between them? What does make the second uncorroborated claim more probable (however slightly) than the first uncorroborated claim?

But none of that should detract from the fact that g together with h are synonymous with empirical knowledge + empirical assumptions ; for whatever we are ignorant about can play no role in our predictions or calculations. — sime

Isn't the Bayesian position that there is no qualitative distinction between assumptions and knowledge? It's all just probabilities with different values.

It's my best guess. Because the claim that claims more eyewitnesses has more persuasive power to some people. Double and triple hearsay is a persuasive enough topic for courts to at least discuss the issue before rejecting the idea of it being valid persuasive evidence.