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Note the word “existence” in the question where I’m trying to be careful with my wording here. This can be better illustrated with an example.

Take the example of the cheating process. Suppose one observes that John has won four straight lotteries, each of which only has a 1 in a 10 million chance of winning. This obviously seems to make it likely that John cheated. If H = John cheated and F = John won by chance, it seems obvious that H is more likely.

But there is additional information here that makes it more obvious that John cheated apart from the fact that his winnings were improbable. For starters, we already know that people cheat and have an incentive to cheat. Secondly, we also know that cheating is possible as a mechanism. But what if we didn’t know this?

What if H instead was = cheating in lotteries is possible. Does these series of observations make it more likely that H is true? Or must this H be independent of any observations or predictions?

If we didn’t know beforehand that cheating was possible and that people have cheated before, should we now believe that cheating as a process is more likely to be true after observing John win many lotteries?

thinkingman
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  • See 'How improbable does an event have to be before we can say it didn't happen by chance?' https://philosophy.stackexchange.com/questions/94079/how-improbable-does-an-event-have-to-be-before-we-can-say-it-didnt-happen-by-ch/94082#94082 TLDR: We develop different standards in different contexts, depending on experience & the risks & hazards of being wrong – CriglCragl Aug 10 '23 at 18:40
  • The wording of the question could do with improvement, otherwise the answer is trivially yes. My birth is antecedently extremely unlikely, especially from a perspective of a few hundred years ago. But given that it has happened, it is evidence of all kinds of things, including the existence of my parents. – Bumble Aug 10 '23 at 19:54
  • I’ve edited the question phrasing. The question was meant to be specifically about the existence of new processes. – thinkingman Aug 10 '23 at 20:19
  • *Suppose one observes that John has won four straight lotteries, each of which only has a 1 in a 10 million chance of winning. This obviously seems to make it likely that John cheated.* This seems like a hasty generalization fallacy. –  Aug 11 '23 at 15:50
  • Further, it introduces a bias in a future investigation of this unlikely event by branding John a cheater. –  Aug 11 '23 at 15:57

5 Answers5

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It depends on your prior. If you are absolutely certain that John doesn't cheat (maybe John is a computer program incapable of cheating), then no amount of wins by John should convince you that he's a cheater, nor suggest that it's even possible to cheat.

On the other hand, it becomes very, very unlikely for John to win fairly multiple times. If there is any possibility that John may cheat, eventually this likelihood of cheating outweighs the likelihood of winning an arbitrary number of times. We may attribute the non-randomness of John's wins to many possible causes, of which cheating may be one.

Nuclear Hoagie
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  • Even if there was a prior possibility of John cheating, the question is HOW this prior should be arrived upon. Let us assume that we never, for sure, find out that John cheated here. Should these observations now INCREASE our prior for cheating as a process to EXIST? It seems that your answer, although great, is addressing moreso whether or not cheating HAPPENED, not confirming whether or not cheating in this kind of lottery is possible – thinkingman Aug 10 '23 at 18:37
  • @thinkingman Priors are set *prior* to observation - by definition they don't depend on data. Adjusted probability estimates based on data are called *posteriors*. A zero prior is problematic in many ways, since there is no way to change it in a posterior - if you are absolutely sure something impossible to begin with, no amount of data should convince you otherwise. (If you could be convinced, you wouldn't have been 100% sure in the first place.) – Nuclear Hoagie Aug 10 '23 at 18:38
  • Yes it seems that a zero prior cannot change your belief, but that assumes that unlikely observations are the only way to change your belief about something. But if the prior doesn’t depend on an unlikely observation, as you yourself admitted to, what is the issue with that? If we assign a non zero prior to every process, such as someone being psychic, then that would imply with enough correct predictions about the future, one should conclude that the person is psychic. But this doesn’t seem correct – thinkingman Aug 10 '23 at 18:41
  • @thinkingman With the psychic example, we must consider other explanations. With many correct predictions by a "psychic" there is a very low probability that they are doing it under the null hypothesis of chance. That just indicates they are not doing it by chance *in some way*, not that they are psychic - maybe they are cheating. At some point, the probability that your experimental apparatus isn't measuring what you think it is may outweigh a vanishingly likely prior even with lots of supporting data. – Nuclear Hoagie Aug 10 '23 at 19:27
  • Basically, if your experiment is indicating something that you believed to be almost impossible, it may just be that you have a bad experiment. – Nuclear Hoagie Aug 10 '23 at 19:28
  • Something being improbable under chance doesn’t imply that occurrence BY chance is improbable. So even a series of unlikely predictions doesn’t imply that chance is unlikely. One must show an alternative hypothesis is more likely. And the question is whether this has to be done independently. Intuitively, it seems like it does – thinkingman Aug 10 '23 at 19:35
  • @thinkingman You may not need an explicit alternative mechanism to chance, you could have a general class of explanations called "not chance". These may include explanations you can't even conceptualize or describe. The alternative hypothesis might include non-random ways of picking the outcome you haven't even thought of. At some point your knowledge of the universe is limited, you might be satisfied with enough wildly improbable data to conclude that "anything else" is the explanation other than chance. – Nuclear Hoagie Aug 10 '23 at 19:58
  • The class of explanations called “not chance” being true does not depend on how improbable an observation is under chance. If John wins five lotteries in a row, it doesn’t make cheating’s existence more likely. It just makes John cheating more likely. This is a subtle but crucial difference. The possibility of cheating purely depends upon the mechanism by which one can cheat, such as a computer hack, or whatever else – thinkingman Aug 10 '23 at 20:16
  • @thinkingman I don't follow that line of thinking. If you're running a lottery that you think is completely secure, and see one person win 100 times in a row, it's very good evidence both that cheating exists and that the person who won was cheating. I don't know how you could conclude that one particular person was cheating with high probability, and still believe that the probability that cheating exists is low. – Nuclear Hoagie Aug 10 '23 at 20:46
  • It’s not evidence that cheating exists. You know apriori that cheating is possible and is atleast plausible in the sense that it’s not breaking any known physical laws. That plausibility is completely independent of how many times the person wins in a row. Once you know, apriori that cheating exists, then the more lotteries he wins, the more likely cheating occurred. But its APRIORI plausibility shouldn’t increase with lottery wins. I fail to see how the same logic can apply for something like psychism for example where the reality of it seems more implausible. – thinkingman Aug 10 '23 at 21:13
  • @thinkingman That reasoning still doesn't make sense to me. You run the lottery facility that generates the winning numbers a day early and holds them in a locked facility, which you estimate has a 1-in-a-million chance of being broken into. If someone keeps winning time after time, you have almost certainly underestimated the probability of cheating, - it's very likely your facility is not as secure as you thought, and that there is a much higher chance than 1-in-a-million of stealing the winning number. This is a posterior based on evidence, you can't change a prior. – Nuclear Hoagie Aug 11 '23 at 11:40
  • @thinkingman The same thing applies to psychics, you'd just need a lot more evidence to be convinced of something you're very, very sure isn't the case. But by putting a 0 prior on psychics, you are saying there is absolutely no amount of evidence that anyone could provide to convince you they're a psychic. But by the time they've guessed the 6-digit number I'm thinking of a million times in a row, I might start to reconsider. – Nuclear Hoagie Aug 11 '23 at 11:43
  • Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/147841/discussion-between-thinkingman-and-nuclear-hoagie). – thinkingman Aug 11 '23 at 16:37
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If you’re looking for a more practical (less mathematical) answer, we need to define “evidence”. For this you might look to the legal system, which has standards like “beyond a reasonable doubt”.

Jl36
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Evidence helps construct a story. And the number of different stories that can be created with evidence is inversely proportional to the quality and quantity of evidence. With only one unlikely event, a plethora of stories can be created that can be supported by that one piece of evidence..

For example, John claims that a psychic provided him with these numbers and the psychic received this information from an alien named Klaatu. The psychic saved Klaatu's life after his ship crashed in remote Washington State and in gratitude began giving the psychic tips about future events. John had rescued the psychic from a brutal assault. In return for his kindness and bravery, the psychic gave John the winning numbers without his knowledge. John didn't cheat.

If it can't be proven that John cheated, then his fantastic tale must be true (how else can his winning be explained) and is evidence of psychics, aliens, and knowledge of future events.

This is as likely as winning four lotteries in a row. Isn't it?

As, more evidence is accumulated, the number of possible stories that can be created from this evidence decreases until, ideally, only one reasonable story can be told.

Unlikely events are meaningless by themselves. More evidence is required to create a likely story: John cheated the lottery.

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This is exactly how new discoveries are made in the scientific world, as follows.

Imagine we understand the root cause behind some physical process- for example, the rules of a card game, which yield a well-defined probability of winning versus losing the game.

Now comes a new player who consistently beats the game and wins at a rate not accounted for by the rules of the game. He has introduced a second mechanism (a "cheat") by which the game can be won, in addition to the first one as established by the rules.

So, the presence of anomalous data which the known mechanism of some physical process cannot account for hints at the existence of an additional mechanism by which the process can proceed, that our existing model of the process does not include. That newly-discovered process then becomes the subject of research.

niels nielsen
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  • The problem is a second mechanism like cheating seems more apriori plausible than some other mechanism, say, supernatural. And that plausibility is still what ends up making it justified to believe in cheating. Without it, I fail to see when it would be ever justified to believe that a current theory needs to be changed despite “anamolous” data. Data can only be anomalous with respect to the plausibility of another alternative – thinkingman Aug 11 '23 at 17:17
  • @thinkingman, are you a physicist? – niels nielsen Aug 12 '23 at 02:41
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Strictly, the probability of possibility can only take 0 or 1. Things can be possible and unlikely, possible and likely, or possible and necessary.

The confusion here comes from importing the common language equivocation of vanishingly small probability and impossible.

If we rephrase your question I think you already know the answer:

There is additional information here that makes it more obvious that John cheated apart from the fact that his winnings were improbable. For starters, we already know that the probability of cheating is nonnegligible. But what if we had assessed that the probability of cheating was negligible?

What if z was the probability of cheating and H instead was z > x for some small but nonnegligible x. Does this series of observations make it more likely that H is true?

g s
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