Infinite Coincidence vs. Causality: A Provable Difference?

Great Debates
1

Let’s start with an example (for the sake of illustration):

Suppose I walk into a room, find a switch in the wall, turn it to the “ON” position, and the room illuminates.

That could be cause-and-effect, but it could also equally be a coincidence.

Now, suppose I do it twice, and it happens twice.
Now suppose I do it ten times, and it happens ten times.
A hundred.
A million.

At what point is causality proven (if ever)?

In other words, how many coincidences have to occur before it demonstrably isn’t a coincidence anymore (I’m well aware that this may not be an answerable question, but I figured I could be wrong…hence my asking)?

Is an infinity of coincidences distinguishible from causality, and if so, how?

If not, should we really be seeing them as different, when they aren’t, really?

Mind you, I’m not taking a position on this question, personally; I just thought this was the forum for its pondering. I’m interested in whatever positions people take on it. If there are “official” answers (from classic philosoph…okay, I’ll admit, I want to say “philosopheers”, but that’s probably not right…“philosophy people” sounds awkward :rolleyes: ), I would like to hear them, but also from Dopers as well. I didn’t study a lot of philosophy…mostly because I already had my own opinions on things.

This question, on the other hand, escapes me entirely.

So, jazz me.

(As an aside, I just realized that both sides of my mustache have grown to the point that I can see them without a mirror. Pardon me whilst I grab some scissors.)

2

Well, speaking of official philosophy and such, you’re essentially outlining a Humean skepticism about causality. Clearly, an infinity of coincidences can’t be distinguished from causality, in the sense that both are observationally equivalent. And this ties in to a lot of other similar problems which aren’t all stated in terms of causality, but in terms of the problem of induction in general (also first notably pursued by Hume). E.g., if you keep flipping a coin and it comes up heads every time, at what point can you conclude the coin is rigged, rather than it being coincidence? [Since, with a fair coin, every particular sequence of coin flips of a certain length has the same probability, if you come in with the assumption that the coin is fair, it would seem bizarre for some sequences to make you squirm more than others]. Or, another related problem, if you do want to conclude that there is a pattern, why conclude the pattern is “Flipping the switch illuminates the room” rather than “Flipping the switch illuminates the room the first 8000 times, then turns on disco music the next 8000 times, then goes back to illumination, etc.” or “Flipping the switch illuminates the room only on odd-numbered years” or such, when you have the same number of confirmatory instances for each? This last point was pursued somewhat by Goodman, in a slightly different form, and was also at the core of much of the philosophy of the later Wittgenstein (in whose terminology it was called “the problem of following a rule”, and was caught up in the problem of how to extract the meaning/use/correctness criteria of a term from a limited number of instances of its use), whose writings on the subject were famously of interest to Kripke, particularly in terms of their implications for the philosophy of mathematics.

So that’s a table of contents of sorts for some of the official philosophy around the topic. Unfortunately, I’ve provided little of actual substance yet, but I’ll try to post more about their particular views on the matter and my own views later.

The short preview of how things will go, though: Nobody has a really great response, of the sort that’ll knock you on your ass and make you say “Ah, yeah, how silly of me to have worried about that”. And lots of people have things to say which only make the problem more vexing, rather than less. Perhaps you’ll hear things in the way of philosophical therapy that dissolve the problem for you anyway. But if you don’t, and remain somewhat worried about it, you’d be far from alone.

3

It is a sort of Humean skepticism, as **Indistinguishable ** says. Of course, as Hume was a radical empiricist, he would say there is no difference between causation and an infinite coincidence.

My naive (having thought little about the problem of causation) response is that they could be distinguished by identifying an underlying mechanism explaining the correlation. This is how causation and mere correlation are usually distinguished. So, for example, a study some years ago showed that children growing up in houses where the family had bottled water delivery were healthier than children who didn’t grow up in such houses. So bottled water causes better health, or is merely correlated with it? The naive Humean would say there is no difference. But the researchers discovered a mechanism: households with bottled water delivery are usually higher-income households, and so can afford better pre- and post-natal medical care, etc., resulting in better health for the children. Thus, I would think that in principle, at least, you could distinguish causation from mere coincidence by identifying the underlying mechanism causing the correlation, and seeing whether it conforms to causation or mere correlation (or coincidence).

4

They would be distinct, since in one case A causes B and in the other B happens to take place after B. But there’d be no way of knowing which was the case. The only thing you can really do is falsify a theory of causality; if for example you say that A causes B because of electricity, and then you find out that the switch isn’t wired up, that would show that theory to be wrong. But you can’t say that A didn’t cause B for some other reason, just as you can’t say the two *aren’t * connected.

I suppose thinking about it we can’t even say that one is more likely than the other. To go from Indistinguishable’s example of coin flipping, it might appear on the surface that flipping a coin 100 times and getting heads indicates the coin’s rigged. Problem is, while the probability of 100 heads in a row is low, that doesn’t mean it can’t happen. And in a theoretical situation where there are potentially infinite coin flips, there is no significant difference in pattern between two heads in a row and thousands. So really no pattern is more or less indicative of a causal relationship.

5

Surely prediction is a key to distinguishing between them. Take the coin toss: eventually, after having flipped the coin and gotten heads for the one millionth straight time, you think, “Maybe this coin is rigged.” At that point, you test your theory by hypothesizing that the next ten flips will also be heads. If they are, then you know that there is only a 1:2[super]10[/super] chance that it is a coincidence. You can’t use the initial observation to confirm the hypothesis, however.

It would be hard to know what the odds are against the light turning on coincident with your flipping the switch every time, but you could still form a very high (though finite) confidence level after a relatively small number of trials.

The problem is what happens when you can’t perform new trials. Can you ever look back and say that having already flipped the switch a million times, we know what the switch does? In the absence of the even the possibility of flipping the switch again, does a claim about what the switch does even make sense, or must all claims be predictive to be intelligible? How do you falsify a hypothesis about the past? These questions are philosophically important to deveolping areas of science that lack experimental methodology, like economics, sociology, and evolutionary biology. (Not every question in these fields lacks an experimental solution, obviously, but many of them do.)

6

It’s intuition. Formally there’s no proof that if you get a hundred tails it’s not incredible luck, but realistically after a little while you’ll demand a new coin. As for when a “little while” is, that depends on the stakes.

7

I think this just breaks down into basic science and statistics that is used in every scientific discipline. You can’t ever prove a statement is true. However, you can prove it false by experimentation. One flip of the switch without the expected results proves the relationship isn’t absolute or at least there is some other factor at play. If you start flipping the switch and running statistics on the probability that they are connected, you will reach a higher and higher probability but it will never reach 100%. You can never prove anything in science. You can just make the conclusions more likely with additional experiments and trials.

8

Right. The best you can say is to give a probability that the results observed are due to chance alone. Now, you can decrease the probability that the results are due to chance by conducting more experiments, but you can also do so by going under the hood, so to speak. If you reveal the wiring, and discover by use of a multimeter that current flows in the wire coincident with the switch being turned on, you increase the number of coincidences that must occur to explain the result. You’ll never eliminate the possibility that coincidences are the cause, and only Mr. Spock can give you an exact probability, but you can certainly reduce the chance beneath epsilon, where epsilon is the probability at which a specific person will accept causality.

Epsilon might be 0 for some people, but they are not fit for polite company.

ETA: What Shagnasty said in many fewer words.

9

But how exactly would you do this? It’s easy to calculate the probability that the results would be as observed were they due to chance alone. It’s difficult to do the converse, to give the probability that the results are due to chance, given that they came out the way they did. You would need to assume some particular probability distribution on the space of probability distributions on the space of coin flip sequences to do that, and I see no reason to assume that to go any which particular way. There’s certainly no observational evidence that could help you do that.

Or perhaps I’m wrong. Feel free to show me how to answer the following questions: Supposing I flip a coin and it comes up heads 10000 times, what is the probability the coin is fair? What’s the probability the coin is biased with 70% heads probability? What is the probability the coin is rigged to always come up heads? What is the probability the coin is rigged to come up heads the first 10000 times, then tails the next 10000 times, then back to heads? What is the probability the coin is rigged to come up heads on every other flip, but the ones inbetween those are fair flips? Etc. It’s easy to do the converses (given the probability distribution, what’s the probability that the coin comes up in a particular sequence), but determining the distribution’s probability from the sequence seems, well, impossible, without assuming things I see no reason to assume.

10

So, you are agreeing that this procedure will not prove anything. The most it can do is suggest a very low probability that it is coincidence.

11

That’s the whole point. That is the way science works. You can only become more confident or find that your hypothesis is invalidated the way you posed it. Newtonian physics stood up to countless experimental results before we found out that it wasn’t quite right after all.

12

[QUOTE=Indistinguishable]
But how exactly would you do this? It’s easy to calculate the probability that the results would be as observed were they due to chance alone. It’s difficult to do the converse, to give the probability that the results are due to chance, given that they came out the way they did. You would need to assume some particular probability distribution on the space of probability distributions on the space of coin flip sequences to do that, and I see no reason to assume that to go any which particular way. There’s certainly no observational evidence that could help you do that.

Or perhaps I’m wrong. Feel free to show me how to answer the following questions: Supposing I flip a coin and it comes up heads 10000 times, what is the probability the coin is fair? What’s the probability the coin is biased with 70% heads probability? What is the probability the coin is rigged to always come up heads? What is the probability the coin is rigged to come up heads the first 10000 times, then tails the next 10000 times, then back to heads? What is the probability the coin is rigged to come up heads on every other flip, but the ones inbetween those are fair flips? Etc. It’s easy to do the converses (given the probability distribution, what’s the probability that the coin comes up in a particular sequence), but determining the distribution’s probability from the sequence seems, well, impossible, without assuming things I see no reason to assume.
[/QUOTE]

That’s the very reason I made the crack about Mr. Spock, who was always giving exact probabilities for things impossible to compute. Anyhow, I thought this thread was about the probability of a supposed causal event being due to chance. I can easily compute (and I suspect you can also) the probabilty that the 10,000 heads is due to chance - where we’d assume such an event was causal. In these things the null hypothesis is usually that chance alone is involved - I haven’t seen any cases where the null hypothesis is that there is a cause, and you are trying to show chance. And you certainly can’t use statistics to generate hypotheses, only to test them.

13

[QUOTE=Keeve]
So, you are agreeing that this procedure will not prove anything. The most it can do is suggest a very low probability that it is coincidence.
[/QUOTE]

In the context of science, demonstrating a very low probability is the preferred method of proving something. In the context of law, providing a mutually corroborating network of testimony, circumstantial evidence, and/or material evidence is considered sufficient to prove something “beyond a reasonable doubt.” Only in mathematics does proof imply logical necessity. I don’t think you can mathematically prove the logical necessity of a coin being rigged or fair, no.

You can’t answer any of those questions retrospectively. That’s why you have to make predictions and test them. Having flipped the coin 10000 times and gotten us heads doesn’t tell us whether the coin is fair or not, you’re right. But flip it again x times, and I’ll tell you with ([1-1/(2^x)]100)% confidence whether it is fair or not.

14

[QUOTE=Voyager]
That’s the very reason I made the crack about Mr. Spock, who was always giving exact probabilities for things impossible to compute. Anyhow, I thought this thread was about the probability of a supposed causal event being due to chance. I can easily compute (and I suspect you can also) the probabilty that the 10,000 heads is due to chance - where we’d assume such an event was causal. In these things the null hypothesis is usually that chance alone is involved - I haven’t seen any cases where the null hypothesis is that there is a cause, and you are trying to show chance. And you certainly can’t use statistics to generate hypotheses, only to test them.
[/QUOTE]

Well, when you speak of reducing probabilities below epsilon to make them ignorable, it seems you must implicitly be computing at least some approximation to the probabilities and getting it below your desired epsilon. At any rate, which probabilities is it that you’re saying you can compute? P(10,000 heads | coin is fair) or P(coin is rigged | 10,000 heads) or… what, exactly?

I understand that what statisticians normally do is formulate a null hypothesis, then calculate P(observations | null hypothesis is true), and reject the null hypothesis if this probability is very low. But this seems totally backwards, especially for confronting such skepticism as put forth in the OP. What you really want to calculate is P(null hypothesis is true | observations), and that’s what seems impossible to determine [without some prior assumed knowledge of the ratio P(observations)/P(null hypothesis is true) to apply Bayes’ Theorem, presumably through prior assumed knowledge of both P(observations) and P(null hypothesis is true); but there really isn’t any good basis to assume either of those to take any particular value, especially the latter when the null hypothesis is “The coin is rigged” or “The coin is fair” or anything like that]. (As usual, Wikipedia is already aware of this point).

[QUOTE=Alan Smithee]
You can’t answer any of those questions retrospectively. That’s why you have to make predictions and test them. Having flipped the coin 10000 times and gotten us heads doesn’t tell us whether the coin is fair or not, you’re right. But flip it again x times, and I’ll tell you with ([1-1/(2^x)]100)% confidence whether it is fair or not.
[/QUOTE]

So 10,000 flips of the coin can’t give us any answers, but 10,000 + x flips can, as long as we make sure to pause after the first 10,000 and mutter some hypotheses to ourselves? And, presumably, you’d say 10,000 flips can’t give us any answers retrospectively, but if we make 5,000 flips, stop and mutter hypotheses, and then make 5,000 more flips, then we could get some answers? I guess you’re trying to test hypotheses on the basis of their predictive power, and get some concrete numbers this way, but anyone embracing the Humean skepticism of the OP wouldn’t be dissuaded by this sort of thing; a limited number of confirmations of a prediction seems unable to confirm the validity of the prediction in general, or even tell us anything about the probability of the prediction holding in general. No matter how many flips you perform, you still never gain any information on the probability that the coin is fair in general; that is, you’ve gained no information on P(coin is fair | observations so far), you’ve just managed to possibly lower P(observations so far | coin is fair) by making more observations.

15

Ack, I was frozen out of the edit window; I wanted to note the point that, in the particular case where our initial hypothesis is “The coin is fair”, any sequence of coin flips of length N is (on this assumption) as likely as any other. Obvious to all of us, right? But if any sequence of coin flips is as likely as any other, how can it be that we should take some sequences to reject the hypothesis and others to confirm it? If I say “I think this coin is fair”, and then toss a million heads, why should I reject the hypothesis? Sure, I observed an event of probability 2^(-million), but any sequence of a million flips, no matter how they came out, would have been an event of equal probability. And surely I wouldn’t have chosen to reject the hypothesis on every sequence of a million flips.

An initial hypothesis of pure randomness can’t be made unlikely by any observations; infinite coincidences that line up neatly aren’t in this context probabilistically distinguishable from infinite chance with no discernible pattern. This is precisely the conundrum of the OP, suitably adapted to probabilistic terms.

16

[QUOTE=Indistinguishable]
Ack, I was frozen out of the edit window; I wanted to note the point that, in the particular case where our initial hypothesis is “The coin is fair”, any sequence of coin flips of length N is (on this assumption) as likely as any other. Obvious to all of us, right? But if any sequence of coin flips is as likely as any other, how can it be that we should take some sequences to reject the hypothesis and others to confirm it? If I say “I think this coin is fair”, and then toss a million heads, why should I reject the hypothesis? Sure, I observed an event of probability 2^(-million), but any sequence of a million flips, no matter how they came out, would have been an event of equal probability. And surely I wouldn’t have chosen to reject the hypothesis on every sequence of a million flips.

[/QUOTE]

If the hypothesis is the coin is fair, and the null hypothesis is that it is rigged, you can’t really say anything, since, as you say, any outcome is equally probable, and one might say that the coin is rigged to produce exactly what is observed. I think you’d have to go with a hypothesis that the coin is rigged to, say, produce the same result on each flip. The null hypothesis then would be that the coin is fair. Given n results that match the hypothesis, it would be easy then to compute if this were due to chance, since the hypothesis expects a certain result.

And I agree that this can never prove that any outcome is causal.

As for epsilon, I got this from John Maynard Keynes’ first book, which I wrote about for my Theory of Knowledge class. Though you can’t compute exact probabilities, you can order them. I couldn’t imagine how to compute the probability that a light would go on by itself, by I know it doing so 100 times is less likely than it doing so 1 time. We can ask the skeptic how many times it goes on when the light switch is used, and not otherwise.

Now for the link, I’m not exactly sure what they mean by P(Null|Data). Was your example an attempt at illustrating this?

17

[QUOTE=Voyager]
If the hypothesis is the coin is fair, and the null hypothesis is that it is rigged, you can’t really say anything, since, as you say, any outcome is equally probable, and one might say that the coin is rigged to produce exactly what is observed. I think you’d have to go with a hypothesis that the coin is rigged to, say, produce the same result on each flip. The null hypothesis then would be that the coin is fair. Given n results that match the hypothesis, it would be easy then to compute if this were due to chance, since the hypothesis expects a certain result.

And I agree that this can never prove that any outcome is causal.
[/QUOTE]

I’m saying, if the null hypothesis is that the coin is fair, then, if things are set up properly, all N-length observations go the same way: they all are events of probability 2^(-N), and thus should either all reject or all fail to reject the null hypothesis. You can’t say “I thought the coin was fair, but then I tossed 50 heads in a row, which is exceedingly unlikely, so I guess it probably wasn’t fair” unless you’re also willing to say “I thought the coin was fair, but then I tossed the sequence {H, T, T, T, H, H, T, H, T, …} of 50 flips, which is exceedingly unlikely, so I guess it probably wasn’t fair”.

On the other hand, if your null hypothesis is that the coin is rigged in some particular nice way (e.g., rigged with a 70% heads bias but separate flips still being independent), then, conceivably, some potential observed sequences would be rejecters while others would be fail-to-rejecters. This whole approach is still pretty backwards, though, as explained below. And if you don’t assume some particular nice rigging, but just ponder the possibility that the coin is rigged at all in any way, then this is just the dual to pondering the possibility that the coin is fair, and the same problems arise with attempting to gain any information on the likelihood of the hypothesis.

[QUOTE=Voyager]
Now for the link, I’m not exactly sure what they mean by P(Null|Data). Was your example an attempt at illustrating this?
[/QUOTE]

P(Null|Data) = P(Null hypothesis is true | The observations we made coming out the way they did) = The probability of the null hypothesis being true, given that the observations come out the way they did. Note that this is different from P(Data|Null) = P(The observations we made coming out the way they did | Null hypothesis is true) = The probability of the observations coming out the way they did, given the null hypothesis. For various choices of null hypothesis, the latter is easy to compute, but it’s the former that we really want, it’s the former that applies to the OP’s problem of differentiation between coincidence and causality, and that’s beyond our access. The whole backwards problem with misapplication of significance testing/p-values/etc. is in conflating these two. You can have a very low p-value with the null hypothesis extremely probable, or a very high p-value with the null hypothesis extremely improbable; there’s no simple connection between the p-value and the probabilities of the null/alternate hypotheses.

To those of you who say “Well, we can never rule out an underlying mechanism of chance and coincidence alone, but we can conclude it unlikely”, well, no, you can’t even do that, not on simple significance testing grounds. Think again about what your tests actually establish. Null hypothesis significance testing is really completely misguided for the problem in the OP, of distinguishing (even probabilistically) between coincidence and causality. Indeed, NHST is really deeply misguided for most purposes towards which it is applied. As put elegantly by Jacob Cohen,

[QUOTE=Jacob Cohen]
What’s wrong with NHST? Well, among many other things, it does not tell us what we want to know, and we so much want to know what we want to know that, out of desperation, we nevertheless believe that it does! What we want to know is “Given these data, what is the probability that H0 [the null hypothesis] is true?” But as most of us know, what it tells us is “Given that H0 is true, what is the probability of these (or more extreme) data?” These are not the same, as has been pointed out many times over the years**…**
[/QUOTE]

18

[QUOTE=Indistinguishable]
I’m saying, if the null hypothesis is that the coin is fair, then, if things are set up properly, all N-length observations go the same way: they all are events of probability 2^(-N), and thus should either all reject or all fail to reject the null hypothesis. You can’t say “I thought the coin was fair, but then I tossed 50 heads in a row, which is exceedingly unlikely, so I guess it probably wasn’t fair” unless you’re also willing to say “I thought the coin was fair, but then I tossed the sequence {H, T, T, T, H, H, T, H, T, …} of 50 flips, which is exceedingly unlikely, so I guess it probably wasn’t fair”.

On the other hand, if your null hypothesis is that the coin is rigged in some particular nice way (e.g., rigged with a 70% heads bias but separate flips still being independent), then, conceivably, some potential observed sequences would be rejecters while others would be fail-to-rejecters. This whole approach is still pretty backwards, though, as explained below. And if you don’t assume some particular nice rigging, but just ponder the possibility that the coin is rigged at all in any way, then this is just the dual to pondering the possibility that the coin is fair, and the same problems arise with attempting to gain any information on the likelihood of the hypothesis.
[/QUOTE]

You are just looking at this an isolated set of statistics though. That is just a tool that helps you figure out the results. It isn’t the whole process though and it certainly isn’t the whole scientific method. For any good experiment, you need a theoretical framework that hopefully goes back deep and wide from other studies to provide context for your hypothesis and validation of the way your are attempting your experiment.

For a coin toss, the theory is simple. Physics states that are fairly smooth, evenly weighted, round about will tend to fall either way when it is tossed and spun into the air. If you wanted to get fancy, you could get a physicist to film and analyze your coin falling and outline what is happening so that you know what you are dealing with but I am sure someone has already worked that out and published it in a journal somewhere. You never need to start from scratch when you do an experiment. Some of the several thousand published journals will have relevant bodies of research that you can build off of.

I think you are making the mistake of conflating pure math and logic with research science when they aren’t the same. The scientific method is much more real-world based and attempts to answer questions the best it can but never claims to be perfect. You do hint at a real problem though. Doing random experiments in isolation with no theoretical framework can easily mean that you won’t understand how and what you are studying enough to interpret the statistics at all. That isn’t a rare problem among some academics and I see it on the SDMB as well.

19

But Shagnasty, as I read it, much of the point of the OP is a skepticism about the inductive aspect of the scientific method. Saying “these laws of physics tell us the coin toss is fair” won’t do much for the man who says “How do we know those ‘laws of physics’ are actually legitimate laws of causality and not merely arbitrary extrapolations from a long string of coincidences?”. It’s true that the way actual research science is done in practice involves saying “Just shut up and go with it; c’mon, we see a nice pattern, it’s worked out every time we tested it, I’m happy to conclude the nice pattern probably works out all or most of the time, and so should be you”, but actual research science in practice wouldn’t give the skepticism in the OP the time of day, and doesn’t seem to be able to do anything to dissuade the man willing to entertain the unorthodox possibilities posed in the OP.

Sure, I’m talking “pure math and logic” and you’re talking “real-world”, but hard-nosed pragmatists aren’t going to engage with the OP’s concerns so much as be content to ignore them. When you start proposing questions like in the OP, you’re already wandering away from “the real world” into some ivory tower philosophizing; it seems most appropriate, then, to meet that discussion in the same context.

20

My apologies for not getting back to this thread sooner.

SO much to say about this (for one thing, if we want to go for the coin flipping rather than the light switching, that’s okay: it’s the same concept to me).

Give me a bit, I’ve got more questions and comments and gratitude here.