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16th February 2011, 08:48 PM | #121 |
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31415926535897932384 Only an idiot would think it was a fair die/toss, correct? But that result is just as likely as any other. What makes it special is that it is one of a small set of results we expect if someone is cheating.
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16th February 2011, 09:08 PM | #122 |
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Originally Posted by Malerin
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16th February 2011, 09:12 PM | #123 |
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There are two questions that need to be separated here.
Q1) Is a sequence of 100 heads on a fair coin possible? A1) Yes. The claim that it isn't can be immediately disposed of by noting that said sequence is no more unlikely than any other, and some sequence must always result. Q2) Is a sequence of 100 heads on a supposedly fair coin more likely to be the result of a truly random fair sequence or of cheating? A2) Cheating, almost certainly. As far as I know, no one other than Piggy disagrees with those. |
16th February 2011, 09:14 PM | #124 |
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I agree with those two answers.
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16th February 2011, 09:59 PM | #125 |
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But that's not quite good enough, for the reasons I mentioned earlier.
A sequence of 100 heads is one of the configurations we can write out on paper as the results of a series of 100 flips. But by the same token "The quick brown fox jumped over the lazy dog" is a series of keystrokes we can write out on paper as the result of a monkey banging on a keyboard. However, experimentation shows that monkeys, in practice, don't produce the entire gamut of possible keystroke combinations, even though on paper this combination is no more unlikely than ttttltlltlllsssiiosssssssssssssssssssssssssssss The question remains whether a series of 100 heads or 100 tails will actually occur in practice, or whether the rough edges of our bumpy world will ensure that the actual outcome has a smaller range of variation and volatility than that. |
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16th February 2011, 10:23 PM | #126 |
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16th February 2011, 10:31 PM | #127 |
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The typing monkey analogy would work if there was approximately a 50% chance of each keystroke being the right keystroke.
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16th February 2011, 10:44 PM | #128 |
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First of all, there is no actual 100-sided die in D&D, so I don't believe that story at all.
As to 6 billion people flipping coins, on average you'd expect at least one of them to get 32 heads in a row on the first go round. Nowhere close to a hundred in a row, but still pretty impressive. As to the OP, you need to have another probability in mind -- the probability that someone is cheating and getting away with it. I don't know how you might determine that. |
16th February 2011, 10:53 PM | #129 |
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16th February 2011, 10:54 PM | #130 |
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16th February 2011, 11:00 PM | #131 |
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I was making your monkey typing analogy fit the chances of getting a streak with a coin.
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17th February 2011, 12:37 AM | #132 |
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17th February 2011, 12:51 AM | #133 |
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17th February 2011, 12:59 AM | #134 |
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I'm just trying to understand what possible mechanism he thinks will cause a string of heads to stop. I understand why long strings don't happen with "small" numbers of iterations, but Piggy seems to think that the likelihood of getting another head is somehow determined by how many heads have already come up, otherwise I really don't understand what makes him think that the number of long strings in practice will be less than the number in theory.
Actually that gives me an idea to test his hypothesis, or at least a variation of it: look at how frequent strings of say 10 heads in a row are in practice. If it's the same as in theory then that's good evidence that the theory is correct. If it's less than in theory (and we can rule out effects like the unfairness of the coins which Sol referred to earlier) then Piggy's ideas can be given a little more credence. |
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17th February 2011, 07:54 AM | #135 |
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You're still not making any sense. No mechanism is needed to ensure extended runs don't happen, they're simply rare enough that we don't expect to see them anyway. Reality predicts exactly the results we see without needing your bizarre contortions, so why do you insist on making them? Yes, flipping 100 heads is possible. No, you shouldn't hold your breath waiting for it to happen.
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17th February 2011, 08:15 AM | #136 |
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17th February 2011, 08:15 AM | #137 |
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If extended runs don't happen in practice, it's only because vast populations don't spend entire lifetimes flipping coins.
The mathematics has already been provided: 2100 (1.2676506 × 1030) tosses would, on average, be expected to yield one sequence which was either all heads or all tails. If you multiply that number of tosses by 100 (1.2676506 × 1032), the result is virtually guaranteed. If you have never seen the result, it's because no one has actually run the experiment. |
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17th February 2011, 09:14 AM | #138 |
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17th February 2011, 09:43 AM | #139 |
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Gargh. This sounds wrong. I'd like to explain why I think this reasoning is faulty.
Basically, it's inferring cause from results, rather than testing a pre-formed cause hypothesis. Bem just got pwnd for this. (See: [Why Psychologists Must Change the Way They Analyze Their Data: The Case of Psi]) This is related to [pareidolia] in that we have a tendency to anthropomorphise 'recognizeable' natural events backwards from the event. ie: we see random numbers, and look for "a pattern" - we're good at finding patterns with completely open criteria. In the world of hypothesis testing, we need to stick to a pre-designated binary criteria, or we've stopped doing science and started doing metaphysics. When a lottery number sequence comes up a winner, to me it's random. To the person who chose it, it's probably a very special set of numbers. Birthdates or something. 31415926535897932384 is 'obviously cheating' to somebody who recognizes pi, but not to other people. I'm sure mathematicians have a personal inventory of irrational numbers they would recognize but the rest of us wouldn't. I think that's what piggy's trying to describe: perhaps a string of identicals is attracting his attention more than other strings because there aren't a lot of special combinations out there he would recognize easily on account of being, well, normal, and not having a math degree or something. And it's not even restricted to math: the problem generalizes to other disciplines like biology. Behe makes a good living reverse-engineering natural sequences in DNA or shapes of molecules for evidence of intelligent cause. |
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17th February 2011, 01:03 PM | #140 |
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I undestand why you're frustrated, but I think the discussion is relevant to the content of your paper.
My interpretation of your paper is that it's a more rigourous version of "extraordinary claims require extraordinary evidence" - a claim with a high prior probability input is easier to accept with a lower confidence interval in the test protocol. I think this is part of the general discussion in other threads about prior probability and its relationship to what constitutes extraordinary claims requiring extraordinary evidence. And I think you're correct that it's something skeptics should be aware of - the take-away from this is that paranormal advocates will use exactly the same formulas (some have) to explain why skeptics are wrong. In their opinion, we have misinterpreted the body of literature and are undervaluing prior probability. They accuse us of insisting on tests with unjustifiably small confidence intervals to 'pass' our challenges. |
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17th February 2011, 05:55 PM | #141 |
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Yes - good - we agree on that - you did not put it that way to start with. Futher: the calculation you showed me does not take this into account. That is what I am saying - you need to adjust the math to take account of the liklyhood of cheating and you didn't. Bayes theorum does, as you point out below:
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You've handled it differently from me since I had to quantify my terms while you are the general idea qualitatively. Crunch the numbers and let me know if you get anything different to me. The term 1/Pr(E) (your notation) is determined by normalization for example and has quite a big impact on the calculation. Pr(H) is the prior probability ... in the article I referred to it as the a-priori suspicion of honesty ... I used the same one. This actually has a strong effect on the outcome. Pr(E|H) (hint: use the "pipe" character - shift+backslash - for the conditional separator) is the forward probability or likelyhood function- which is the same for any specific sequence of heads and tails. C'mon, you know this! This is the term that is (0.5)^n for n tosses. That is the figure you kept quoting to me ... as it turns out - out of context. If you use this figure by itself for your hypothesis testing, you will reject the hypothesis too soon for a particular prior. Now - you don't have to do it this way. Instead you can just rig the number of trials in a test to be so large it makes no difference since Pr(H|E)--->Pr(E|H) for n --> \inf ... as you point out The trouble is (read first post) I don't want very big numbers of trials. I want to avoid the argument that I need a much larger number of trials than I have performed in order to be certain. To do this I need to model what happens for small numbers of trials ... in particular, where is the threshold? How do I adjust the liklyhood while evidence is still rolling (or tossing) in? Bayesian stats is great at this. One of the principle results in the article is that the rejection thresholds change in the manner of a learning curve (go look at the graph) ... for high priors one may be quite justified in maintaining a strong belief for a counter-intuitively long time... but the prior has to be pretty strong (0.9999 was the strongest considered).
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17th February 2011, 06:01 PM | #142 |
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No, that's not it. It's a long string of physical events occurring in a turbulent environment which reach the same end state. This puts it in a separate category from the types of strings of events -- of many various configurations -- which we would expect, for good reason, to result from actions within such an environment.
An extremely long string of identical end-states is not the kind of behavior that results from a situation such as a fair coin toss in the real world, where randomness is the primary forcer. It's like if someone drops a metal cup and it goes clanking down a long stairway. If it hits flat on the bottom of the cup two steps in a row, that's a coincidence. But if it does exactly three flips and hits on the bottom of the cup at each step all the way down the flight, then something's up, because the natural world doesn't work that way. Now, you can do all your math and determine some small probability for that occurrence, but that doesn't matter unless you can demonstrate that your math completely describes the real world and isn't in any way idealized -- which, of course, you can't because it is idealized, as is the math being presented for the coin toss case. |
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17th February 2011, 06:11 PM | #143 |
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It doesn't much matter how long the string of events is, or how turbulent the environment is, the "end state" of a flipped coin is constrained by gravity, the shape of a coin, and the nature of flat surfaces to be either "heads" or "tails."
All your handwaving about "reality's rough edges" doesn't mean squat in this system. If you flipped a coin 1,000,000,000,000,000,000,000,000,000,000,000 times, there would be a run of 100 in there somewhere. |
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17th February 2011, 06:18 PM | #144 |
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It's a fascinating question, and in a way it's one I don't have an answer for, but in another way I do have an answer.
Let's go back to the monkey example for a moment. There is nothing that prevents a monkey from pressing the "u" key after he presses the "q" key. And there's nothing that prevents a monkey from pressing the space bar after pressing the "n" key. Or from pressing the dot key after pressing the "g" key. Yet despite that fact, observation of actual monkey-and-keyboard setups demonstrates that they will never produce the string "The quick brown fox jumped over the lazy dog." no matter how long we give them to do it. Each pair is possible, and yet the extended string will never occur, because in reality they have a much narrower range of output than that at that length, with a lot of repetition of identical or similar patterns. Similarly, in the case of the metal cup dropped down the long flight of stairs, it may hit on the same spot on two consecutive steps, but if it's a random drop (and not a carefully orchestrated effort) it will not make identical strikes all the way down, no matter how many times you drop it, because our rough and bumpy world doesn't work like that. Now, precisely why is that the case? That, I couldn't tell you. But a truly fair toss of a fair coin, in the real world, when repeated many times in sequence, is exactly the kind of thing that we should expect to reflect the messiness of physical reality. If we find that this messiness is absent -- which would certainly be the case if we got 100 heads or tails in a row -- we would be wise to conclude that the exercise had somehow been shielded from it, or in other words, that it was rigged. |
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17th February 2011, 06:21 PM | #145 |
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17th February 2011, 06:22 PM | #146 |
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I suspect you'll get a Nobel prize in physics or a Fields Medal or something if you can prove what you are arguing for, that the laws of physics makes things of this likelihood impossible.
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17th February 2011, 06:22 PM | #147 |
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17th February 2011, 06:26 PM | #148 |
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It is possible to get 10 heads in a row, right?
Why is it literally impossible to get 10 heads in a row, ten times in a row? Or one heads in a row, 100 times in a row. Each flip is approximately 50/50. It doesn't mysteriously change at some point to 0% tails, 100% heads. |
17th February 2011, 06:27 PM | #149 |
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Oh, I'm sure I would,too. I would love to be able to prove it.
But I guess the short answer to your earlier question is that any combination of 2 results is consistent with the randomness inherent in the physical setup, while a series of 100 identical results is not. By the way, can you prove that a string of fair flips of fair coins is adequately described by your math, that it is in fact a system which will eventually result in all possible full-length combinations? |
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17th February 2011, 06:28 PM | #150 |
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17th February 2011, 06:30 PM | #151 |
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According to the cited paper, we wouldn't expect a string of 10 unless we were performing more than 2,000 flips.
When it comes to flips, the terrain of results is bumpy on a large scale. When the terrain we observe is, instead, smooth on a large scale, we conclude that we're no longer looking at a series of fair tosses. Similarly, if we're looking at a large body of water and it lacks turbulence, currents, and tides, we conclude we're not observing the ocean. |
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17th February 2011, 06:31 PM | #152 |
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17th February 2011, 06:35 PM | #153 |
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Reality shows a rugged terrain of results at large expanses of coin-toss space.
Why is this? I don't know if there's a satisfactory answer to be had at this point. But nevertheless, that's what we see. If we view a coin-toss space with a perfectly smooth terrain at large expanses, it is at odds with all experience for fair tosses of fair coins; however, it is perfectly consistent with coin-toss space for rigged coins or rigged tosses. |
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17th February 2011, 06:37 PM | #154 |
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17th February 2011, 06:39 PM | #155 |
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17th February 2011, 06:41 PM | #156 |
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17th February 2011, 06:41 PM | #157 |
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Randomness can include clumping, of 100 or of any other number. So yes, 100 can be consistent with randomness.
Something does not need to eventually result in order to be possible. And fair flips of fair coins? A computer system is the way to test that out. But I know that's not what you mean to ask. I have to go soon so I won't respond in more detail at the moment. |
17th February 2011, 06:43 PM | #158 |
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I hope you realize that the above is utter nonsense. There is nothing about real sequences of coin flips that indicates anything other than that they behave as expected based on the standard laws of physics and probability. Long sequences of heads are rare, because that's what standard theory tells you. But it also tells you that they are not impossible, and it also explains why you've never seen one.
Not to put to fine a point on it, the position you are arguing for is unscientific, magical thinking based on your failure to comprehend basic statistics. |
17th February 2011, 06:43 PM | #159 |
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[quote=blutoski;6888084]I undestand why you're frustrated, but I think the discussion is relevant to the content of your paper.
My interpretation of your paper is that it's a more rigourous version of "extraordinary claims require extraordinary evidence" - a claim with a high prior probability input is easier to accept with a lower confidence interval in the test protocol.[\quote]Hurray! That was one of the things I'd hoped the curves illustrate - except that the confidence interval does not change with the prior. In the prediction example, a believer in precognition will reject chance as an explanation for their results too soon. We could technically work the system backwards and produce the effective prior the person is using (unconsciously). For example - this skeptical audience will intuitively reject chance after 5 or so tosses ... which corresponds to a prior of 0.5. Probably this is due to experience of paranormal testing. 95% of the time we see someone saying "look at me I can predict 5 tosses in a row" they are cheating - the other 5% we'd expect them to repeat the performance what: about one time in 20? We have to be careful about this though - our intuitive estimate of the prior kinda depends, amongst other things, on our assumptions about the context of the experiment.
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Cearly you will need to have some justification, and the A Fair Coin Revisted has an example of how to go about it. But the main point of A Fair Coin? is to examine how assumptions about the prior affect the overall outcome. This can short-cut a lot of the debate in those other threads because, once you quantify the effect, you may find it is not such a big gap. Using the model as an illustration: a mundane claim would have a low prior ... at p=0.5 you'd reject the hypothesis after only 5 or so contrary results. We don't need much evidence. Perhaps it fits well with everything else we feel we can prove? This is why pseudoscience exists - to give the appearance that an odd claim is actually quite mundane. Then you can be tricked into giving it less scrutiny than it deserves. An extraordinary claim is at the other end, see the graph for p=0.9999. We would want to see a lot of careful testing before we start to accept it but once those tests have been done, the acceptance gets high quite fast with subsequent supporting evidence. The claim is never completely accepted ... but at the top end of the curve it is very flat. If, in the coin-toss thing, we have rigged the experiment so that we have eliminated every possible way of cheating that has ever been used, then we are going to need longer to conclude that something we don't already know about is going on with the coin. You'll see that emphasis on controlling variables in the experiment has already been mentioned.
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You still have the argument about what counts as a reasonable prior... fortunately, it is very unusual for a claimant to actually perform so far over odds as to make this sort of thing a problem in an actual test... but showing them the curves will help some of them understand why we expect longer tests. You still have suspicion of cheating the other way though ... a believer may come to the conclusion that the test is biased in some underhand way because we are part of a cover up or something ... the model does not do much for irrational fears. But it may help guide us to an assessment of what counts as irrational. Which is where I started. It is not conclusive - it is not intended to be. Just one more discussion tool. Help me make it more useful... maybe I can turn it into a book later: I'll give you all credit |
17th February 2011, 06:45 PM | #160 |
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Indeed, and the question is just as pertinent there. Your position is analogous to asserting that such a perfectly calm state is physically impossible rather than simply very unlikely.
You have no basis for that claim, just as you have no basis for your claim about coin flips. We, on the other hand, have a very strong basis for ours (that it is possible). |
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