This is mostly nonsense; the work is being done by this assumption that is mentioned quickly in passing:
> Let's suppose you can be sure that one of the three particular Sticky/Switchy/Steady hypotheses in Figures 1–3 are true, but you can't be sure which.
In reality, if you don't know what's true, you also don't know that it must be one of a small set of convenient models.
Right, so in that case it isn't clear that the committing the Gambler's Fallacy is going to fare any worse than other prediction models.
This is IMHO the best kind of nonsense -- by its own "nonsensical" arguments, it shows that the textbook-derived arguments aren't really that great either. Seriously, as the article points out, a lot about probability are just tautological based on artificial assumptions -- eg. "assume a fair coin", but of course coin tosses in the physical world aren't 100% fair.
I think the author will agree with your sentiment that people assuming convenient models is the problem. If I understand the article correctly, it wasn't to make the assumption as a general statement about coin/koin tosses, but to illustrate alternative processes/models over the textbook "fair coin toss", which in itself, has been demonstrated to be not so fair in empirical studies.
TL;DR - pages and pages of word salad and pointless math which boils down to – the gambler's fallacy is a fallacy if the coin is unbiased, but maybe it isn't.
Doesn't this itself prove the article's point though? If you don't know anything, it seems rather presumptuous to say somebody has committed a fallacy.
I think it's slightly more interesting than that. It's not talking about bias (usually understood to mean that there's >50% chance that it lands on one specific side) but about independence.
Many things aren't independent. The article mentions the odds that it will rain on day depends on whether it rained the previous day. Likewise, the odds of a bus arriving in a city with a well-organized strict bus schedule is less if a bus just recently arrived.
How about blackjack? If I get an ace, are the odds still 1/52 that my next card will be an ace? Of course not.
How about the lottery? Are the odds that a scratch-off ticket is a winner different if the previous ticket on the roll was a winner? I have no idea, but I wouldn't be surprised if the lottery runners introduced such shenanigans.
I think that's the author's only point - that we shouldn't always assume independence.
But it falls apart a little because it seems to be talking about coins (even if it's actually talking about koins), which is a domain that starts from the premise that events are independent.
> But it falls apart a little because it seems to be talking about coins (even if it's actually talking about koins), which is a domain that starts from the premise that events are independent.
And that premise is empirically false as noted by the author.
Now the question is whether it's meaningful to accuse people of committing fallacies in hypothetical scenarios where perfectly fair coins are real.
This word you used, "independent" has a term in these kind of probabilistic systems, its called "memory". A system is memoryless if the next outcome is not dependent on previous outcomes. A coin flip is such a memoryless system. The odds it will rain on a particular day is not independent of pervious outcomes; such a system is not memoryless. All the examples in your comment share this property and so the reasoning doesn't strictly apply to memoryless systems.
The author is not making this point, and early on, he calls coin flips "Markovian" which is synonymous with "memoryless". Independence is assumed in this article.
Specifically, Markovian means that the probabilities of potential future states are dependent on the current state only. This is memoryless (more specifically, it has what's called the Markovian property) because no past states have anything to do with the next state.
A whole lot of words just to say that you can make any assumptions you want and get any predictions you want, in the face of ambiguous unreliable information.
But the author doesn't even realize that's what they are arguing.
I'm surprised that HN is so quiet on Sunday morning that 8 pts is enough to make #5 frontpage post for an article that fails the Wikipedia Test (”is the article better than the Wikipedia post on the same topic?”)
Actually the 'Gambler's Fallacy' is an example of Cognitive Dissonance. You are holding two 'truths/fallacies' in your head at the same time.
Truth/Fallacy 1: There is a 50/50 probability that the next flip will be either Heads or Tails.
Truth/Fallacy 2: In the long string of flips, the probability is that there will be a 50:50 distribution of Heads and Tails, so the next flip be a value which will tend to regress the overall value to the mean of 50:50.
If you hold to Truth No. 1, you have no difficulty in believing that 1000 flips can come up 1000 Tails just as easily as the 50:50 result.
If you hold to Truth No. 2 you are guilty of subscribing to the 'Gambler's Fallacy'.
> If you hold to Truth No. 1, you have no difficulty in believing that 1000 flips can come up 1000 Tails just as easily as the 50:50 result.
There are (1000 choose 500) ways to an equal number of heads/tails and 1 way to get all tails. If you get all tails, you should probably question the correctness of your model.
In the standard gambler's fallacy situation, it's assumed known that the coin is fair.
However, in real life there's always some probability that the assumption is incorrect.
One way to think about it is in terms of likelihoods, priors, and posteriors over models, in addition to the probability of an outcome conditional on a model.
So, the classical assumption is something like P(X | Mf) = 0.5 for a "fair" model Mf, and you're asking someone "what's the probability of heads?". However, there's also the possibility that the coin is actually biased, under Mb. So the actual probability of an observed sequence is something like
P(X|Mf)P(Mf) + P(X|Mb)P(Mb).
Usually we assume that P(Mf) >> P(Mb) but there must be some point at which P(Mb) becomes great enough that it would be rational to start to question that.
Implicitly there's some Bayesian estimate of P(Mb|X) that could be estimated, and some decision point where you decide P(Mb|X) > P(Mf|X).
this is an absolutely valid Bayesian approach to this problem. sadly, this point is not even touched upon in this article, which deals solely with minutiae of the probability distributions (kurtosis, etc) of coins that are indeed fair.
> But this can't be right––for no one commits that fallacy
I most certainly cannot agree with this premise. I’ve met many people who make this mistake all the time. I even have a friend who is amongst the smartest people I know who honest to god thinks he’s lucky. He believes that there is some force that allows him to either effect the next said coin toss or allows him to devine the next coin toss. It’s wild, he’s even really good at board games too, so it’d be easy to think he might be lucky too.
Yeah, if people truly think that they've never been to a casino in their life. Go walk to any roulette table that shows the results of previous spins. You'll find multiple people going "it's been reds for a while, black is due" or "this machine gets more reds, you should bet on red".
22 comments
[ 2.4 ms ] story [ 27.0 ms ] thread> Let's suppose you can be sure that one of the three particular Sticky/Switchy/Steady hypotheses in Figures 1–3 are true, but you can't be sure which.
In reality, if you don't know what's true, you also don't know that it must be one of a small set of convenient models.
Right, so in that case it isn't clear that the committing the Gambler's Fallacy is going to fare any worse than other prediction models.
This is IMHO the best kind of nonsense -- by its own "nonsensical" arguments, it shows that the textbook-derived arguments aren't really that great either. Seriously, as the article points out, a lot about probability are just tautological based on artificial assumptions -- eg. "assume a fair coin", but of course coin tosses in the physical world aren't 100% fair.
I think the author will agree with your sentiment that people assuming convenient models is the problem. If I understand the article correctly, it wasn't to make the assumption as a general statement about coin/koin tosses, but to illustrate alternative processes/models over the textbook "fair coin toss", which in itself, has been demonstrated to be not so fair in empirical studies.
Many things aren't independent. The article mentions the odds that it will rain on day depends on whether it rained the previous day. Likewise, the odds of a bus arriving in a city with a well-organized strict bus schedule is less if a bus just recently arrived.
How about blackjack? If I get an ace, are the odds still 1/52 that my next card will be an ace? Of course not.
How about the lottery? Are the odds that a scratch-off ticket is a winner different if the previous ticket on the roll was a winner? I have no idea, but I wouldn't be surprised if the lottery runners introduced such shenanigans.
I think that's the author's only point - that we shouldn't always assume independence.
But it falls apart a little because it seems to be talking about coins (even if it's actually talking about koins), which is a domain that starts from the premise that events are independent.
And that premise is empirically false as noted by the author.
Now the question is whether it's meaningful to accuse people of committing fallacies in hypothetical scenarios where perfectly fair coins are real.
The author is not making this point, and early on, he calls coin flips "Markovian" which is synonymous with "memoryless". Independence is assumed in this article.
Independence is not assumed in the "sticky" or "switchy" states in the article. The probability of the next flip depends on the previous flip.
You're right about the article.
I'm surprised that HN is so quiet on Sunday morning that 8 pts is enough to make #5 frontpage post for an article that fails the Wikipedia Test (”is the article better than the Wikipedia post on the same topic?”)
https://en.m.wikipedia.org/wiki/Gambler%27s_fallacy
Truth/Fallacy 1: There is a 50/50 probability that the next flip will be either Heads or Tails.
Truth/Fallacy 2: In the long string of flips, the probability is that there will be a 50:50 distribution of Heads and Tails, so the next flip be a value which will tend to regress the overall value to the mean of 50:50.
If you hold to Truth No. 1, you have no difficulty in believing that 1000 flips can come up 1000 Tails just as easily as the 50:50 result.
If you hold to Truth No. 2 you are guilty of subscribing to the 'Gambler's Fallacy'.
Which way do you jump? :)
There are (1000 choose 500) ways to an equal number of heads/tails and 1 way to get all tails. If you get all tails, you should probably question the correctness of your model.
In the standard gambler's fallacy situation, it's assumed known that the coin is fair.
However, in real life there's always some probability that the assumption is incorrect.
One way to think about it is in terms of likelihoods, priors, and posteriors over models, in addition to the probability of an outcome conditional on a model.
So, the classical assumption is something like P(X | Mf) = 0.5 for a "fair" model Mf, and you're asking someone "what's the probability of heads?". However, there's also the possibility that the coin is actually biased, under Mb. So the actual probability of an observed sequence is something like
P(X|Mf)P(Mf) + P(X|Mb)P(Mb).
Usually we assume that P(Mf) >> P(Mb) but there must be some point at which P(Mb) becomes great enough that it would be rational to start to question that.
Implicitly there's some Bayesian estimate of P(Mb|X) that could be estimated, and some decision point where you decide P(Mb|X) > P(Mf|X).
"The 'koin' is fixed. Bet Tails."
I have incomplete information on what a "koin" is -- I am told that it's like 50/50.
But PURELY from what I have seen, it clearly likes "tails."
Exactly why shouldn't I bet "tails?" This is pretty much the same as "yes, you can't PROVE that the sun won't rise tomorrow but that's how we act."
I most certainly cannot agree with this premise. I’ve met many people who make this mistake all the time. I even have a friend who is amongst the smartest people I know who honest to god thinks he’s lucky. He believes that there is some force that allows him to either effect the next said coin toss or allows him to devine the next coin toss. It’s wild, he’s even really good at board games too, so it’d be easy to think he might be lucky too.