“The gambler’s fallacy” refers to the situation where successive outcomes are known to be independent (i.e. known to be Steady, and neither Switchy nor Streaky), and yet the gambler acts according to Switchiness anyway. The gambler in that situation is wrong and the Bayesian will not commit that error.
When Switchiness and Streakiness are open possibilities, then of course the evidence so far may favour either of them; but if evidence accumulates for either of them it is no longer an error to favour it.
Would it have helped if I added the attached paragraphs (in the paper, page 3, cut for brevity)?
Frame the conclusion as a disjunction: “either we construe ‘gambler’s fallacy’ narrowly (as by definition irrational) or broadly (as used in the blog post, for expecting switches). If the former, we have little evidence that real people commit the gambler’s fallacy. If the latter, then the gambler’s fallacy is not a fallacy.”
This seems to be an argument against the very idea of an error. How can people possibly make errors of reasoning? If the gambler knows the die rolls are independent, how could they believe in streaks? How could someone who knows the spelling of a word possibly mistype it? There seems to be a presumption of logical omniscience and consistency.
I am not sure that is right. A very large percentage of people really don’t think the rolls are independent. Have you ever met anyone who believed in fate, Karma, horoscopes , lucky objects or prayer? They don’t think its (fully) random and independent. I think the majority of the human population believe in one or more of those things.
If someone spells a word wrong in a spelling test, then its possible they mistyped, but if its a word most people can’t spell correctly then the hypothesis “they don’t know the spelling’ should dominate. Similarly, I think it is fair to say that a very large fraction of humans (over 50%?) don’t actually think dice rolls or coin tosses are independent and random.
I am not sure that is right. A very large percentage of people really don’t think the rolls are independent. Have you ever met anyone who believed in fate, Karma, horoscopes , lucky objects or prayer? They don’t think its (fully) random and independent. I think the majority of the human population believe in one or more of those things.
“The gambler’s fallacy” refers to the situation where successive outcomes are known to be independent (i.e. known to be Steady, and neither Switchy nor Streaky), and yet the gambler acts according to Switchiness anyway. The gambler in that situation is wrong and the Bayesian will not commit that error.
When Switchiness and Streakiness are open possibilities, then of course the evidence so far may favour either of them; but if evidence accumulates for either of them it is no longer an error to favour it.
Would it have helped if I added the attached paragraphs (in the paper, page 3, cut for brevity)?
Frame the conclusion as a disjunction: “either we construe ‘gambler’s fallacy’ narrowly (as by definition irrational) or broadly (as used in the blog post, for expecting switches). If the former, we have little evidence that real people commit the gambler’s fallacy. If the latter, then the gambler’s fallacy is not a fallacy.”
This seems to be an argument against the very idea of an error. How can people possibly make errors of reasoning? If the gambler knows the die rolls are independent, how could they believe in streaks? How could someone who knows the spelling of a word possibly mistype it? There seems to be a presumption of logical omniscience and consistency.
I am not sure that is right. A very large percentage of people really don’t think the rolls are independent. Have you ever met anyone who believed in fate, Karma, horoscopes , lucky objects or prayer? They don’t think its (fully) random and independent. I think the majority of the human population believe in one or more of those things.
If someone spells a word wrong in a spelling test, then its possible they mistyped, but if its a word most people can’t spell correctly then the hypothesis “they don’t know the spelling’ should dominate. Similarly, I think it is fair to say that a very large fraction of humans (over 50%?) don’t actually think dice rolls or coin tosses are independent and random.
They may well do. But they are wrong.