“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.
“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.