If I’m understanding the paper correctly—and I’ve only looked at it very briefly so there’s an excellent chance I haven’t—there’s an important asymmetry here which is worth drawing attention to.
The paper is concerned with two quite specific “Sticky” and “Switchy” models. They look, from glancing at the transition probability matrices, as if there’s a symmetry between sticking and switching that interchanges the models—but there isn’t.
The state spaces of the two models are defined by “length of recent streak”, and this notion is not invariant under e.g. the prefix-XOR operation mentioned by James Camacho.
What does coincidental evidence for Sticky look like? Well, e.g., 1⁄8 of the time we will begin with HHHH or TTTT. The Sticky:Switchy likelihood ratio for this, in the “5-step 90%” model whose matrices are given in the OP, is 1 (first coin) times . 58/.42 (second coin) times .66/.34 (third coin) times .74/.26 (fourth coin), as the streak builds up.
What does coincidental evidence for Switchy look like? Well, the switchiest behaviour we can see would be HTHT or THTH. In this case we get a likelihood ratio of 1 (first coin) times .58/.42 (second coin) times .58/.42 (third coin) times .58/.42 (third coin), which is much smaller.
It’s hard to get strong evidence for Switchy over Sticky, because a particular result can only be really strong if it was preceded by a long run of sticking, which itself will have been evidence for Sticky.
This seems like a pretty satisfactory explanation for why this set of models produces the results described in the paper. (I see that Mlxa says they tried to reproduce those results and failed, but I’ll assume for the moment that the correct results are as described. I wonder whether perhaps Mlxa made a model that doesn’t have the multi-step streak-dependence of the model in the paper.)
What’s not so obvious to me is whether this explanation makes it any less true, or any less interesting if true, that “Bayesians commit the gambler’s fallacy”. It seems like the bias reported here is a consequence of choosing these particular “sticky” and “switchy” hypotheses, and I can’t quite figure out whether a better moral would be “Bayesians who are only going to consider one hypothesis of each type should pick different ones from these”, or whether actually this sort of “depends on length of latest streak, but not e.g. on length of latest alternating run” hypothesis is an entirely natural one.
If I’m understanding the paper correctly—and I’ve only looked at it very briefly so there’s an excellent chance I haven’t—there’s an important asymmetry here which is worth drawing attention to.
The paper is concerned with two quite specific “Sticky” and “Switchy” models. They look, from glancing at the transition probability matrices, as if there’s a symmetry between sticking and switching that interchanges the models—but there isn’t.
The state spaces of the two models are defined by “length of recent streak”, and this notion is not invariant under e.g. the prefix-XOR operation mentioned by James Camacho.
What does coincidental evidence for Sticky look like? Well, e.g., 1⁄8 of the time we will begin with HHHH or TTTT. The Sticky:Switchy likelihood ratio for this, in the “5-step 90%” model whose matrices are given in the OP, is 1 (first coin) times . 58/.42 (second coin) times .66/.34 (third coin) times .74/.26 (fourth coin), as the streak builds up.
What does coincidental evidence for Switchy look like? Well, the switchiest behaviour we can see would be HTHT or THTH. In this case we get a likelihood ratio of 1 (first coin) times .58/.42 (second coin) times .58/.42 (third coin) times .58/.42 (third coin), which is much smaller.
It’s hard to get strong evidence for Switchy over Sticky, because a particular result can only be really strong if it was preceded by a long run of sticking, which itself will have been evidence for Sticky.
This seems like a pretty satisfactory explanation for why this set of models produces the results described in the paper. (I see that Mlxa says they tried to reproduce those results and failed, but I’ll assume for the moment that the correct results are as described. I wonder whether perhaps Mlxa made a model that doesn’t have the multi-step streak-dependence of the model in the paper.)
What’s not so obvious to me is whether this explanation makes it any less true, or any less interesting if true, that “Bayesians commit the gambler’s fallacy”. It seems like the bias reported here is a consequence of choosing these particular “sticky” and “switchy” hypotheses, and I can’t quite figure out whether a better moral would be “Bayesians who are only going to consider one hypothesis of each type should pick different ones from these”, or whether actually this sort of “depends on length of latest streak, but not e.g. on length of latest alternating run” hypothesis is an entirely natural one.