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Well obviously Silver isn't actually offering to bet at those odds. But complaining that a model is incorrect because it enables arbitrage is a standard compla
by ikeboy 8y ago
Well obviously Silver isn't actually offering to bet at those odds.
But complaining that a model is incorrect because it enables arbitrage is a standard complaint, it's how you can prove all non-Bayesian models are irrational. That doesn't mean all Bayesian ones automatically pass the test though.
Error bars and spreads are already outside the Bayesian paradigm, which demands a single number.
- et2o 8y agoCurious about this. > It's how you can prove all non-Bayesian models are irrational Can you expand or link? > Error bars and spreads are already outside the Bayesian paradigm, which demands a single number Not following at all. What do you mean?
- ikeboy 8y agohttps://plato.stanford.edu/entries/epistemology-bayesian/supplement2.html https://plato.stanford.edu/entries/epistemology-bayesian/sup... >Not following at all. What do you mean? One assumption is that given a bet, you must choose to take a side of it. If you won't bet for something at X%, you must bet against it at 1-X%.
- et2o 8y agoAh, interesting link. I think I ran into some confusion since 'rational' and 'Bayesian' are somewhat overloaded terms. When I think of Bayesian analysis, I always think of Bayesian models which attempt to model uncertainty and always have a posterior distribution (spread) of outcomes. For example, in the Dutch Book page, instead of having a point-prior (P=0.51 (delta on P=0.51?)), each degree of belief could have some (different) distribution which could yield a posterior with a distribution and change the math here.
- galby 8y agoCan you expand on this? Bayesian data analysis looks at the full posterior, and reducing that to a single number is frowned upon by most.
- ikeboy 8y agoIf you're looking at a single proposition then the probability of it being true is a single number. To say that the probability of it being true is a range is incoherent.
- galby 8y agoI think most bayesians would disagree with you on this point :) It's not incoherent to analyze the posterior distribution by reporting a credibility interval instead of a single number (for example the posterior median).
- ikeboy 8y agoAn interval to estimate a parameter is different from an interval to represent a probability. You can't have a range for a probability, you can have a range for a parameter you're trying to estimate.
- comex 8y agoWell, in theory you can divide the uncertainty into two parts, by supposing that the outcome is driven by a fundamentally stochastic process and you also have imperfect information about the parameters of that process. For example, the outcome could be determined by a coin flip but you’re not sure whether the coin is fair. In that case, there is a “true” probability of heads based on the nature of the coin, and separately, a Bayesian observer can have a probability distribution for (i.e. representing their beliefs about) the value of the true probability. The observer could then come up with a single number representing their belief in heads by taking the expected value of that probability distribution, and if they just want to gamble on the outcome, that number would be all they need. But in order to correctly update their beliefs given future information about the parameters, they have to remember the original probability distribution; they also might just be curious about the nature of the underlying stochastic process, in addition to the final outcome. How well that models an actual election is debatable, of course, but I think it does model it to some degree. In reality, there are not two stages but multiple, and none of those stages are necessarily fundamentally stochastic; rather, you just need exponentially more information to predict one stage than to predict the previous one, and without that information you may as well treat it as stochastic. For example, if I’m about to flip a coin, a god with exact knowledge of the state of my body and brain, the air currents in the room, etc. might be able to predict how I’ll throw it and how it will fall, but mere mortals have to treat a coin flip as random. Similarly, a god with exact knowledge about the state of the universe might be able to predict an election result eons in advance… though quantum randomness might trip them up. Getting more down to earth, if you just could poll every American about their political beliefs, you could make much better predictions than you can with real polls, which have to take random samples and thus accept some level of stochastic polling error. On the other hand, polls can also suffer from methodological error, which is fundamentally different in nature; it can be highly pernicious, but does require a smaller quantity of information to correct for. And so on.