If over time, your forecasts predictably change, then you can be arbitraged.
If e.g. I know that you'll have a probability below 40% at some point in the future, and I know you'll have a number above 41% at some point in the future, then it's trivial to create a strategy over time that is guaranteed to make money.
Taleb's claim, as I understand it, is that Silver is incorporating random noise into his model and so will exhibit such predictably swings, which enables arbitrage.
If over time, your forecasts predictably change, then they are not the best forecasts you could make with the information on hand (I take it that by "predictably change", you mean that they will change in a predictable way, not just that it is predictable that they will change somehow.)
If I am not mistaken, an opportunity for arbitrage is not created merely by a certainty that they will change somehow; it would require that they change in a specific way, and deterministically so. Absent this, a bet would just be speculation, not arbitrage.
>If over time, your forecasts predictably change, then they are not the best forecasts you could make with the information on hand.
And this is precisely the complaint Taleb is making.
An arbitrage is just a strategy that's guaranteed to make money. If you know for sure that something will go up and down at different points you could make money risk free. Simply buy at any point below the current price and sell at any point above.
> An arbitrage is just a strategy that's guaranteed to make money. If you know for sure that something will go up and down at different points you could make money risk free. Simply buy at any point below the current price and sell at any point above.
This is not arbitrage. Also this is precisely the kind of strategy Taleb would have argued against in his book about black swan events.
If someone was flipping a coin and raising their price when it landed heads and lowering it when tails, arbitraging using reversion to the means would be a perfectly legitimate arb strategy.
Taleb's argument is that Silver's estimates are based on random noise, and are thus vulnerable to this form of arbitrage.
It think it’s… sort of the reverse? Flipping a coin results in Brownian motion, which is an example of a “martingale”, a function whose expected future value is equal to its current value; and the evolution of a share price being a martingale (at least under a modified probability measure) is exactly the condition for the market to not have arbitrage:
I’m not sure exactly how the arbitrage works, though.
But the argument in the paper seems to be not that there’s noise but that there’s too much of it. In fact, it assumes that the expected vote share over time is subject to Brownian motion with specific parameters (which is a rather flawed assumption), but then concludes that 538’s probability distribution is more swingy what he’d calculate from those parameters.
The point is that Silver's predictions are the result of coin flips, but that the underlying probability isn't. If he's updating further than he should then you can make money by updating the correct amount and betting against him.
A price set by coin-flips does not revert to any mean. It's a random walk and it diverges. The idea that heads and tails must eventually even out is a common fallacy.
Oh so the coin-flipper is not setting the price, but trying to predict it? I think at least I understand what you're explaining. I don't see how that is a useful model for what is going on though.
Predicting the result of a vote for a day when it's not going to be held is a form of entertainment that can't be verified. Is Silver claiming any more than that?
Yes, Silver is absolutely trying to forecast elections months in advance. Taleb's argument is that the polls at that point are largely noise, and so Silver is updating too much on them when he should basically ignore most of them until shortly before the election.
Silver's forecasts do predictably change, more precisely in that they swing deterministically in favour of one candidate whenever the most recent poll to come in favours that candidate heavily.
Each time the probability drops below 40% (in your example) it is likely because the fair odds swung in the latest polls. To attempt to take advantage of this is to ignore the recent evidence (conditional upon that evidence being quite different to the current concensus).
It's not at all unreasonable to do so, but isn't arbitrage. What if all polls started being increasingly conclusive leading up to election? 40, 39, ..., 0. No surprises and no arbitrage.
Perhaps another way to express this critique of Silver's methodology is that it lends too much weight on recent outlier observations? Sure if you knew this you could profit off these swings, but making the arbitrage argument doesn't add much to the discussion to my mind.
So what you're saying, I guess, is that Silver, by dint of making point probability predictions, is offering a two-sided market with infinite liquidity trading at that value for a binary option, and you could make arbitrage gains by trading off of volatility?
I'm having a fair bit of trouble picking up that from either Taleb's or Clayton's article, but let's say that it's a fair assessment.
Is the complaint that the observed variance is high enough to enable these arbitrages? So in order to make his model arbitrage-free, Silver would have to provide a spread rather than a point price, with his forecast as the midpoint of the spread, and the width of the spread wide enough that .5 is always inside the spread, thus making the forecast economically useless?
Or is it that Silver does not provide error bars that would estimate that uncertainty?
Or is it that because of intrinsic uncertainty in the election?
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.
Ah, 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.
I 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).
An 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.
Well, 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.
Sure, but as a good Bayesian you should be willing to bet based on that expected value, not based on the spread.
Re polls: still more complicated than that, even with 100% polling you'd still have response error and non-voters. Much of the difference between polls is their assumptions about demographics of voters.
Sure, but that doesn’t mean 538 can’t include information about the spread, in addition to their top-line number which is just a single probability.
And yes, my discussion of polling error was oversimplified, but I was mainly just trying to explain what I meant about having to treat something as random if it’s theoretically predictable given enough information.
EDIT: Hold on; I don’t think 538 does include error bars for their probabilities. They have error bars for measurables like the vote share, but that’s normal. So I don’t understand what this argument is even about :)
Someone several comments up suggested Silver should include error bars and should bet based on a spread, I said that was not proper Bayesian methods, then we had this back and forth.
If e.g. I know that you'll have a probability below 40% at some point in the future, and I know you'll have a number above 41% at some point in the future, then it's trivial to create a strategy over time that is guaranteed to make money.
Taleb's claim, as I understand it, is that Silver is incorporating random noise into his model and so will exhibit such predictably swings, which enables arbitrage.