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So... I wrote this. Maybe I can clarify a little what I meant by the statement "Since Silver’s forecasts begin with probability models, it’s safe to assume they obey all the rules, including Bayes’, and would be arbitrage-free.", since this seems to be confusing some people: What I'm addressing here is Taleb's claim that Silver's probabilities would allow for arbitrage (i.e., riskless profit, not just profit on average) if turned into betting prices. This is in the sense of arbitrage-through-time, buying low and then selling high. As I discussed in the piece, an old argument due to de Finetti says that if prices are arbitrage-free they must satisfy the equations of probability, meaning you can in some sense think of the price as giving a probability of the outcome. For time-dynamic arbitrage the relevant equation is Bayes' Theorem. All I meant by my statement above is that the converse to de Finetti's argument is also true, trivially. If prices obey the equations of probability they are automatically arbitrage-free. And since Silver's probabilities begin life as probabilities, they satisfy all the relevant equations (one would expect). Technically, the way we'd express this in modern finance is through the Fundamental Theorem of Asset Pricing, which says a (complete) market is arbitrage-free if and only if there exists an equivalent measure under which asset prices are martingales. Silver's probabilities are necessarily martingales, just because of the way conditional probability math works, so unless Taleb can claim that his and Silver's probabilities aren't equivalent, meaning they disagree on what events have probability zero, then there is no chance of arbitrage. That's just the mathy way of saying the same thing I said in the post. There are many other possible errors Silver could be making, and many other possible criticisms Taleb could have made but did not. In this case he wrote a paper claiming a mathematical result that just isn't true. Hope that's helpful! |