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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 i

by galby 8y ago

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).

ikeboy 8y ago
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.
- comex 8y ago
  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.
  - ikeboy 8y ago
    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.
    - comex 8y ago
      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 :)
      - ikeboy 8y ago
        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.

by galby 8y ago

ikeboy 8y ago

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.

comex 8y ago
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.
- ikeboy 8y ago
  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.
  - comex 8y ago
    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 :)
    - ikeboy 8y ago
      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.