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> The probability was .018 under their model. This doesn't mean that this is the true probability. Well of course different models will give you different prob

by yuliyp 8y ago

> The probability was .018 under their model. This doesn't mean that this is the true probability. Well of course different models will give you different probabilities. But given that this is a past event, talking about "true probability" is a bit weird. The only "true" probabilities are 0 and 1. There's nothing nondeterministic about the past. Someone wrote the song.

azernik 8y ago
That's not how Bayesian statistics work. In this model (on which statistical learning and information theory based) a probability does not require nondeterminism, just incomplete information; and a probability is defined with respect to a certain amount/type of information. (Actually information is defined in terms of things that change probability, and probability is the more "core" concept.) So for example, if someone rolls a six-sided die, hides it from you, and asks what the probability of a "6" is, under Bayesian theory it's 1/6; if they then tell you that the number rolled was odd, the probability of a "6" drops to zero, even though nothing physically changed, purely because you now have more information. This is generally a much more useful and intuitive definition of probability than to say "the probability of it being a 6 is either 0 or 1, I just don't know yet".
- gweinberg 8y ago
  Bayeseian probability is always probability within a model. For things like dice where it's pretty uncontroversial to say the only sensible model is one where the a priori probability of the die landing on each face is equal. For questions like "who wrote the song" it is much less clear what the proper model should be.
  - nonbel 8y ago
    >"Bayeseian probability is always probability within a model." Is there a conception of probability that is not conditional on a model?
    - antidesitter 8y ago
      Perhaps the most general model is that of algorithmic probability (the probability of a string being output when inputting random strings into a Universal Turing Machine): https://en.wikipedia.org/wiki/Algorithmic_probability https://en.wikipedia.org/wiki/Algorithmic_probability. See also Solomonoff induction and the universal semimeasure.
      - nonbel 8y ago
        Sorry, but I really have no patience for these purely "philosophical" discussions of probability. I don't see a single equation on that page but I can assure you that to compute a probability you are assuming some form of the likelihood, ie the way you calculate P(D|H), applies to your problem.
        antidesitter 8y ago
        Sorry, but I really have no patience for these purely "philosophical" discussions of probability. I’m not sure what you mean by this. Can you elaborate? I don't see a single equation on that page If equations are what you’re looking for, here are a few good starting points: https://link.springer.com/chapter/10.1007/978-0-387-84816-7_1 https://link.springer.com/chapter/10.1007/978-0-387-84816-7_... (Ray Solomonoff, founder of algorithmic information theory) https://www.springer.com/gp/book/9783540221395 https://www.springer.com/gp/book/9783540221395 (Marcus Hutter) https://link.springer.com/book/10.1007/978-3-642-44958-1 https://link.springer.com/book/10.1007/978-3-642-44958-1 but I can assure you that to compute a probability you are assuming some form of the likelihood, ie the way you calculate P(D|H), applies to your problem. You should really take the time to read the Scholarpedia article (or above sources) on algorithmic probability and Solomonoff induction. It’s one of the most beautiful and elegant examples of mathematically formalizing perennial philosophical intuitions. The reason I brought up algorithmic probability is because there is a precise mathematical sense in which it is “optimal”. Again, I encourage you to read about universal semimeasures.
        nonbel 8y ago
  - azernik 8y ago
    Sure. GGP was claiming that it's only 0.018 "under this model", GP claimed no models mattered because this is a past event and probability is undefined, and I (parent comment) said that it does make sense to talk about probability in this context with some model. In this case, a good model would probably include things like "a songwriter tends to write things similar to their other works", "this is the set of works written by Paul McCartney", "this is the set of works written by John Lennon", "these are the works they did together at varying degrees of cooperation", and "this is how a creative process works". All very hard to quantify and program, but possible with expert help.
- lottin 8y ago
  The problem I have with the Bayesian definition of probability is that it isn't a definition, is it? Since probability is a ratio, you need to define two amounts. So what does the 1 in 1/6 actually mean? And the 6? What does it mean that someone is 1/6 certain of something? How does one measure certainty? The more I think about it the more I'm convinced Bayesian probabilities are a flawed concept.
  - nonbel 8y ago
    >"The problem I have with the Bayesian definition of probability is that it isn't a definition, is it?" You'll have to clarify what exactly your problem with the concept is here. >"So what does the 1 in 1/6 actually mean? And the 6?" It means there are six possible outcomes, one of which is of special interest and five others. You have no reason to think any one possibility is more likely than any other. >"What does it mean that someone is 1/6 certain of something?" You would take any bet on it with better than 6 to 1 odds. If you have to decide between a 1/6 opportunity and a 1/60th opportunity for the same reward you should choose the first. You can use such probabilities to perform a cost-benefit analysis amongst various possible actions with various costs and rewards. >"How does one measure certainty?" I assume you are using "certainty" as a synonym for "probability", in which case one way is to ask people to bet. >"The more I think about it the more I'm convinced Bayesian probabilities are a flawed concept." Ok, but your objections here don't seem to have much thought/experience behind them.
  - azernik 8y ago
    I'm not sure what you mean by "isn't a definition." Mathematically, it's extremely well defined - as a ratio of the subjective likelihood of one possibility, compared to the subjective likelihoods of all possibilities. Your exact physical interpretation can vary - from a decision-theoretic (at what odds would it be worth it to make a bet?), to an information-theoretic (how many bits does it take to encode that it's this possibility rather than any other?), etc. Generally, like quantum mechanics, while there are many interpretations of the underlying reality corresponding to the theory, in practice it is impossible to interpret the world accurately (correlation/independence, inference/learning, etc.) with previous (in this case frequentist) theories.
antidesitter 8y ago
Under the Bayesian definition, probability is a degree of belief. Probabilities belong “inside” an agent, not somewhere “out there” in the world. For a justification of this, see the Dutch Book Argument: “The main point of the Dutch Book Argument is to show that rational people must have subjective probabilities for random events, and that these probabilities must satisfy the standard axioms of probability.”
posterboy 8y ago
the probability that we know the true probababillity is what's concerned here.

by yuliyp 8y ago

azernik 8y ago

That's not how Bayesian statistics work. In this model (on which statistical learning and information theory based) a probability does not require nondeterminism, just incomplete information; and a probability is defined with respect to a certain amount/type of information. (Actually information is defined in terms of things that change probability, and probability is the more "core" concept.) So for example, if someone rolls a six-sided die, hides it from you, and asks what the probability of a "6" is, under Bayesian theory it's 1/6; if they then tell you that the number rolled was odd, the probability of a "6" drops to zero, even though nothing physically changed, purely because you now have more information. This is generally a much more useful and intuitive definition of probability than to say "the probability of it being a 6 is either 0 or 1, I just don't know yet".

gweinberg 8y ago
Bayeseian probability is always probability within a model. For things like dice where it's pretty uncontroversial to say the only sensible model is one where the a priori probability of the die landing on each face is equal. For questions like "who wrote the song" it is much less clear what the proper model should be.
- nonbel 8y ago
  >"Bayeseian probability is always probability within a model." Is there a conception of probability that is not conditional on a model?
  - antidesitter 8y ago
    Perhaps the most general model is that of algorithmic probability (the probability of a string being output when inputting random strings into a Universal Turing Machine): https://en.wikipedia.org/wiki/Algorithmic_probability https://en.wikipedia.org/wiki/Algorithmic_probability. See also Solomonoff induction and the universal semimeasure.
    - nonbel 8y ago
      Sorry, but I really have no patience for these purely "philosophical" discussions of probability. I don't see a single equation on that page but I can assure you that to compute a probability you are assuming some form of the likelihood, ie the way you calculate P(D|H), applies to your problem.
      - antidesitter 8y ago
        Sorry, but I really have no patience for these purely "philosophical" discussions of probability. I’m not sure what you mean by this. Can you elaborate? I don't see a single equation on that page If equations are what you’re looking for, here are a few good starting points: https://link.springer.com/chapter/10.1007/978-0-387-84816-7_1 https://link.springer.com/chapter/10.1007/978-0-387-84816-7_... (Ray Solomonoff, founder of algorithmic information theory) https://www.springer.com/gp/book/9783540221395 https://www.springer.com/gp/book/9783540221395 (Marcus Hutter) https://link.springer.com/book/10.1007/978-3-642-44958-1 https://link.springer.com/book/10.1007/978-3-642-44958-1 but I can assure you that to compute a probability you are assuming some form of the likelihood, ie the way you calculate P(D|H), applies to your problem. You should really take the time to read the Scholarpedia article (or above sources) on algorithmic probability and Solomonoff induction. It’s one of the most beautiful and elegant examples of mathematically formalizing perennial philosophical intuitions. The reason I brought up algorithmic probability is because there is a precise mathematical sense in which it is “optimal”. Again, I encourage you to read about universal semimeasures.
        nonbel 8y ago
- azernik 8y ago
  Sure. GGP was claiming that it's only 0.018 "under this model", GP claimed no models mattered because this is a past event and probability is undefined, and I (parent comment) said that it does make sense to talk about probability in this context with some model. In this case, a good model would probably include things like "a songwriter tends to write things similar to their other works", "this is the set of works written by Paul McCartney", "this is the set of works written by John Lennon", "these are the works they did together at varying degrees of cooperation", and "this is how a creative process works". All very hard to quantify and program, but possible with expert help.
lottin 8y ago
The problem I have with the Bayesian definition of probability is that it isn't a definition, is it? Since probability is a ratio, you need to define two amounts. So what does the 1 in 1/6 actually mean? And the 6? What does it mean that someone is 1/6 certain of something? How does one measure certainty? The more I think about it the more I'm convinced Bayesian probabilities are a flawed concept.
- nonbel 8y ago
  >"The problem I have with the Bayesian definition of probability is that it isn't a definition, is it?" You'll have to clarify what exactly your problem with the concept is here. >"So what does the 1 in 1/6 actually mean? And the 6?" It means there are six possible outcomes, one of which is of special interest and five others. You have no reason to think any one possibility is more likely than any other. >"What does it mean that someone is 1/6 certain of something?" You would take any bet on it with better than 6 to 1 odds. If you have to decide between a 1/6 opportunity and a 1/60th opportunity for the same reward you should choose the first. You can use such probabilities to perform a cost-benefit analysis amongst various possible actions with various costs and rewards. >"How does one measure certainty?" I assume you are using "certainty" as a synonym for "probability", in which case one way is to ask people to bet. >"The more I think about it the more I'm convinced Bayesian probabilities are a flawed concept." Ok, but your objections here don't seem to have much thought/experience behind them.
- azernik 8y ago
  I'm not sure what you mean by "isn't a definition." Mathematically, it's extremely well defined - as a ratio of the subjective likelihood of one possibility, compared to the subjective likelihoods of all possibilities. Your exact physical interpretation can vary - from a decision-theoretic (at what odds would it be worth it to make a bet?), to an information-theoretic (how many bits does it take to encode that it's this possibility rather than any other?), etc. Generally, like quantum mechanics, while there are many interpretations of the underlying reality corresponding to the theory, in practice it is impossible to interpret the world accurately (correlation/independence, inference/learning, etc.) with previous (in this case frequentist) theories.

antidesitter 8y ago

Under the Bayesian definition, probability is a degree of belief. Probabilities belong “inside” an agent, not somewhere “out there” in the world. For a justification of this, see the Dutch Book Argument: “The main point of the Dutch Book Argument is to show that rational people must have subjective probabilities for random events, and that these probabilities must satisfy the standard axioms of probability.”

posterboy 8y ago

the probability that we know the true probababillity is what's concerned here.