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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 amount

by 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.
- lottin 8y ago
  We agree that a probability is a ratio, right? In order to make sense of what a ratio means, we must know what the numerator and the denominator mean. From what I gather, Bayesians say probability is a degree of certainty, so a subjective belief. Okay, but in their calculations, mathematically, it's still a ratio. So what are the terms in this ratio? This is what I'm saying. They don't say it. If you say that these terms are "subjective likelihoods", then you have to provide a definition for this, because it isn't clear at all what a "likelihood" is.
  - nanis 8y ago
    > We agree that a probability is a ratio, right? No, probability is a normalized denumerably additive measure defined over a σ-algebra of subset of an abstract space. (see Kolmogorov). Probability is a completely mathematical construct which is devoid of empirical content. Now, we have used to to varying degrees of success in various situations, by assuming various things which make varying degrees of sense, but it is important keep in mind that all of that follows from a handful of axioms none of which states that probability is a ratio. Neither classical or Bayesian attempts to give probability empirical meaning make sense in these types of one off situations, though.
    - antidesitter 8y ago
      Bayesian attempts to give probability empirical meaning [do not] make sense in these types of one off situations, though. I agreed with your comment until this. What do you mean?
      - nanis 8y ago
        Either John wrote the song or Paul did. Therefore P(Paul wrote it) either one or zero. It cannot be anything else. We do not know which one and we are not going to find out by counting or doing arithmetic. After listening to the song one more time, and reading the article again, I've updated my beliefs that this senior lecturer in Stats fancies himself some kind of cool guy, and is trying to make name for himself by immersing himself in the study of women's beach volleyball[1] and Beatles. Also, it sounds to me like this something Paul would have written. Now, can you please explain what it means to calculate that P(Paul wrote it) = 0.018? Note that I am not asking about how the calculations work. [1]: http://www.glicko.net/ http://www.glicko.net/

by lottin 8y ago

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.

lottin 8y ago
We agree that a probability is a ratio, right? In order to make sense of what a ratio means, we must know what the numerator and the denominator mean. From what I gather, Bayesians say probability is a degree of certainty, so a subjective belief. Okay, but in their calculations, mathematically, it's still a ratio. So what are the terms in this ratio? This is what I'm saying. They don't say it. If you say that these terms are "subjective likelihoods", then you have to provide a definition for this, because it isn't clear at all what a "likelihood" is.
- nanis 8y ago
  > We agree that a probability is a ratio, right? No, probability is a normalized denumerably additive measure defined over a σ-algebra of subset of an abstract space. (see Kolmogorov). Probability is a completely mathematical construct which is devoid of empirical content. Now, we have used to to varying degrees of success in various situations, by assuming various things which make varying degrees of sense, but it is important keep in mind that all of that follows from a handful of axioms none of which states that probability is a ratio. Neither classical or Bayesian attempts to give probability empirical meaning make sense in these types of one off situations, though.
  - antidesitter 8y ago
    Bayesian attempts to give probability empirical meaning [do not] make sense in these types of one off situations, though. I agreed with your comment until this. What do you mean?
    - nanis 8y ago
      Either John wrote the song or Paul did. Therefore P(Paul wrote it) either one or zero. It cannot be anything else. We do not know which one and we are not going to find out by counting or doing arithmetic. After listening to the song one more time, and reading the article again, I've updated my beliefs that this senior lecturer in Stats fancies himself some kind of cool guy, and is trying to make name for himself by immersing himself in the study of women's beach volleyball[1] and Beatles. Also, it sounds to me like this something Paul would have written. Now, can you please explain what it means to calculate that P(Paul wrote it) = 0.018? Note that I am not asking about how the calculations work. [1]: http://www.glicko.net/ http://www.glicko.net/