5 ms·

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, co

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/
      - antidesitter 8y ago
        Either John wrote the song or Paul did. Therefore P(Paul wrote it) either one or zero. That’s not how Bayesian probability works. See https://news.ycombinator.com/item?id=17743049 https://news.ycombinator.com/item?id=17743049 https://news.ycombinator.com/item?id=17742927 https://news.ycombinator.com/item?id=17742927 By your logic, every proposition is either true or false, and therefore the concept of probability is useless. Now, can you please explain what it means to calculate that P(Paul wrote it) = 0.018? That depends on their model, which I’m not necessarily defending.
        nanis 8y ago
        > By your logic, every proposition is either true or false, and therefore the concept of probability is useless. That would be nonsensical interpretation of my logic. In cases where it is possible to conceive of an experiment being repeated in the future, a "degree of belief" interpretation is appealing. If the answer were going to be revealed objectively and someone were taking bets, sure, fine, I'll go along with that. But, don't make an appeal to the Dutch book argument in a unique one-off case where no one has anything at stake other than free publicity which they have chosen to pursue by counting some stuff and dividing those counts by some numbers and stuff.
- azernik 8y ago
  Mathematically, probability is a function over the domain of values of a random variable, where the sum (for discrete variables) or integral (for continuous ones) over the entire domain is equal to 1. Division happens for some random variable operations (for example, conditional probabilities) but generally multiplication and addition are more common In this case, its value at 6 happens to be 1/6, because there are six possible values and for a fair die all have equal values of P[Val].

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/
    - antidesitter 8y ago
      Either John wrote the song or Paul did. Therefore P(Paul wrote it) either one or zero. That’s not how Bayesian probability works. See https://news.ycombinator.com/item?id=17743049 https://news.ycombinator.com/item?id=17743049 https://news.ycombinator.com/item?id=17742927 https://news.ycombinator.com/item?id=17742927 By your logic, every proposition is either true or false, and therefore the concept of probability is useless. Now, can you please explain what it means to calculate that P(Paul wrote it) = 0.018? That depends on their model, which I’m not necessarily defending.
      - nanis 8y ago
        > By your logic, every proposition is either true or false, and therefore the concept of probability is useless. That would be nonsensical interpretation of my logic. In cases where it is possible to conceive of an experiment being repeated in the future, a "degree of belief" interpretation is appealing. If the answer were going to be revealed objectively and someone were taking bets, sure, fine, I'll go along with that. But, don't make an appeal to the Dutch book argument in a unique one-off case where no one has anything at stake other than free publicity which they have chosen to pursue by counting some stuff and dividing those counts by some numbers and stuff.
azernik 8y ago
Mathematically, probability is a function over the domain of values of a random variable, where the sum (for discrete variables) or integral (for continuous ones) over the entire domain is equal to 1. Division happens for some random variable operations (for example, conditional probabilities) but generally multiplication and addition are more common In this case, its value at 6 happens to be 1/6, because there are six possible values and for a fair die all have equal values of P[Val].