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.
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".
Is there a conception of probability that is not conditional on a model?
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_... (Ray Solomonoff, founder of algorithmic information theory)
https://www.springer.com/gp/book/9783540221395 (Marcus Hutter)
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.
I’m not sure what you mean by this. Can you elaborate?"
There is nothing in that link that I can apply to a problem. It is all too general and abstract.
What I wanted to see is any example of algorithmic probability being applied to a real world problem. I am certain that to calculate the probability they need to define/derive a model of the phenomenon being modeled to use as the likelihood.
Also, I don't have easy access to those books, nor do I want to get them and flip through looking for what you are talking about, but if I did I am sure I would find that the probability calculation requires a model.
This is also a good guide: https://arxiv.org/pdf/cs/0703024.pdf.
There is nothing in that link that I can apply to a problem. It is all too general and abstract.
Algorithmic probability is a framework for general prediction, so it has applications in reinforcement learning (see AIXI), grammar discovery in linguistics, text compression, continuation of integer sequences, etc. See http://www.scholarpedia.org/article/Applications_of_algorith....
I am certain that to calculate the probability they need to define/derive a model of the phenomenon being modeled to use as the likelihood... I am sure I would find that the probability calculation requires a model
Yes, and that model is a universal computer, as stated in the articles I linked to.
This basically says "the model is anything at all". From the first example in the first link, I see the probability in that case is conditional on the binomial model correctly describing the phenomenon (equation 8). This is what I expected is going on.
No, no, no! It says the model is powerful enough to compute any computable function. This is where the optimality of universal measures comes from.
From the first example in the first link, I see the probability in that case is conditional on the binomial model correctly describing the phenomenon (equation 8).
Equation 8 is presenting the binomial model as an example that is different from the universal machine model.
Then the model must be whatever function it is computing/approximating for that case. Otherwise I can't make sense of what you keep saying. You must be using "model" in a sense that is so general as to have no meaning to me.
>"Equation 8 is presenting the binomial model as an example that is different from the universal machine model."
Ok, where is an actual probability is calculated from the 'universal machine model"? That is all I want to see at this point (which was the original topic). I want to see some sort of real life phenomenon for which this procedure calculates a probability, so I can tell what it means. Not a general proof or something like that, some real data or simple data generating process.
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.
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.
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.
I agreed with your comment until this. What do you mean?
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.
That’s not how Bayesian probability works. See
https://news.ycombinator.com/item?id=17743049
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.
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.
That's just the thing: Bayesian probability is emphatically not about an experiment being repeated in the future. That's the frequentist interpretation of probability.
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.
???
The Dutch book argument shows that rational people must have subjective probabilities which behave according to the axioms of probability. I don't see the relevance of your statement here at all, particularly the bits about "anything at stake" and "free publicity".
And what are you trying to say by the phrase "counting some stuff and dividing those counts by some numbers and stuff"?
I am saying something different, but this seems like a futile discussion. What does it mean to have P(Paul wrote it) equal to some number in (0,1)? Let's call that number q. Explain what q means in words.
If the truth were ever going to be revealed, or if I could conceive of the experiment being repeated in the future, I could explain it as "I would be willing to pay up to $q to buy an asset that pays $1 if Paul wrote it and $0 otherwise."
> And what are you trying to say by the phrase "counting some stuff and dividing those counts by some numbers and stuff"?
I mean it is very hard for me to take a person seriously who is doing a whole bunch of interviews without even a working paper somewhere.
It seems that if you use some arithmetic, people are inclined to accept what you did without questioning whether it makes sense to apply such models in this case.
> The Dutch book argument shows that rational people ...
You need to do better than just quoting passages from Wikipedia if you want to understand what that means.
I don’t object to your comments about the absence of a working paper.
You need to do better than just quoting passages from Wikipedia if you want to understand what that means.
You’re accusing me of not understanding the Dutch book argument. Do you have any reason for this, or is it just a baseless accusation?
Isn't it obvious that interpretation does not apply in this situation?
> You’re accusing me of not understanding the Dutch book argument.
It is not an accusation. It is a statement.
The Dutch book argument is basically a "no arbitrage" or "no free lunch" argument. Isn't it obvious then that it cannot be used to justify anything where no money is at stake and no bet regarding the outcome can ever be resolved?
PS: This will be my last comment in this thread because I get the distinct feeling that I am talking to Eliza instead of a human being.
Why not?
> It is not an accusation. It is a statement.
It is an accusation. It is also a false statement.
> Isn't it obvious then that it cannot be used to justify anything where no money is at stake and no bet regarding the outcome can ever be resolved?
This is so ridiculous I don't know where to begin. It's like saying the expected value of a dice roll doesn't exist because no roll ever equals 3.5, or the average of any number of trials is not necessarily 3.5. It's claiming a counterfactual is false because its antecedent is false.
> PS: This will be my last comment in this thread because I get the distinct feeling that I am talking to Eliza instead of a human being.
You're welcome to run away with your tail between your legs after making baseless accusations and misunderstanding the concept of probability.
LOL!
As I mentioned elsethread, you misunderstand what I am saying. But, just to help you reason this out, answer me this: Where do you think the data to update your beliefs come from?
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].
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.”