4 ms·

I agree, there are problems and non-intuitive aspects to both approaches to probability. Otherwise there wouldn't be two approaches. I think your objection to

I agree, there are problems and non-intuitive aspects to both approaches to probability. Otherwise there wouldn't be two approaches. I think your objection to frequentism unfairly conflates the idealized world with the physical world. Only in the idealized world can you assert a priori that a coin has 50% probability of coming up heads. In that world you can toss the coin infinite times and it almost surely comes up heads close to 50% of the time. In the physical world you merely have a model stating that heads will come up 50% of the time. In this world you could toss the coin millions of times and have it come up heads 66% of the time, but all you've done is provide really strong evidence that your model is wrong. Also, you left out 2 arguments against frequentism: it allows inconsistent beliefs, and in practice it has allowed bad approaches in scientific papers. As for the Bayseian view, being non-intuitive isn't the same as problematic. The Twins Paradox https://en.wikipedia.org/wiki/Twin_paradox https://en.wikipedia.org/wiki/Twin_paradox violates our intuitive understanding of how time works, but that's because our intuition is wrong in some conditions. You thought when I said the coin had 50% chance of coming up heads that I was making a statement about the coin, but really I was making a subjective statement about how I would bet. If you believe it's 25% then there's a clear way for us to resolve our different beliefs.

danbruc 10y ago
I think the interesting part is not that the relative frequency might converge to something other than 50 % under non-ideal conditions, but that it might not converge at all, admittedly only very, very rarely. And this seems to force you into an infinite regress. 50 % probability for heads means that if you toss the coin infinitely often the relative frequency will converge to 50 %. But not quite, in very rare cases it won't. So you have to toss a coin infinitely often an infinite number of times and then you will see that all but a tiny fraction of the experiments indeed show the relative frequency converge to 50 %. But now you have to quantify that this tiny fraction is something of probability zero. And even worse, it is still possible that none of your repeated experiments showed convergence, you seem right back where you started. I would love to know to what you are referring with the inconsistent beliefs. I would not say that Bayesian view is non-intuitive, I would say it fails to account for important things. A priori probabilities have to be rooted somewhere. Where does you beliefe in a 50 % probability for heads come from? Because you have previously observed the relative frequencies of coin tosses? Because you made some theoretical observations about symmetries? There must be, at least so it seems to me, something about the probabilities associated with coin tosses that is independent of any individual, otherwise it would become rather difficult to explain how different individuals would arrive at similar probabilities independent of each other. So banning probabilities into the realm of beliefs does not cut it in my opinion.
- p1esk 10y ago
  Well, you can build a physical model of a coin, and all forces acting on it during toss/fall, and show that in an ideal scenario (perfect coin/perfect landing surface), there would be roughly half of the initial conditions leading to heads, and roughly half leading to tails.
  - danbruc 10y ago
    If you want to go down this road, then we will have to switch to a nuclear decay based coin or something like that. In the case of a coin toss there never was any real randomness, as you say we were just ignorant of the initial conditions. Given a distribution over the possible initial conditions, we can determine the probabilities for heads and tails. The possibly huge number of degrees of freedom and deterministic chaos will of course make this an unpleasant exercise.
    - p1esk 10y ago
      Can we base our belief about coin toss probability on the belief that about half of the initial conditions lead to heads and half to tails, without verifying it?
- yellowstuff 10y ago
  > I would love to know to what you are referring with the inconsistent beliefs. You could build frequentist models for the odds of 0 to 1 inches of rain falling in Cleveland tomorrow, 1 to 2 inches, and 0 to 2 inches, and the odds of 0 to 1 plus 1 to 2 don't need to add up to 0 to 2 inches. You can justify all 3 models, but they are inconsistent. Bayesian models don't allow that. I read about this in Aaron Brown's "Red-Blooded Risk". Here's an online source asserting "frequentists can have two different unbiased estimators under the same likelihood functions." https://chenghanyu.wordpress.com/2014/03/26/the-strengths-and-weaknesses-of-the-frequentist-and-bayesian-paradigms/ https://chenghanyu.wordpress.com/2014/03/26/the-strengths-an... Here's one saying that Bayesian models are self-consistent: http://bactra.org/notebooks/bayesian-consistency.html http://bactra.org/notebooks/bayesian-consistency.html Here's Eliezer Yudkowsky giving examples of my other point, that frequentist models can encourage sloppy behavior from scientists: http://lesswrong.com/lw/1gc/frequentist_statistics_are_frequently_subjective/ http://lesswrong.com/lw/1gc/frequentist_statistics_are_frequ...

danbruc 10y ago

I think the interesting part is not that the relative frequency might converge to something other than 50 % under non-ideal conditions, but that it might not converge at all, admittedly only very, very rarely. And this seems to force you into an infinite regress. 50 % probability for heads means that if you toss the coin infinitely often the relative frequency will converge to 50 %. But not quite, in very rare cases it won't. So you have to toss a coin infinitely often an infinite number of times and then you will see that all but a tiny fraction of the experiments indeed show the relative frequency converge to 50 %. But now you have to quantify that this tiny fraction is something of probability zero. And even worse, it is still possible that none of your repeated experiments showed convergence, you seem right back where you started. I would love to know to what you are referring with the inconsistent beliefs. I would not say that Bayesian view is non-intuitive, I would say it fails to account for important things. A priori probabilities have to be rooted somewhere. Where does you beliefe in a 50 % probability for heads come from? Because you have previously observed the relative frequencies of coin tosses? Because you made some theoretical observations about symmetries? There must be, at least so it seems to me, something about the probabilities associated with coin tosses that is independent of any individual, otherwise it would become rather difficult to explain how different individuals would arrive at similar probabilities independent of each other. So banning probabilities into the realm of beliefs does not cut it in my opinion.

p1esk 10y ago
Well, you can build a physical model of a coin, and all forces acting on it during toss/fall, and show that in an ideal scenario (perfect coin/perfect landing surface), there would be roughly half of the initial conditions leading to heads, and roughly half leading to tails.
- danbruc 10y ago
  If you want to go down this road, then we will have to switch to a nuclear decay based coin or something like that. In the case of a coin toss there never was any real randomness, as you say we were just ignorant of the initial conditions. Given a distribution over the possible initial conditions, we can determine the probabilities for heads and tails. The possibly huge number of degrees of freedom and deterministic chaos will of course make this an unpleasant exercise.
  - p1esk 10y ago
    Can we base our belief about coin toss probability on the belief that about half of the initial conditions lead to heads and half to tails, without verifying it?
yellowstuff 10y ago
> I would love to know to what you are referring with the inconsistent beliefs. You could build frequentist models for the odds of 0 to 1 inches of rain falling in Cleveland tomorrow, 1 to 2 inches, and 0 to 2 inches, and the odds of 0 to 1 plus 1 to 2 don't need to add up to 0 to 2 inches. You can justify all 3 models, but they are inconsistent. Bayesian models don't allow that. I read about this in Aaron Brown's "Red-Blooded Risk". Here's an online source asserting "frequentists can have two different unbiased estimators under the same likelihood functions." https://chenghanyu.wordpress.com/2014/03/26/the-strengths-and-weaknesses-of-the-frequentist-and-bayesian-paradigms/ https://chenghanyu.wordpress.com/2014/03/26/the-strengths-an... Here's one saying that Bayesian models are self-consistent: http://bactra.org/notebooks/bayesian-consistency.html http://bactra.org/notebooks/bayesian-consistency.html Here's Eliezer Yudkowsky giving examples of my other point, that frequentist models can encourage sloppy behavior from scientists: http://lesswrong.com/lw/1gc/frequentist_statistics_are_frequently_subjective/ http://lesswrong.com/lw/1gc/frequentist_statistics_are_frequ...