3 ms·

Wait: the entire point of Bayes's idea was that under very sensible and straightforward rules, "plausibilities" behave exactly like probabilities. So why multip

by cscheid 12y ago

Wait: the entire point of Bayes's idea was that under very sensible and straightforward rules, "plausibilities" behave exactly like probabilities. So why multiply concepts? What is the difference, in your proposal, between "plausibilities" and "probabilities"?

jerrytsai 12y ago
You are correct: the way these mathematical creatures work, plausibilities behave like probabilties. What tjradcliife is trying to do, by differentiating between plausibilities and probabilities is to address the concern of frequentism with Bayesianism: namely, the "P(h)"-- what is usually termed as a "prior probability" in statistical jargon-- usually does not arise from objective data. Accordingly, Bayesianism can be treated with suspicion-- since the P(h) is arbitrarily specified, one could "load the dice" and obtain a P(h|e) ("posterior probability") guaranteed to suit the analyst's conclusions. By differentiating the two creatures, tjradcliffe hopes to resolve the dis-ease with calling P(h|e) a "probability". By frequentist philosophy, probabilities should be grounded in some sort of firm soil, e.g., the relative frequency of what would be observed if one could conduct an experiment ad infinitum. A Bayesian "probability" is not grounded this way, so calling it a "plausibility" instead would resolve this tension. Another way of putting this-- P(h) to a frequentist feels like a tainted probability-- really not a probability at all-- so even after updating the prior probability with data [i.e., P(e|h)/P(e)], the posterior probability P(h|e) still retains the influence of that original taint. Solution: stop calling the Bayesian probabilities "probabilities"; instead, call them "plausibilities".
captainmuon 12y ago
Well, one problem is that some people, including myself, object to assigning probabilities to facts of nature. What is the probability that the mass of the Higgs Boson is 126 GeV +- 0.1 GeV? That doesn't make sense, it is either 100% or 0%, true or false, as the mass is a natural constant. It's like asking for the probability that 3 is between 2 and 5. Of course, you can make statements like P(m=126) sensible, by recognizing P as degree of belief, or plausibility. These don't describe facts, but rather knowledge, or your uncertainty. The wonderful thing is that they fulfill the same axioms as ordinary probabilities, and thus you can use Bayes theorem and so on with them. Distinct from degrees of belief are probabilities (or frequencies in some Bayesian literature). These do not describe your uncertainty, but are really physical properties, or facts of reality, in disguise. If you have a fair die, P(1) = P(2) = ... = P(6) = 1/6, the equality of the numbers means that all faces have the same area (under the plausible assumption that the die has a uniform density). The area of a face is (for small deviations from P=1/6) proportional to the probability to land on that face. So a probability of P != 1/6 means you have a deviation from the perfect cube form. One reason I think it is very important to be aware of this distinction is that people object to "Bayesianism" because most Bayesians don't make the distinction. People do not reject Bayesianism because it's methods don't work, they reject it because it seems philosophically unsound. If they would stop calling plausibilities probabilities, a lot of discussion and confusion would go away IMHO. (Maybe we should forgo all the p words and call the Bayesian quantity just "awesome score" or something. Less people would have problems with evaluating an arbitrary formula that doesn't have all the philosophical and political connotations that probabilities have.)
- nkurz 12y ago
  one problem is that some people, including myself, object to assigning probabilities to facts of nature What do you feel you gain from this philosophical objection? Is there a difference between a "fact of nature" and a one time event that has already happened? Consider a coin flip that has happens a light second away. After the flip, but before the result has reached you, much like a physical constant, isn't it also 100% heads or tails but just unknown to you? Does it change from being a probability to a plausibility at the time of the flip? (asked with ignorance but genuine curiosity)
  - captainmuon 12y ago
    That's actually a good question. I don't know if you gain much from that objection besides clarity. After all, both "kinds" of probabilities share the same mathematics. I think it's the other way around. People have certain intuitions, or prejudices if you wish, and it's useful to have terms that match your intuitions - even if they are slightly redundant. I find it helps me understand and explain statistical problems better. About the coin: I think you have to distinguish between one concrete flip of the coin, and the coin itself. The coin itself has certain probabilities for generating heads and tails, and they don't change. A concrete flipping of the coin is a different situation. You might see the coin, and know with near 100% certainty what it shows. Or you might have some machine which detects whether it is heads or tails, and this machine has a certain accuracy. When it signals heads, you might say it has a 95% plausibility to really be heads. And before you measured the state of the coin, you might say the plausibility is 50%, assuming the coin is fair. So the plausibility does change at time of observation (or when you get new information). What about the time of the flip? Well, after the flip has occurred, this concrete instance is either 100% heads or 100% tails of course. If I repeat this exact coin toss (which I can't do in reality), I will always get the same result. And the probability to have heads, when you have heads, is 100% (P(toss #1 is heads | toss #1 is heads) = 100%). This statement is a bit silly, but I think it shows that the probability of a thrown, lying coin to be heads or tails is not a very interesting quantity. Our degree of belief, given certain facts, that it is heads or tails, is interesting. And the probability of the coin in general, determined by its geometry, is interesting. And especially interesting is how you get from one to the other, which is where Bayesian inference, hypothesis testing, and so on come in (The coin is known to be rigged with heads=99%. My flawed detector says it is tails. What should I believe is the state of the coin, or what should I bet on?). Finally, I should note that you should take what I've said with a grain of salt. I'm at best an armchair statistician, although I do get to think about stuff like this a lot at work (as a physicist).

by cscheid 12y ago

jerrytsai 12y ago

You are correct: the way these mathematical creatures work, plausibilities behave like probabilties. What tjradcliife is trying to do, by differentiating between plausibilities and probabilities is to address the concern of frequentism with Bayesianism: namely, the "P(h)"-- what is usually termed as a "prior probability" in statistical jargon-- usually does not arise from objective data. Accordingly, Bayesianism can be treated with suspicion-- since the P(h) is arbitrarily specified, one could "load the dice" and obtain a P(h|e) ("posterior probability") guaranteed to suit the analyst's conclusions. By differentiating the two creatures, tjradcliffe hopes to resolve the dis-ease with calling P(h|e) a "probability". By frequentist philosophy, probabilities should be grounded in some sort of firm soil, e.g., the relative frequency of what would be observed if one could conduct an experiment ad infinitum. A Bayesian "probability" is not grounded this way, so calling it a "plausibility" instead would resolve this tension. Another way of putting this-- P(h) to a frequentist feels like a tainted probability-- really not a probability at all-- so even after updating the prior probability with data [i.e., P(e|h)/P(e)], the posterior probability P(h|e) still retains the influence of that original taint. Solution: stop calling the Bayesian probabilities "probabilities"; instead, call them "plausibilities".

captainmuon 12y ago

Well, one problem is that some people, including myself, object to assigning probabilities to facts of nature. What is the probability that the mass of the Higgs Boson is 126 GeV +- 0.1 GeV? That doesn't make sense, it is either 100% or 0%, true or false, as the mass is a natural constant. It's like asking for the probability that 3 is between 2 and 5. Of course, you can make statements like P(m=126) sensible, by recognizing P as degree of belief, or plausibility. These don't describe facts, but rather knowledge, or your uncertainty. The wonderful thing is that they fulfill the same axioms as ordinary probabilities, and thus you can use Bayes theorem and so on with them. Distinct from degrees of belief are probabilities (or frequencies in some Bayesian literature). These do not describe your uncertainty, but are really physical properties, or facts of reality, in disguise. If you have a fair die, P(1) = P(2) = ... = P(6) = 1/6, the equality of the numbers means that all faces have the same area (under the plausible assumption that the die has a uniform density). The area of a face is (for small deviations from P=1/6) proportional to the probability to land on that face. So a probability of P != 1/6 means you have a deviation from the perfect cube form. One reason I think it is very important to be aware of this distinction is that people object to "Bayesianism" because most Bayesians don't make the distinction. People do not reject Bayesianism because it's methods don't work, they reject it because it seems philosophically unsound. If they would stop calling plausibilities probabilities, a lot of discussion and confusion would go away IMHO. (Maybe we should forgo all the p words and call the Bayesian quantity just "awesome score" or something. Less people would have problems with evaluating an arbitrary formula that doesn't have all the philosophical and political connotations that probabilities have.)

nkurz 12y ago
one problem is that some people, including myself, object to assigning probabilities to facts of nature What do you feel you gain from this philosophical objection? Is there a difference between a "fact of nature" and a one time event that has already happened? Consider a coin flip that has happens a light second away. After the flip, but before the result has reached you, much like a physical constant, isn't it also 100% heads or tails but just unknown to you? Does it change from being a probability to a plausibility at the time of the flip? (asked with ignorance but genuine curiosity)
- captainmuon 12y ago
  That's actually a good question. I don't know if you gain much from that objection besides clarity. After all, both "kinds" of probabilities share the same mathematics. I think it's the other way around. People have certain intuitions, or prejudices if you wish, and it's useful to have terms that match your intuitions - even if they are slightly redundant. I find it helps me understand and explain statistical problems better. About the coin: I think you have to distinguish between one concrete flip of the coin, and the coin itself. The coin itself has certain probabilities for generating heads and tails, and they don't change. A concrete flipping of the coin is a different situation. You might see the coin, and know with near 100% certainty what it shows. Or you might have some machine which detects whether it is heads or tails, and this machine has a certain accuracy. When it signals heads, you might say it has a 95% plausibility to really be heads. And before you measured the state of the coin, you might say the plausibility is 50%, assuming the coin is fair. So the plausibility does change at time of observation (or when you get new information). What about the time of the flip? Well, after the flip has occurred, this concrete instance is either 100% heads or 100% tails of course. If I repeat this exact coin toss (which I can't do in reality), I will always get the same result. And the probability to have heads, when you have heads, is 100% (P(toss #1 is heads | toss #1 is heads) = 100%). This statement is a bit silly, but I think it shows that the probability of a thrown, lying coin to be heads or tails is not a very interesting quantity. Our degree of belief, given certain facts, that it is heads or tails, is interesting. And the probability of the coin in general, determined by its geometry, is interesting. And especially interesting is how you get from one to the other, which is where Bayesian inference, hypothesis testing, and so on come in (The coin is known to be rigged with heads=99%. My flawed detector says it is tails. What should I believe is the state of the coin, or what should I bet on?). Finally, I should note that you should take what I've said with a grain of salt. I'm at best an armchair statistician, although I do get to think about stuff like this a lot at work (as a physicist).