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I'd have a hard time doing better than Jaynes: > Suppose you have a penny and you are allowed to examine it carefully, convince yourself that it's an honest co

I'd have a hard time doing better than Jaynes: > Suppose you have a penny and you are allowed to examine it carefully, convince yourself that it's an honest coin; i.e. accurately round, with head and tail, and a center of gravity where it ought to be. Then, you're asked to assign a probability that this coin will come up heads on the first toss. I'm sure you'll say 1/2. > Now, suppose you are asked to assign a probability to the proposition that there was once life on Mars. Well, I don't know what your opinion is there but on the basis of all the things that I have read on the subject, I would again say about 1/2 for the probability. But, even though I have assigned the same "external" probabilities to them, I have a very different "internal" state of knowledge about those propositions. > To see this, imagine the effect of getting new information. Suppose we tossed the coin 5 times and it comes up tails every time. You ask me what's my probability for heads on the next throw; I'll still say 1/2. But if you tell me one more fact about Mars, I'm ready to change my probability assignment completely. There is something which makes my state of belief very stable in the case of the penny, but very unstable in the case of Mars. Source: https://books.google.com/books?id=tTN4HuUNXjgC&pg=PA553&lpg=PA553#v=onepage&q&f=false https://books.google.com/books?id=tTN4HuUNXjgC&pg=PA553&lpg=... Ultimately it's a question of how much each bit of evidence would cause you to update (that is, difference of degree, rather than kind).

tshaddox 4y ago
But both of those cases are concerning the “Bayesian epistemology” side of things, not using probability to reason about how to bet in games of chance with well-established rules and mechanics. The usage of probability Deutsch is approving of is the latter, and is unrelated to a particular observer’s level of confidence that the cards are designed fairly or that the dealer and other players are playing fairly.
mhb 4y ago
> Ultimately it's a question of how much each bit of evidence would cause you to update (that is, difference of degree, rather than kind). Isn't this only because you have strong priors for the coin and weak ones for Mars life? Suppose you didn't have your example's information about the coin. Then the situations are the same and a new piece of evidence in either case (e.g., discovering whether the coin is round, has two heads, or where its center of gravity is).
- deleted 4y ago
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- lisper 4y ago
  There is no such thing as a "strong" prior or a "weak" prior. A prior is jut a number between 0 and 1. But the coin example is really not very apt because there is potential information that will change your belief about the coin. For example, you might discover that the coin is magnetic, and there is an external field being generated somewhere in just the right way to make the coin land heads every time. Or something like that. Whatever the case, if you get (say) 100 heads in a row you should start to have some serious doubts about your initial analysis of the situation.
  - unholiness 4y ago
    But ultimately, it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? So, the prior which is strong/weak is the probability distribution of probability distributions. There is a <<1% chance that I will ever believe there is a >90% chance of flipping heads. But there is a much greater probability that I will one day believe there is a >90% chance of life on Mars. What terminology would you use to describe that, if not strong/weak priors?
    - lisper 4y ago
      > it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? I would say it's the exact opposite. Encountering an unfair coin would not surprise me a bit because I know, for example, that double-headed coins exist. But encountering life on Mars would surprise me a lot because I've accumulated a lot of evidence already that there isn't any. (There might have been in the past, but there isn't any now.) The only thing that I consider "surprising" is encountering evidence that moves one of my priors from being close to 0 to being close to 1 or vice versa. That happens less and less as time goes by because (I hypothesize) my beliefs converge to reality, and reality is stable in the sense that there is something out there towards which it is possible to converge. But at no stage in this process are my beliefs about propositions modeled by anything other than numbers between 0 and 1, with 0 meaning, "I am absolutely certain this proposition is false", 1 meaning, "I am absolutely certain this proposition is true", and 0.5 meaning, "I have absolutely no idea whether or not this proposition is true."
thomasahle 4y ago
> suppose you are asked to assign a probability to the proposition that there was once life on Mars As soon as you accept this question as meaningful, you have already come out as a Bayesian. Deutsch would simply say (if I understand him right) that this question is not a matter of probabilities, whereas the coin question is.
- ElevenLathe 4y ago
  Suppose you were a bookie and somebody wanted to make a million dollar bet on it. You'd love to take the money, but obviously you need a line. How do you pick it?
  - MostlyStable 4y ago
    A bookie would not care at all what the real likelihood of life on mars would be. A bookie would care about the likelihood that would attract approximately equal bets on both sides of the line, since that way they guarantee that they make money. So they would do whatever sociological research would be required to estimate that and set the line accordingly and probably not do an ounce of research on Mars itself.
    - ElevenLathe 4y ago
      This assumes that there is a chance to create a matched book. This is usually possible for something "liquid" like a big league sporting event, but not always for poorly-followed sports or weird prop bets on low-interest games. Even for liquid games, the book is not always perfectly matched and they may end up with some actual exposure to the outcome of the event. In any case, in this scenario its just one guy looking to make a bet. There is definitely SOME line, possibly million-to-one or higher, that a smart book (or any rational actor) should be willing to take the bet.

tshaddox 4y ago

But both of those cases are concerning the “Bayesian epistemology” side of things, not using probability to reason about how to bet in games of chance with well-established rules and mechanics. The usage of probability Deutsch is approving of is the latter, and is unrelated to a particular observer’s level of confidence that the cards are designed fairly or that the dealer and other players are playing fairly.

mhb 4y ago

> Ultimately it's a question of how much each bit of evidence would cause you to update (that is, difference of degree, rather than kind). Isn't this only because you have strong priors for the coin and weak ones for Mars life? Suppose you didn't have your example's information about the coin. Then the situations are the same and a new piece of evidence in either case (e.g., discovering whether the coin is round, has two heads, or where its center of gravity is).

deleted 4y ago
[deleted]
lisper 4y ago
There is no such thing as a "strong" prior or a "weak" prior. A prior is jut a number between 0 and 1. But the coin example is really not very apt because there is potential information that will change your belief about the coin. For example, you might discover that the coin is magnetic, and there is an external field being generated somewhere in just the right way to make the coin land heads every time. Or something like that. Whatever the case, if you get (say) 100 heads in a row you should start to have some serious doubts about your initial analysis of the situation.
- unholiness 4y ago
  But ultimately, it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? So, the prior which is strong/weak is the probability distribution of probability distributions. There is a <<1% chance that I will ever believe there is a >90% chance of flipping heads. But there is a much greater probability that I will one day believe there is a >90% chance of life on Mars. What terminology would you use to describe that, if not strong/weak priors?
  - lisper 4y ago
    > it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right? I would say it's the exact opposite. Encountering an unfair coin would not surprise me a bit because I know, for example, that double-headed coins exist. But encountering life on Mars would surprise me a lot because I've accumulated a lot of evidence already that there isn't any. (There might have been in the past, but there isn't any now.) The only thing that I consider "surprising" is encountering evidence that moves one of my priors from being close to 0 to being close to 1 or vice versa. That happens less and less as time goes by because (I hypothesize) my beliefs converge to reality, and reality is stable in the sense that there is something out there towards which it is possible to converge. But at no stage in this process are my beliefs about propositions modeled by anything other than numbers between 0 and 1, with 0 meaning, "I am absolutely certain this proposition is false", 1 meaning, "I am absolutely certain this proposition is true", and 0.5 meaning, "I have absolutely no idea whether or not this proposition is true."

thomasahle 4y ago

> suppose you are asked to assign a probability to the proposition that there was once life on Mars As soon as you accept this question as meaningful, you have already come out as a Bayesian. Deutsch would simply say (if I understand him right) that this question is not a matter of probabilities, whereas the coin question is.

ElevenLathe 4y ago
Suppose you were a bookie and somebody wanted to make a million dollar bet on it. You'd love to take the money, but obviously you need a line. How do you pick it?
- MostlyStable 4y ago
  A bookie would not care at all what the real likelihood of life on mars would be. A bookie would care about the likelihood that would attract approximately equal bets on both sides of the line, since that way they guarantee that they make money. So they would do whatever sociological research would be required to estimate that and set the line accordingly and probably not do an ounce of research on Mars itself.
  - ElevenLathe 4y ago
    This assumes that there is a chance to create a matched book. This is usually possible for something "liquid" like a big league sporting event, but not always for poorly-followed sports or weird prop bets on low-interest games. Even for liquid games, the book is not always perfectly matched and they may end up with some actual exposure to the outcome of the event. In any case, in this scenario its just one guy looking to make a bet. There is definitely SOME line, possibly million-to-one or higher, that a smart book (or any rational actor) should be willing to take the bet.