10/10 scale

[Kevin_Lowe]

You've said this a few times, but I don't see that you've demonstrated it.

Let's look at your two scenarios again:

1. Two runners of equal ability are facing off against each other.
2. Two runners of unknown ability are facing off against each other.

Someone offers you a favorable bet. In which circumstance is it necessarily correct to accept it?

Neither. In the first circumstance, the offerer may have privileged information that you don't that has led him to an understand that one runner is more likely to win than the other. In a deterministic universe such information must exist.

Okay, look everybody. I'll explain this one last time.

The point of my example is that in world one you have the information that the runners are evenly matched. In world two you do not have any information about the runners.

What people keep doing is saying either "suppose in world one your information is wrong", or "suppose in world two you have information about the runners". You can do that if you like, but any such modified scenario simply is not relevant to the scenario I am actually proposing.

Now in my example, in world one, you know that a favourable-odds bet on either of the runners is a bet with a positive expectation. In fact, you can calculate exactly what the expected return on the bet will be if you are so inclined.

Again in my example, in world two, you do not know that a favourable-odds bet on either of the runners is a bet with a positive expectation. It's possible the person offering the bet knows more than you, and it's possible they don't. (I think it's ambiguous whether or not it's rational to take such a bet, and since dragging in rationality muddies the issue I'm inclined to drop that aspect of the story. The important point is that you don't know for certain that the bet has a positive expectation).

From this it seems to me to follow that there is a difference between ignorance of the odds, and knowledge that the odds are 50/50. If so there is something wrong with Bayesianism, which equates the two. it may well be a purely philosophical problem but I think it's still there.

 

[69dodge]

Okay, look everybody. I'll explain this one last time.

You sound annoyed. I'm not trying to annoy anyone. Sorry if I've done so. Just having a friendly discussion, that's all.

Now in my example, in world one, you know that a favourable-odds bet on either of the runners is a bet with a positive expectation. In fact, you can calculate exactly what the expected return on the bet will be if you are so inclined.

Again in my example, in world two, you do not know that a favourable-odds bet on either of the runners is a bet with a positive expectation.

You're considering here the expectation of a bet, as you were considering before the probability of an event, to be an objective property of the world: a person might know it or he might not know it, but in any case, there's something there to know, which is independent of whatever information anyone might have about it.

Does an event really have a probability, or a bet an expectation, independent of anyone's knowledge? Consider this example:

I intend to roll a standard six-sided die and then look at the number of dots on top. I will tell person A whether the number is even. I will tell person B whether the number is a multiple of three. I will tell person C whether the number is a four. None of them will see the die, or hear what I tell any of the others, and they all know beforehand that this what I plan to do.

I roll the die, and a five happens to come up. So I tell A that it's not even, I tell B that it's not a multiple of three, and I tell C that it's not a four.

I ask each person, privately, what the probability is that it's a five. Person A says 1/3, person B says 1/4, person C says 1/5. If anyone asked me, I of course would say 1.

Based on these different probabilities, each person would likewise calculate different values, in general, for the expectation of various bets about the result of the roll.

Who's "really" right? Does the question even make any sense?

When person B, for example, says that the probability of a five is 1/4, he's not saying, "there's this objective property of the world called 'the probability that the number is five' and I know for certain that its value is 1/4." He's saying, "there's this objective property of the world called 'whether the number is five'. I don't know for certain whether it's true or not. The limited amount of information I have leads me to be 1/4 sure it's true and 3/4 sure it's false."

(edited to make the numbers right)

 

[Kevin_Lowe]

You sound annoyed. I'm not trying to annoy anyone. Sorry if I've done so. Just having a friendly discussion, that's all.

You're considering here the expectation of a bet, as you were considering before the probability of an event, to be an objective property of the world: a person might know it or he might not know it, but in any case, there's something there to know, which is independent of whatever information anyone might have about it.

No I'm not.

In world one our hero has enough information to be highly certain that a bet either way would be an even money proposition. That's not an objective property of the world, it's just that our hero knows some objective properties of the world, specifically ones that lead him to conclude that a bet either way would be an even money proposition as far as anyone he is betting with can know (in his world at that time).

Philosophically speaking people could have all sorts of other sets of information about the race and draw other conclusions from it. Our hero just happens, in this particular instance, to know that his information is correct and is the best available.

In world two our hero doesn't.

 

[69dodge]

Philosophically speaking people could have all sorts of other sets of information about the race and draw other conclusions from it. Our hero just happens, in this particular instance, to know that his information is correct and is the best available.

Why does it matter that he knows it's the best information available?

In one world, he knows that the runners are evenly matched. This means that he can't tell who will win the race and furthermore he knows that no one else can tell who will win the race either. In the other world, he also has no idea who will win the race, but other people might have a better idea than he does. But why does that matter? He can choose to bet or not. Who cares what other people know? (We're supposing that he can't conclude anything about the runners from others' willingness to bet against him, correct? He always has the choice to bet or not; it's totally up to him.)

 

[drkitten]

I didn't say the runners had run a lot of races. I said that the runners were such that if they did, then they would win about half each and nobody could predict in advance which one would win any individual race.

Then the situation you describe is impossible by assumption. There is always more information available that would allow you to place a more informed bet. Under no theory of probability does your scenario qualify as rational -- hence no "rational" bets can be made.

 

[JamesDillon]

Earth isn't round. It's roughly spherical... a lumpy sort of sphere, actually.

Good scale, though!

A good point, and a perfect example of why you should never assign anything a 10 with the possible exception of mathematical truths. 9.5 is about as high as you should go for anything else, and I dare say that most "common sense" truths ought to fall in the 7-8.5 range.

That aside, your scale looks like a good way of making a point to students that our judgments of "truth" are subject to different degrees of certainty depending on context.

 

[GreedyAlgorithm]

From this it seems to me to follow that there is a difference between ignorance of the odds, and knowledge that the odds are 50/50. If so there is something wrong with Bayesianism, which equates the two. it may well be a purely philosophical problem but I think it's still there.

It's not a problem at all. You're confusing a model of all reality that proposes to be useful in certain ways with a model of all reality that proposes to be useful in every way. Bayesianism is the first such model, not the second. In fact you'll find that only reality is the second such model.

Bayesianism specifies that every event has a probability. Not "a chance to occur", just a number between 0 and 1 that corresponds at least loosely to our belief that the event occurs. This does not might not equate exactly to reality, and does not claim to, but it's an extraordinarily useful framework to deal with predictions.

Edited to avoid ruling out Bayesianism as exactly equivalent to reality. Maybe it is!

 

[delphi_ote]

This thread has been a strawman from the very beginning. Beyesian statistics are always of the form:

P(H|E)

The E stands for evidence. Bayesian inferences are always made and adjusted based on evidence.

A Bayesian is free to assign the probabilities here as he sees fit. Kevin is forcing them to choose P(win) = P(Runner1) = P(Runner2). But the Bayesian is also free to assign P(win), which does not have to be the same as P(Runner1) = P(Runner2). Based on their past experience with gambling with no or very little information, they are probably less confident about winning the bet and can assign P(win|evidence)=.001. They are perfectly free to do this.

 

[Kevin_Lowe]

Why does it matter that he knows it's the best information available?

That was just to stop people saying "But if he had more information he would know who would win, therefore there is no problem!".

In one world, he knows that the runners are evenly matched. This means that he can't tell who will win the race and furthermore he knows that no one else can tell who will win the race either. In the other world, he also has no idea who will win the race, but other people might have a better idea than he does. But why does that matter?

Because in one world he knows what the chances are that either runner will win, and in the other he does not know what the chances are.

He can choose to bet or not. Who cares what other people know? (We're supposing that he can't conclude anything about the runners from others' willingness to bet against him, correct? He always has the choice to bet or not; it's totally up to him.)

The fact he has the choice to bet or not is immaterial to what he knows, and what he knows about is what I'm talking about.

Then the situation you describe is impossible by assumption. There is always more information available that would allow you to place a more informed bet. Under no theory of probability does your scenario qualify as rational -- hence no "rational" bets can be made.

The scenario is not "impossible by assumption". Maybe our hero is just the world's greatest judge of runners, or maybe he has an amazing scanning gadget that measures bone lengths and muscle mass and such and predicts with greater accuracy than any other device in the world how fast a person can run a given distance. Nothing is the least bit impossible about a person knowing that their information is the best available to anybody at that time, nor is it impossible that their information might indicate that a race was equally likely to go either way, unless you are going to go the radical skepticism route and say that nobody ever really knows anythying.

This thread has been a strawman from the very beginning. Beyesian statistics are always of the form:

P(H|E)

The E stands for evidence. Bayesian inferences are always made and adjusted based on evidence.

A Bayesian is free to assign the probabilities here as he sees fit. Kevin is forcing them to choose P(win) = P(Runner1) = P(Runner2). But the Bayesian is also free to assign P(win), which does not have to be the same as P(Runner1) = P(Runner2). Based on their past experience with gambling with no or very little information, they are probably less confident about winning the bet and can assign P(win|evidence)=.001. They are perfectly free to do this.

So in other words the people saying that, for Bayesians, no evidence equals evidence of a 50/50 proposition were just wrong?

That would indeed explain why the claim didn't make sense.

 

[delphi_ote]

So in other words the people saying that, for Bayesians, no evidence equals evidence of a 50/50 proposition were just wrong?

That would indeed explain why the claim didn't make sense.

No. The Bayesian could assign even probabilities for the runners, but not to winning the bet.

"As far as I know, either runner is as likely as the other to win the race. Being that I don't know anything about the runners and I know that this gambling establishment makes a profit, I probably have one in a hundred chance of being right about the odds. I'd put my chances of winning the bet at one in a thousand."

 

[Kevin_Lowe]

No. The Bayesian could assign even probabilities for the runners, but not to winning the bet.

"As far as I know, either runner is as likely as the other to win the race. Being that I don't know anything about the runners and I know that this gambling establishment makes a profit, I probably have one in a hundred chance of being right about the odds. I'd put my chances of winning the bet at one in a thousand."

This is another instance of assuming the person in world #2 has extra information, which I specified he does not have.

 

[delphi_ote]

This is another instance of assuming the person in world #2 has extra information, which I specified he does not have.

So your Bayesian has to make a wager without knowing he's making a wager. You know, I can't imagine why this scenario breaks down into absurdity...

 

[delphi_ote]

*bump*

 

[Kevin_Lowe]

So your Bayesian has to make a wager without knowing he's making a wager. You know, I can't imagine why this scenario breaks down into absurdity...

Sorry, I've been travelling interstate. I won't be responding to posts promptly or reliably for a while yet.

The point of the thought experiment was just to contrast a known 50/50 chance with a two-horse race about which nothing is known. You can make up reasons why you think the known 50/50 chance shouldn't really be a known 50/50. Similarly, you can make up reasons why you think we should know something about the two-horse race. In neither case does doing so speak to the actual problem I was asking about.

Nonetheless, it's all anyone seems to want to do. I suppose I will just resign myself to never getting a straight answer.

 

[drkitten]

The point of the thought experiment was just to contrast a known 50/50 chance with a two-horse race about which nothing is known.

I suppose I will just resign myself to never getting a straight answer.

You got a straight answer. "Your thought experiment is incoherent."

 

[69dodge]

No. The Bayesian could assign even probabilities for the runners, but not to winning the bet.

Someone who bets on runner A will win his bet if and only if runner A wins the race. So, how can he assign different probabilities to him winning his bet and to runner A winning the race?

"As far as I know, either runner is as likely as the other to win the race. Being that I don't know anything about the runners and I know that this gambling establishment makes a profit, I probably have one in a hundred chance of being right about the odds. I'd put my chances of winning the bet at one in a thousand."

You're comparing probabilities that are based on different information, which is confusing.

Perhaps you didn't know anything about the runners initially, but after you find out what odds have been put on them by a gambling establishment that generally makes a profit, you do know something about them. So, you should update your probabilities accordingly.

 

[69dodge]

Sorry, I've been travelling interstate. I won't be responding to posts promptly or reliably for a while yet.

Sure, no problem.

The point of the thought experiment was just to contrast a known 50/50 chance with a two-horse race about which nothing is known. You can make up reasons why you think the known 50/50 chance shouldn't really be a known 50/50. Similarly, you can make up reasons why you think we should know something about the two-horse race. In neither case does doing so speak to the actual problem I was asking about.

You're simply assuming that the two are in fact different. I say, "what's the difference? In both cases you have no idea who will win." And you reply, "yeah, but in one case I know what the chances are that each will win." And I say, "Right. But, what does that mean, actually?"

And round and round we go...

If you say that "the chances are 50:50" means something more than just "I don't know who will win this race," I think that any attempt to describe exactly what you do mean by it will have to involve what you think will happen in a real or hypothetical string of similar races between the two runners. When talking about a string of races, the two statements are different. But we're not talking about a string of races, and what fraction of them will be won by each runner; we're just talking about this single race, and which runner will win it. That's all we're interested in: who will win this race. (Because that's all that determines whether we win our bet or lose it.)

If you know nothing about the runners, have you any more reason to believe that one will win than the other? No.

If you know that the runners are evenly matched, have you any more reason to believe that one will win than the other? No.

So, what's the difference?

 

[delphi_ote]

Someone who bets on runner A will win his bet if and only if runner A wins the race. So, how can he assign different probabilities to him winning his bet and to runner A winning the race?

The bettor cannot distiguish between runner A and runner B before placing his bet. He has no information about them, so they are totally interchangable until he makes his choice. The bettor wins if and only if his choice is the correct one. He's free after he places the bet on one horse or the other to say that because he placed his bet on that particular horse A or B, he believes it is less likely to win.

Perhaps you didn't know anything about the runners initially, but after you find out what odds have been put on them by a gambling establishment that generally makes a profit, you do know something about them. So, you should update your probabilities accordingly.

Of course. That was very clearly stated. Thanks.

 

[69dodge]

The bettor cannot distiguish between runner A and runner B before placing his bet. He has no information about them, so they are totally interchangable until he makes his choice. The bettor wins if and only if his choice is the correct one. He's free after he places the bet on one horse or the other to say that because he placed his bet on that particular horse A or B, he believes it is less likely to win.

I don't get it. How did he decide which one to bet on? I guess, since he has no information about them, he just chose randomly. Correct? Suppose he flipped a coin. How can a coin flip tell him anything useful about who will win a race? Surely, the result of the coin flip is independent of the result of the race.

 

[delphi_ote]

I don't get it. How did he decide which one to bet on? I guess, since he has no information about them, he just chose randomly. Correct? Suppose he flipped a coin. How can a coin flip tell him anything useful about who will win a race? Surely, the result of the coin flip is independent of the result of the race.

This still seems okay to me, but I need to think about it more. There are many related questions:

1. What is the probability that a person will win a bet given that they choose flipping a coin (depends on our assumptions. My inner cynic and pessimist says <1/2)?

2. What is the probability that A wins there race? (If we know nothing, I say 1/2)?

3. What is the probability a person flipping a coin chooses A (I'm sure we all agree 1/2)?

4. Given that a person chose A by flipping a coin, what is the probability they will win the bet (you could use either 1 or 2 here)?

5. Given that a person chose A by flipping a coin, what is the probability that A wins the race (intuitively, we suspect these events are independent, but if a person is betting with no information)?

6. Given that A wins the race, what is the probability a person chose A by flipping a coin?

Bayes's Theorem says:
P(A|B)=P(B|A)P(A)/P(B)

So many of these are related. Unfortunately, I've had a long day and can't quite work it all out yet.