I’ve always been really confused by this but it isn’t clear that an event with P=0 is an impossible event unless we’re talking about the probability of an event in a finite set of possible events. (Edit again: You can skip the rest of this paragraph and the next if you are smarter than me and already get continuous probability distributions. I’m obviously behind today.)This is how it was explained to me: Think of a dart board with a geometric line across it. That line represents probability space. An event with P=.5 is modeled by marking the middle of the line. If someone throws a dart at the line there is an equal chance that it lands at any point along the line. However, at any given point the probability that the dart lands there is zero.
I think the probability of any particular complex entity, event or law existing can be said to have a probability of zero absent a creator or natural selection or some other mechanism for enabling complexity. Of course this is really counterintuitive since our evolved understanding of probability deals with finite sets of possibilities. Also it means that ‘impossible’ can’t be assigned a probability. (Edit: Also, the converse is true. The probability that the dart lands anywhere other than the spot you pick is 1 so certainty can’t be mapped as 1 either.)
Also, imperfect Bayesians will sometimes assign less than ideal probabilities to things. A perfect Bayesian would presumably never wrongly declare something impossible because it could envision possible future evidence that would render the thing possible. But regular people are going to misinterpret evidence and fail to generate hypotheses so they might sometimes think something is impossible only to later have it’s possibility thrown in their faces.
Interesting point. Since physics does appear on the surface to be continuous, I can’t rule out continuous propositions. Perhaps the amended saying should read “0 and 1 are not probability masses, and 0 is not a probability density.”
I’ve always been really confused by this but it isn’t clear that an event with P=0 is an impossible event unless we’re talking about the probability of an event in a finite set of possible events. This is how it was explained to me: Think of a dart board with a geometric line across it. That line represents probability space. An event with P=.5 is modeled by marking the middle of the line. If someone throws a dart at the line there is an equal chance that it lands at any point along the line. However, at any given point the probability that the dart lands there is zero.
And, having assigned p(A) = 0 to such an event A I will not be able to rationally update away from zero without completely discarding my former model. There is no evidence that can cause me to rationally update p(A) = 0 to something else ever. Discard it and overwrite with something completely unrelated perhaps, but never update.
I’ve always been really confused by this but it isn’t clear that an event with P=0 is an impossible event unless we’re talking about the probability of an event in a finite set of possible events. This is how it was explained to me: Think of a dart board with a geometric line across it. That line represents probability space. An event with P=.5 is modeled by marking the middle of the line. If someone throws a dart at the line there is an equal chance that it lands at any point along the line. However, at any given point the probability that the dart lands there is zero.
And, having assigned p(A) = 0 to such an event A I will not be able to rationally update away from zero without completely discarding my former model. There is no evidence that can cause me to rationally update p(A) = 0 to something else ever. Discard it and overwrite with something completely unrelated perhaps, but never update. p(A) is right there as the numerator!
(But yes, I take your point and tentatively withdraw the use of ‘impossible’ to refer to p=0.)
ETA: Well, maybe you’re allowed to use some mathematical magic to cancel out the 0 if p(B) = 0 too. But then, the chance of that ever happening is, well, 0.
Er, my bad. I missed your point. I see it now, duh.
So my friend thinks something S has a probability of zero but I know otherwise and point out that it is possible give assumption which I know my friend believes has a .1 chance of being true. He says “Oh right. I guess S is possible after all.” What has just happened? What do we say when we see the dart land at a specific point on the line?
Your friend had incorrectly computed the implications of his prior to the problem in question. On your prompting he re-ran the computation, and got the right answer (or at least a different answer) this time.
Perfect Bayesians are normally assumed to be logically omniscient, so this just wouldn’t happen to them in the first place.
What do we say when we see the dart land at a specific point on the line?
In order to specify a point on the line you need an infinite amount of evidence, which is sufficient to counteract the infinitesimal prior. (The dart won’t hit a rational number or anything else that has a finite exact description.)
Or if you only have a finite precision observation, then you have only narrowed the dart’s position to some finite interval, and each point in that interval still has probability 0.
So my friend thinks something S has a probability of zero but I know otherwise and point out that it is possible give assumption which I know my friend believes has a .1 chance of being true. He says “Oh right. I guess S is possible after all.” What has just happened?
You wasted a great gambling opportunity.
Pengvado gives one good answer. I’ll add that your friend saying something has a probability of zero most likely means a different thing than what a Bayesian agent means when it says the same thing. Often people give probability estimates that don’t take their own fallibility into account without actually intending to imply that they do not need to. That is, if asked to actually bet on something they will essentially use a different probability figure that incorporates their confidence in their reasoning. In fact, I’ve engaged with philosophers who insist that you have to do it that way.
What do we say when we see the dart land at a specific point on the line?
“Did not! Look closer, you missed by 1/infinity miles!”
I’ve always been really confused by this but it isn’t clear that an event with P=0 is an impossible event unless we’re talking about the probability of an event in a finite set of possible events. (Edit again: You can skip the rest of this paragraph and the next if you are smarter than me and already get continuous probability distributions. I’m obviously behind today.)This is how it was explained to me: Think of a dart board with a geometric line across it. That line represents probability space. An event with P=.5 is modeled by marking the middle of the line. If someone throws a dart at the line there is an equal chance that it lands at any point along the line. However, at any given point the probability that the dart lands there is zero.
I think the probability of any particular complex entity, event or law existing can be said to have a probability of zero absent a creator or natural selection or some other mechanism for enabling complexity. Of course this is really counterintuitive since our evolved understanding of probability deals with finite sets of possibilities. Also it means that ‘impossible’ can’t be assigned a probability. (Edit: Also, the converse is true. The probability that the dart lands anywhere other than the spot you pick is 1 so certainty can’t be mapped as 1 either.)
Also, imperfect Bayesians will sometimes assign less than ideal probabilities to things. A perfect Bayesian would presumably never wrongly declare something impossible because it could envision possible future evidence that would render the thing possible. But regular people are going to misinterpret evidence and fail to generate hypotheses so they might sometimes think something is impossible only to later have it’s possibility thrown in their faces.
Interesting point. Since physics does appear on the surface to be continuous, I can’t rule out continuous propositions. Perhaps the amended saying should read “0 and 1 are not probability masses, and 0 is not a probability density.”
Oh. I was expecting your belief to be as with infinite-set atheism: that we never actually see an infinitely precise measurement.
We don’t, but what if there are infinitely precise truths nonetheless? The math of Bayesianism would require assigning them probabilities.
And, having assigned p(A) = 0 to such an event A I will not be able to rationally update away from zero without completely discarding my former model. There is no evidence that can cause me to rationally update p(A) = 0 to something else ever. Discard it and overwrite with something completely unrelated perhaps, but never update.
And, having assigned p(A) = 0 to such an event A I will not be able to rationally update away from zero without completely discarding my former model. There is no evidence that can cause me to rationally update p(A) = 0 to something else ever. Discard it and overwrite with something completely unrelated perhaps, but never update. p(A) is right there as the numerator!
(But yes, I take your point and tentatively withdraw the use of ‘impossible’ to refer to p=0.)
ETA: Well, maybe you’re allowed to use some mathematical magic to cancel out the 0 if p(B) = 0 too. But then, the chance of that ever happening is, well, 0.
Er, my bad. I missed your point. I see it now, duh.
So my friend thinks something S has a probability of zero but I know otherwise and point out that it is possible give assumption which I know my friend believes has a .1 chance of being true. He says “Oh right. I guess S is possible after all.” What has just happened? What do we say when we see the dart land at a specific point on the line?
Your friend had incorrectly computed the implications of his prior to the problem in question. On your prompting he re-ran the computation, and got the right answer (or at least a different answer) this time.
Perfect Bayesians are normally assumed to be logically omniscient, so this just wouldn’t happen to them in the first place.
In order to specify a point on the line you need an infinite amount of evidence, which is sufficient to counteract the infinitesimal prior. (The dart won’t hit a rational number or anything else that has a finite exact description.)
Or if you only have a finite precision observation, then you have only narrowed the dart’s position to some finite interval, and each point in that interval still has probability 0.
You wasted a great gambling opportunity.
Pengvado gives one good answer. I’ll add that your friend saying something has a probability of zero most likely means a different thing than what a Bayesian agent means when it says the same thing. Often people give probability estimates that don’t take their own fallibility into account without actually intending to imply that they do not need to. That is, if asked to actually bet on something they will essentially use a different probability figure that incorporates their confidence in their reasoning. In fact, I’ve engaged with philosophers who insist that you have to do it that way.
“Did not! Look closer, you missed by 1/infinity miles!”