So the prior that you’re updating for each point the clever arguer makes starts out low. It crosses 0.5 at the point where his argument is about as strong as you would expect given a 50⁄50 chance of A or B.
I don’t believe this is exactly correct. After all, when you’re just about to start listening to the clever arguer, do you really believe that box B is almost certain not to contain the diamond? Why would you listen to him, then? Rather, when you start out, you have a spectrum of expectations for how long the clever arguer might go on—to the extent you believe box A contains the diamond, you expect box B not to have many positive portents, so you expect the clever arguer to shut up soon; to the extent you believe box B contains the diamond, you expect him to go on for a while.
The key event is when the clever arguer stops talking; until then you have a probability distribution over how long he might go on.
The quantity that slowly goes from 0.1 to 0.9 is the estimate you would have if the clever arguer suddenly stopped talking at that moment; it is not your actual probability that box B contains the diamond.
Your actual probability starts out at 0.5, rises steadily as the clever arguer talks (starting with his very first point, because that excludes the possibility he has 0 points), and then suddenly drops precipitously as soon as he says “Therefore...” (because that excludes the possibility he has more points).
It is very possible I don’t understand this properly, but assuming you have knowledge of what strength of evidence is possible, could you start at 0.5 and consider strong arguments (relative to possible strength) as increasing the possibility and weak arguments as decreasing the possibility instead? With each piece of evidence you could increase the point at which weak arguments are viewed as having a positive effect, so numerous weak arguments could still add up to a decently high probability of the box containing the diamond.
For example, if arguments are rated in strength from 0 to 1, and most arguments would not be stronger than .5, my approach would be as follows for each piece of evidence:
Piece 1: Probability += (strength-.25)
Piece 2: probability += (strength-.22)
Piece 3: probability += (strength-.20)
etc.
I am of course oversimplifying the math, and looking at how you are approaching stoppage, perhaps this isn’t actually effectively much different from your approach. But this approach is more intuitive to me than considering stopping a separate event on its own. If he is struck by lightning, as mentioned several times throughout this discussion, it is hard to view this in the same light as if he had stopped on his own as an independent event, but I am not sure the difference is enough that the probability of the diamond being in the box should be substantially different in the two cases.
Can someone clear up what issues there are with my approach? It makes more sense to me and if it is wrong, I would like to know where.
So the prior that you’re updating for each point the clever arguer makes starts out low. It crosses 0.5 at the point where his argument is about as strong as you would expect given a 50⁄50 chance of A or B.
I don’t believe this is exactly correct. After all, when you’re just about to start listening to the clever arguer, do you really believe that box B is almost certain not to contain the diamond? Why would you listen to him, then? Rather, when you start out, you have a spectrum of expectations for how long the clever arguer might go on—to the extent you believe box A contains the diamond, you expect box B not to have many positive portents, so you expect the clever arguer to shut up soon; to the extent you believe box B contains the diamond, you expect him to go on for a while.
The key event is when the clever arguer stops talking; until then you have a probability distribution over how long he might go on.
The quantity that slowly goes from 0.1 to 0.9 is the estimate you would have if the clever arguer suddenly stopped talking at that moment; it is not your actual probability that box B contains the diamond.
Your actual probability starts out at 0.5, rises steadily as the clever arguer talks (starting with his very first point, because that excludes the possibility he has 0 points), and then suddenly drops precipitously as soon as he says “Therefore...” (because that excludes the possibility he has more points).
It is very possible I don’t understand this properly, but assuming you have knowledge of what strength of evidence is possible, could you start at 0.5 and consider strong arguments (relative to possible strength) as increasing the possibility and weak arguments as decreasing the possibility instead? With each piece of evidence you could increase the point at which weak arguments are viewed as having a positive effect, so numerous weak arguments could still add up to a decently high probability of the box containing the diamond.
For example, if arguments are rated in strength from 0 to 1, and most arguments would not be stronger than .5, my approach would be as follows for each piece of evidence:
Piece 1: Probability += (strength-.25)
Piece 2: probability += (strength-.22)
Piece 3: probability += (strength-.20)
etc.
I am of course oversimplifying the math, and looking at how you are approaching stoppage, perhaps this isn’t actually effectively much different from your approach. But this approach is more intuitive to me than considering stopping a separate event on its own. If he is struck by lightning, as mentioned several times throughout this discussion, it is hard to view this in the same light as if he had stopped on his own as an independent event, but I am not sure the difference is enough that the probability of the diamond being in the box should be substantially different in the two cases.
Can someone clear up what issues there are with my approach? It makes more sense to me and if it is wrong, I would like to know where.