## Tuesday, June 15, 2010

### Best guess at objective probability

I think I may have once thought that epistemic probabilities are something like one's best guess at the objective probability. But that's obviously mistaken. Suppose Sally tells me that the objective probability of some future event is 0.8. Sam tells me the probability is 0.2. God tells me that either Sally or Sam is right. Moreover, I know that that Sally tends to be right 60% of the time. What should my epistemic probability of the event be? Well, my "best guess at the objective probability" is 0.8—what Sally says, since she's right 60% of the time. But obviously my epistemic probability should be (0.6)(0.8)+(0.4)(0.2)=0.56. Which I know is at least 0.24 away from the actual objectively probability.

#### 2 comments:

Mike Almeida said...

Well, my "best guess at the objective probability" is 0.8—what Sally says, since she's right 60% of the time. But obviously my epistemic probability should be (0.6)(0.8)+(0.4)(0.2)=0.56. Which I know is at least 0.24 away from the actual objectively probability.

That's a clever (and tricky!) example. I can use the same sort of argument to show that the proper objective probability I should assign an event is not the best estimate of the objective probability of that event. Take a simple event such as a fair coin falling tails. Smith tells me it the chances are .5. Jones tells me the chances are .5. Both have about the same chances of being right. Suppose God tells me that one of them is right (after all, the coin is fair!). God has already tossed the coin and it is under his right hand. Either the coin fell tails (objective probability 1)or fell heads (objective probability 1). What's my epistemic probability that it fell tails? It's .5. What objective probability should I assign to it fell tails? That's also .5. So the proper objective probability I should assign is at least .5 off of the objective probability of it fell tails (which it either 0 or 1)!

Mike Almeida said...

Or maybe it's easier to make the proposition about the future (though I'm sure it matters). You know the coin is fair and will be tossed at 10am. Before 10am, God (or some clairvoyent or soothsayer or something) tells you truthfully that he knows whether will fall heads or fall tails. So it is true right now that it will fall tails or that it will fall heads. But the chance I should assign to each outcome stays the same, .5. I shoould not conclude that, no matter which is right, I know my assignment of objective chance is off by .5.

So knowing that it is right that it will come up heads or it is right that it will come up tails doesn't make my epistemic probability pull apart from the objective probabiliyt about what will come up. That is, I don't conclude from that that the objective probability is 1 or 0 for tails, therefore my epistemic probabiliyt for tails must be off by .5