Timeline for Conditional probability of continuous variable
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2015 at 13:51 | comment | added | whuber♦ | When theory and intuition diverge, one cure is to change the theory. But first you should try to understand the discrepancy. All that "theory" amounts to, in the end, is drawing ineluctable conclusions from a set of axioms that are taken to be "intuitively obvious" or incontrovertibly demonstrated. When one of those conclusions seems intuitively false, then we should seriously consider that our intuition is inconsistent. Indeed, that's the normal state of affairs for human beings. Thus our default reaction should first be to recalibrate our intuition and not be hasty to abandon the theory. | |
| Mar 28, 2015 at 0:52 | comment | added | Neil G | @whuber: You're right. I really liked Potato's answer to one of my questions. I personally think that if there is discrepancy between theory and intuition, that we should seek new, more complete theories. Maybe the "principle of indifference" isn't quite right, or isn't generally workable, but I have a natural desire for probability theory to answer questions for which we have an intuitive understanding. Maybe Lebesgue had the same kind of angst about Riemann integration when he created his integral? | |
| Mar 28, 2015 at 0:20 | comment | added | whuber♦ | That is a much more interesting argument. I want to give it some thought, because densities are tricky to work with. For instance, if $5$ and $6$ had been replaced by $0$ and $10$, the answer might be different. And if the uniform density on $[0,10]$ had been replaced by an equal mixture of uniform densities on $[0,5)$ and $(5,10]$, some delicate reasoning would be needed--even though the underlying distribution in both cases is exactly the same. | |
| Mar 27, 2015 at 21:50 | comment | added | Neil G | @whuber: it replaced one distribution with another whereby the uniform regions around the 5 and 6 were unchanged — in the same way I think that zooming out tries to leave densities unchanged in the original circles in the Bertrand paradox. | |
| Mar 27, 2015 at 21:45 | comment | added | whuber♦ | Sure it does: there are plenty of ways to transform one continuous distribution to another that swap two values. Actually, your "flipping" didn't even preserve the original distribution. (It changed its support altogether.) So it would appear that all you're doing is replacing one distribution by another one. There doesn't seem to be any principle operating here at all. | |
| Mar 27, 2015 at 21:39 | comment | added | Neil G | @whuber: The flipping argument wouldn't work for a Cauchy distribution, unless you flipped around its mode. | |
| Mar 27, 2015 at 21:37 | comment | added | whuber♦ | Thank you for a thoughtful post. I, for one, seriously doubt the "principle of indifference" will ever be mainstream, because it is not workable. Your argument falls apart when the underlying values are re-expressed. The uniform distribution on $[0,10]$ might thereby become, say, a Cauchy distribution, $5$ could become $0$, and $6$ become $\sqrt{1-\frac{2}{\sqrt{5}}}$. Your "principle of indifference" now produces a completely different answer. (I used the probability transforms to work out this example.) | |
| Mar 27, 2015 at 21:08 | history | answered | Neil G | CC BY-SA 3.0 |