tag:blogger.com,1999:blog-3891434218564545511.post8124645336640246488..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: More on discounting small probabilitiesAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3891434218564545511.post-71753490824767524232022-11-15T10:50:18.639-06:002022-11-15T10:50:18.639-06:00Ah, "block" is my term. :-)
Here's ...Ah, "block" is my term. :-)<br /><br />Here's the background for why I am interested in expected values with respect to non-probabilities. The credences or degrees of belief of real human beings are unlikely to be consistent. In particular, they are unlikely to satisfy the axioms of probability, especially additivity. At the same time, real human beings need a way of making predictions. Mathematical expectation is out, because that requires at least a finitely-additive measure (normally Lebesgue integrals are defined with respect to a countably-additive measure but they can also be defined with respect to a finitely-additive one). So we need some other method for making predictions or generating expectations when the credences do not satisfy the axioms of probability.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-10622521438312814832022-11-14T09:42:02.899-06:002022-11-14T09:42:02.899-06:00Fubini's theorem applies to expected values de...Fubini's theorem applies to expected values defined with respect to a measure. The credence function P_e is not a measure in general, because in general it fails finite additivity. Thus, the standard Lebesgue integral with respect to P_e is undefined. I don't know what a "block integral" is.<br /><br />The point of level-set integrals for me is that they allow one to define a fairly well-behaved expectation or prevision with respect to credence assignments that are not probabilities because instead of additivity they only satisfy monotonicity (P(A) is less than or equal to P(B) if A is a subset of B).Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-91687400275271748222022-11-12T17:43:30.419-06:002022-11-12T17:43:30.419-06:00Why calculate the expected utility via "level...Why calculate the expected utility via <i>"level-set integrals"</i>, when according to <a href="https://proofwiki.org/wiki/Fubini%27s_Theorem" rel="nofollow">Fubini</a> you can also calculate the expected utility with "block integrals" or any other appropriate transformations of the x and y coordinates gaining the same exact result?<br />Why not calculate it with <a href="https://en.wikipedia.org/wiki/Polar_coordinate_system" rel="nofollow">polar coordinates</a>?<br />Sure, that would be more difficult to do without having any rotational symmetries here.<br />But you can do that and by doing that properly you or we should gain the same result for the expected utility.<br />Sooo...<br />What exactly makes <i>"level-set integrals"</i> so special here?!?<br />I don't see any particular good reason for this specific approach for calculating expected utilities over "block integrals" here.Kritschhttps://www.blogger.com/profile/13025223721628879816noreply@blogger.com