Doesn’t the necessity of this half-plane containing the actions restore a difference, even just within this decision algorithm (leaving aside the uniqueness of probability in learning and reasoning), between probability and utility? Probability is the direction perpendicular to this invisible line, and utility is the slope relative to this invisible line.
Sure. And given the rescalability, for any set of values you can rescale everything so that almost any line is possible. But then everything is in some suspiciously narrow band, which again lends itself to a principal component sort of coordinate system.
When inferring such a division, interestingly the ordering of value remains unchanged, but the ordering of the inferred probability can be different than the ordering of the original probability, because average-value events are interpreted as more probable.
Doesn’t the necessity of this half-plane containing the actions restore a difference, even just within this decision algorithm (leaving aside the uniqueness of probability in learning and reasoning), between probability and utility? Probability is the direction perpendicular to this invisible line, and utility is the slope relative to this invisible line.
It can, if there is a unique line. There isn’t a unique line in general—you can draw several lines, getting different probability directions for each.
Sure. And given the rescalability, for any set of values you can rescale everything so that almost any line is possible. But then everything is in some suspiciously narrow band, which again lends itself to a principal component sort of coordinate system.
When inferring such a division, interestingly the ordering of value remains unchanged, but the ordering of the inferred probability can be different than the ordering of the original probability, because average-value events are interpreted as more probable.