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>since simulating random points on a thousand dimensional sphere doesn't seem to give points near these axes. that is kind of circular argument as "random" rea
by trhway 5y ago
>since simulating random points on a thousand dimensional sphere doesn't seem to give points near these axes.
that is kind of circular argument as "random" really depends on the density measure underlying the chosen sampling distribution.
- tgb 5y agoIt's no more ill defined than volume itself (and less so since it doesn't need an arbitrary scale).
- eru 5y agoI don't understand your 'less so' comment. Could you please elaborate?
- tgb 5y agoIf you want to define volume in D-dimensional space then you need to choose what the unit volume is, for example that a unit hyper cube has unit volume one. (That choice is relevant for these discussions and you could equally have chosen a unit ball to have volume one since you're defining D-dimensional volume you get to choose.) But a probability distribution is always normalized to integrate to one, which means either choice of volume definition will give you the same uniform probability distribution on the ball. So there's an extra arbitrary constant to choose for volume that isn't important in the probability distribution even though you define the probability density in terms of the volume.
- eru 5y agoOK. Though you could leave your volume definition parameterized, instead of picking a specific scale, couldn't you? I do agree that volume and probability distribution are essentially the same. (That's also why at my university measure theory and probability theory were handled in the same course.)
- kgwgk 5y agoIf only there was some kind of natural density measure to use in an Euclidean space R^D…