Sniffnoy

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I'm Harry Altman. I do strange sorts of math.

Posts I'd recommend:

A summary of Savage's foundations for probability and utility -- if the arguments used to ground probability and utility seem circular to you, here's a non-circular way of doing it.
Underappreciated points about utility functions
Should

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4196

1167

16y

Sniffnoy

Should correlation coefficients be expressed as angles?

Edit 11/28: Edited note at bottom to note that the random variables should have finite variance, and that this is essentially just L². Also some formatting changes. This is something that has been bugging me for a while. The correlation coefficient between two random variables can be interpreted as the cosine of the angle between them[0]. The higher the correlation, the more "in the same direction" they are. A correlation coefficient of one means they point in exactly the same direction, while -1 means they point in exactly opposite directions. More generally, a positive correlation coefficient means the two random variables make an acute angle, while a negative correlation means they make an obtuse angle. A correlation coefficient of zero means that they are quite literally orthogonal. Everything I have said above is completely standard. So why aren't correlation coefficients commonly expressed as angles instead of as their cosines? It seems to me that this would make them more intuitive to process. Certainly it would make various statements about them more intuitive. For instance "Even if A is positive correlated with B and B is positively correlated with C, A might be negatively correlated with C." This sounds counterintuitive, until you rephrase it as "Even if A makes an acute angle with B and B makes an acute angle with C, A might make an obtuse angle with C." Similarly, the geometric viewpoint makes it easier to make observations like "If A and B have correlation exceeding 1/√2 and so do B and C, then A and C are positively correlated" -- because this is just the statement that if A and B make an angle of less than 45° and so do B and C, then A and C make an angle of less than 90°. Now when further processing is to be done with the correlation coefficients, one wants to leave them as correlation coefficients, rather than take their inverse cosines just to have to take their cosines again later. (I don't know that the angles you get this way are ac

101Nov 28, 2012

Sniffnoy

Message

I'm Harry Altman. I do strange sorts of math.

Posts I'd recommend:

A summary of Savage's foundations for probability and utility -- if the arguments used to ground probability and utility seem circular to you, here's a non-circular way of doing it.
Underappreciated points about utility functions
Should

...

4196

1167

16y

What do we mean by "impossible"?

(I'm reposting this here from an old Dreamwidth post of mine, since I've seen people reference it occasionally and figure it would be easier to find here.) So people throw around the word "impossible" a lot, but oftentimes they actually mean different things by it. (I'm assuming here we're talking...

Jan 1023

More on policy arguments and the AB problem

This is a followup to this earlier post, which I realized today didn't quite cover everything I wanted to. To recap, the AB problem (note: I don't know who coined this name, does anyone know the origin?) is a problem that arises in arguments that leads people mischaracterizing their opponent's...

Jun 11, 202510

Suggesting some revisions to Graham's hierarchy of disagreement

(This is once again me taking what old material from my personal blog and reposting it here with some revisions.) Graham's hierarchy of disagreement is pretty well known and fairly useful, but I think the upper levels have some issues. (Are the lower levels necessarily the best they could be?...

Apr 2, 20258

Workflow vs interface vs implementation

(Found this old draft that was more or less complete; figured I should publish it. It's adapted from this old blog post of mine.) It's commonly appreciated that when explaining how to use something, there's a distinction between the high-level usage instructions and the low-level details. But what I've noticed...

Mar 27, 202512

Quick thoughts on the difficulty of widely conveying a non-stereotyped position

(This is probably covering subjects already widely discussed here (haven't read LW much recently), but I wanted to get this down quickly and this seemed an appropriate place to post it.) Here's a problem to which I don't have a solution. How do you get large numbers of people past...

Mar 27, 202512

Doing principle-of-charity better

(Thought I'd try posting here some various rationality "quick thoughts" that I feel like I haven't seen expressed overtly enough -- sometimes I've already written about these elsewhere, but thought would be good to get them down here. Not claiming any originality here.) I think the principle of charity is...

Mar 27, 202522

X as phenomenon vs as policy, Goodhart, and the AB problem

Mar 27, 202514

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Should correlation coefficients be expressed as angles?

A summary of Savage's foundations for probability and utility.

Underappreciated points about utility functions (of both sorts)

Draft/wiki: Infinities and measuring infinite sets: A quick reference

Sniffnoy

What do we mean by "impossible"?

More on policy arguments and the AB problem

Suggesting some revisions to Graham's hierarchy of disagreement

Workflow vs interface vs implementation

Quick thoughts on the difficulty of widely conveying a non-stereotyped position

Doing principle-of-charity better

X as phenomenon vs as policy, Goodhart, and the AB problem

Should correlation coefficients be expressed as angles?

A summary of Savage's foundations for probability and utility.

Underappreciated points about utility functions (of both sorts)

Draft/wiki: Infinities and measuring infinite sets: A quick reference

What do we mean by "impossible"?

More on policy arguments and the AB problem

Suggesting some revisions to Graham's hierarchy of disagreement

Workflow vs interface vs implementation

Quick thoughts on the difficulty of widely conveying a non-stereotyped position

Doing principle-of-charity better

X as phenomenon vs as policy, Goodhart, and the AB problem