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PipFoweraker comments on Open Thread May 23 - May 29, 2016 - Less Wrong Discussion
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Reminiscing over one of my favourite passages from Anathem, I've been enjoying looking through visual, wordless proofs of late. The low-hanging fruit is mostly classical geomety, but a few examples of logical proofs have popped up as well.
This got me wondering if it's possible to communicate the fundamental idea of Bayes' Theorem in an entirely visual format, without written language or symbols needing translation. I'd welcome thoughts from anyone else on this.
Challenge accepted.
https://i.imgsafe.org/914f428.png
This looks great and I can see that it should work, but I can't seem to find a formal proof. Can you explain a bit?
http://lesswrong.com/lw/nhi/geometric_bayesian_update/
If I am reading this correctly:
I saw some footprints; I know that there are 1/3 humans around and 2/3 cats around. there is a 3/4 likelyhood that humans made the human shaped footprint; there is a 1/4 chance that cats in boots made the human shaped footprints. Therefore my belief is that humans are more likely to have made the footprints than cats.
(I think it needs a little work, but it's an excellent diagram so far)
A suggestion: modify the number of creatures on the left to equal a count of the frequency of the priors? And the number on the right to account for frequency of belief.
Yup.
I don't buy "frequency of belief". Maybe instead, I'd put those in thought bubbles, and change scaling of the bubbles.
C̶a̶n̶ ̶y̶o̶u̶ ̶a̶l̶s̶o̶ ̶a̶d̶d̶ ̶a̶ ̶w̶a̶t̶e̶r̶m̶a̶r̶k̶ ̶s̶o̶ ̶t̶h̶a̶t̶ ̶y̶o̶u̶ ̶g̶e̶t̶ ̶c̶r̶e̶d̶i̶t̶s̶ ̶i̶f̶ ̶I̶ ̶r̶e̶p̶o̶s̶t̶ ̶t̶h̶e̶ ̶i̶m̶a̶g̶e̶?̶ Edit: woops there is a watermark, I just didn't see it.
I was thinking more specficially, "I live with 1 humans and 2 cats. therefore my priors of who could have made these footprints are represented by one human and two cats". not exactly frequency of belief but a "belief of frequency"?
Edit: also can it be a square not a rectangle? Is there a reason it was a rectangle to begin with? Something about strength of evidence maybe?
One last edit: Can you make the "cat in boots" less likely? How many cats in boots do other people have in normal priors??
It's not supposed to be realistic - real frequency of cats in boots is way too low for that. But I adjusted it a little for you: https://i.imgsafe.org/5876a8e.png
Edit: and about the shape, it matters not, as long as you think in odds ratios.
I like this version much better. Yes the shape does not matter; it does help me think about it though. I think this is generally an excellent visual representation. Well done!
Whoah. That gets many points. What an excellent layout! We need to know what boots are for it to translate, but that's a lot closer to an ideal solution than I've worked through.
Edit - I thought the diagram looked familiar!
Bayes is mostly about conditioning, and so I think you can draw a Venn Diagram that makes it fairly clear.
Thanks! I've been playing around with it for a week or so but can't elegantly find a way to do it that meets my arbitrary standards of elegance and cool design :-)
Becomes easier when using non-circular shapes for Venn-ing, but my efforts look a little hacky.
I prefer a diagram like this with just overlapping circles. And you can kind of see how the portion of the hypothesis that exists in the evidence circle represents it's probability.
Arbital also has some nice visualizations: https://arbital.com/p/bayes_rule_waterfall/?l=1x1 https://arbital.com/p/bayes_rule_proportional/ https://arbital.com/p/bayes_log_odds/ and https://arbital.com/p/bayes_rule_proof/?l=1yd
Fivethirtyeight also made a neat graphic: https://espnfivethirtyeight.files.wordpress.com/2016/05/hobson-theranos-1-rk.png?w=1024&h=767
Was considering something like a tshirt of p(smoke|fire) and p(fire|smoke). never came to fruition; feel free to take the idea if you like.