Sideways comments on Bayes' Theorem Illustrated (My Way) - Less Wrong

126 03 June 2010 04:40AM

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Comment author: 04 June 2010 05:02:04PM 3 points [-]

Your analogy doesn't hold, because each spin of the roulette wheel is a separate trial, while choosing a door and then having the option to choose another are causally linked.

If you've really thought about XiXiDu's analogies and they haven't helped, here's another; this is the one that made it obvious to me.

Omega transmutes a single grain of sand in a sandbag into a diamond, then pours the sand equally into three buckets. You choose one bucket for yourself. Omega then pours the sand from one of his two buckets into the other one, throws away the empty bucket, and offers to let you trade buckets.

Each bucket analogizes to a door that you may choose; the sand analogizes to probability mass. Seen this way, it's clear that what you want is to get as much sand (probability mass) as possible, and Omega's bucket has more sand in it. Monty's unopened door doesn't inherit anything tangible from the opened door, but it does inherit the opened door's probability mass.

Comment author: 04 June 2010 05:08:36PM 2 points [-]

That works better for you? That's deeply surprising. Using entities like Omega and transmutation seems to make things more abstract and much harder to understand what the heck is going on. I must need to massively update my notions about what sort of descriptors can make things clear to people.

Comment author: 04 June 2010 05:19:15PM 1 point [-]

I use entities outside human experience in thought experiments for the sake of preventing Clever Humans from trying to game the analogy with their inferences.

"If Monty 'replaced' a grain of sand with a diamond then the diamond might be near the top, so I choose the first bucket."

"Monty wants to keep the diamond for himself, so if he's offering to trade with me, he probably thinks I have it and wants to get it back."

It might seem paradoxical, but using 'transmute at random' instead of 'replace', or 'Omega' instead of 'Monty Hall', actually simplifies the problem for me by establishing that all relevant facts to the problem have already been included. That never seems to happen in the real world, so the world of the analogy is usefully unreal.

Comment author: 04 June 2010 09:14:33PM 1 point [-]

I really like this technique.

Comment author: 05 June 2010 01:06:38AM 0 points [-]

I'm not keen on this analogy because you're comparing the effect of the new information to an agent freely choosing to pour sand in a particular way. A confused person won't understand why Omega couldn't decide to distribute sand some other way - e.g. equally between the two remaining buckets.

Anyway, I think JoshuaZ's explanation is the clearest I've ever seen.