Comment on Open Thread: January 2010

Wed, 06 Jan 2010 22:14:21 GMT

It wasn't: when a certain form of argument is asserted to be valid, it suffices to demonstrate a single counterexample to falsify the assertion.

Not for probabilistic claims.

It's kind of funny -- you wrote

Valid reasoning. The problem lies in the failure to include all relevant knowledge [].

But the the failure to include all relevant knowledge is exactly why the reasoning isn't valid.

No. The reasoning can be valid even though, given additional information, the conclusion would be changed.

Example:

Bob is accused of murder.
Then, Bob's fingerprints are the only ones found on the murder weapon.
Bob has an ironclad alibi: 30 witnesses and video footage of where he was.

O(guilty|accused of murder) = 1:3
P(prints on weapon|guilty) / P(prints on weapon|~guilty) = 1000
O(guilty|accused of murder, prints on weapon) = 1000*(1:3) = 1000:3
P(guilty| ....) > 99%.

If Bob is accused of murder, he has a moderate chance of being guilty.
Bob's prints are much more likely to later be the only ones found on the murder weapon if he were guilty than if he were not.
Bob's prints are the only ones on the murder weapon.
Therefore, there is a very high probability Bob is guilty.
Bob probably isn't guilty.
Therefore the Bayes Theorem is invalid reasoning. (???)

See the problem? The form of the reasoning presented originally is valid. That is what I was defending. But obviously, you can show the conclusion is invalid if you include additional information. In the general case, reasoning that

If a person is an X, then he is probably not a Y.
This person is a Y.
Therefore, he is probably not an X.

is valid, if that is all you know. But you can only invert the conclusion by assuming a higher level of knowledge than what is presented (in the quoted model above) -- specifically, that you have an additional low-entropy point in your probability distribution for "Y implies high probability of X". But again, this assumes a probability distribution of lower entropy (higher informativeness) than you can justifiably claim to have.

So you can actually form a valid probabilistic inference without looking up the specific p(H)/p(E) ratio applying to this specific situation -- just use your max entropy distribution for those values, which favors the reasoning I was defending.

I'm actually writing up an article for LW about the "Fallacy Fallacy" that touches on these issues -- I think it would be worthwhile to finish it and post it. (So no, I'm not just arguing this point to save face -- there's an important lesson here that ties into the Bertrand Paradox and Jaynes's work.)

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