Wednesday, February 19, 2014

Bayesian inference

[from another conference--a friend asked for a quick explanation of what Bayesian inference is]

The basic idea behind Bayesian reasoning is quite simple, it's simply quantifying with a formula what you do anyway instinctively. Here goes, without the math:

When you estimate probabilities, it's fair to say that you're guessing at the ratio of possible universes. Saying "Three out of four, Richard Nixon is a crook" means that of all the possible universes you can see from where you are today, Richard Nixon is a crook in three of them. "Fifty-fifty it comes up heads" means you think the possible universes are evenly balanced. That's the Bayesian view of probability, that it's a subjective estimate instead of something real.

Bayesian inference is about how your view of a given probability changes over time as you see new things. When you see something new, you delete all the universes in which that thing would not have happened, and then look at the ratio of remaining possibilities for the new probability. Example: say Nixon says "I am not a crook." If I estimate that Nixon would always say that in universes where he is a crook, and he wouldn't bother to say it in half the universes where he isn't a crook, then as soon as he says it, half of the universes where he was innocent vanish. Instead of 3/4 chance he's guilty, it's now a 6/7 chance. (Remember, the chance of being innocent drops from 1/4 to 1/8, and it's the ratio of 1/8 to 3/4 that matters because any universe not in that ratio has already been disproven.)

This doesn't just work for yes/no questions, it could be that I'm trying to decide between "Richard Nixon is a crook" vs. "Richard Nixon is incompetent." Either way I'm simply eliminating possible universes with each piece of evidence.

So far so good, but the great weakness of Bayesian inference is this: I'm simply narrowing down my hypotheses each time based on evidence. What if the truth isn't in any of my hypotheses? What if the real explanation is that "Richard Nixon is an alien"? Since Bayesian inference is all about ELIMINATING possibilities, I will never, ever tumble to the truth. Even if I see him beam himself up into his spaceship, Bayesianism has no way for me to deal with that. And yet in the real world, I would indeed change my mind if I saw his spaceship--I'd generate a new hypothesis on the fly. So my mind isn't Bayesian, it isn't deductive. What is my mind doing?

Bayesian inference is demonstrably the most reasonable way of deciding between two possibilities, but somewhere inside my head I must have a container that says "infinite number of other possibilities."

Deck thyself now with majesty and excellency; and array thyself with glory and beauty.
Cast abroad the rage of thy wrath: and behold every one that is proud, and abase him.
Look on every one that is proud, and bring him low; and tread down the wicked in their place.
Hide them in the dust together; and bind their faces in secret.
Then will I also confess unto thee that thine own right hand can save thee.

I could not love thee, dear, so much,
Loved I not Honor more.

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