Considering all scenarios when using Bayes' theorem.

post by Alexei · 2011-06-20T18:11:34.810Z · LW · GW · Legacy · 7 comments

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Disclaimer: this post is directed at people who, like me, are not Bayesian/probability gurus.

Recently I found an opportunity to use the Bayes' theorem in real life to help myself update in the following situation (presented in gender-neutral way):

Let's say you are wondering if a person is interested in you romantically. And they bought you a drink.
A = they are interested in you.
B = they bought you a drink.
P(A) = 0.3 (Just an assumption.)
P(B) = 0.05 (Approximately 1 out of 20 people, who might be at all interested in you, will buy you a drink for some unknown reason.)
P(B|A) = 0.2 (Approximately 1 out of 5 people, who are interested in you, will buy you a drink for some unknown reason. Though it's more likely they will buy you a drink because they are interested in you.)

These numbers seem valid to me, and I can't see anything that's obviously wrong. But when I actually use Bayes' theorem:
P(A|B) = P(B|A) * P(A) / P(B) = 1.2
Uh-oh! Where did I go wrong? See if you can spot the error before continuing.

Turns out:
P(B|A) = P(A∩B) / P(A) ≤ P(B) / P(A) = 0.1667
BUT
P(B|A) = 0.2 > 0.1667

I've made a mistake in estimating my probabilities, even though it felt intuitive. Yet, I don't immediately see where I went wrong when I look at the original estimates! What's the best way to prevent this kind of mistake?
I feel pretty confident in my estimates of P(A) and P(B|A). However, estimating P(B) is rather difficult because I need to consider many scenarios.

I can compute P(B) more precisely by considering all the scenarios that would lead to B happening (see wiki article):

P(B) = ∑_i P(B|H_i) * P(H_i)

Let's do a quick breakdown of everyone who would want to buy you a drink (out of the pool of people who might be at all interested in you):
P(misc. reasons) = 0.05; P(B|misc) = 0.01
P(they are just friendly and buy drinks for everyone they meet) = 0.05; P(B|friendly) = 0.8
P(they want to be friends) = 0.3; P(B|friends) = 0.1
P(they are interested in you) = 0.6; P(B|interested) = P(B|A) = 0.2
So, P(B) = 0.1905
And, P(A|B) = 0.315 (very different from 1.2!)

Once I started thinking about all possible scenarios, I found one I haven't considered explicitly -- some people buy drinks for everyone they meet -- which adds a good amount of probability (0.04) to B happening. (Those types of people are rare, but they WILL buy you a drink.) There are also other interesting assumptions that are made explicit:

• Out of all the people under consideration in this problem, there are twice as many people who would be romantically interested in you vs. people who would want to be your friend.
• People who are interested in you will buy you a drink twice as often as people who want to be your friend.

The moral of the story is to consider all possible scenarios (models/hypothesis) which can lead to the event you have observed. It's possible you are missing some scenarios, which under consideration will significantly alter your probability estimates.

Do you know any other ways to make the use of Bayes' theorem more accurate? (Please post in comments, links to previous posts of this sort are welcome.)

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