Daily Archives: January 18, 2012

The Benefit Of Thinking Like A Bayesian

So this following quote is from an article I posted in the last paragraph of my “Why Are Smart People Ugly” post called The Marvels And The Flaws Of Intuitive Thinking:

Now, what can’t [System 1; “Intuitive Thinking”] do? It cannot deal with multiple possibilities at once. Dealing with multiple possibilities at once is something we do consciously and deliberately. System 1 is bound to the suppression of ambiguity, which means one interpretation. It cannot do sum-like variables. Sum-like variables demand another kind of thinking. It is not going to do probability properly, it is not going to do economic value properly, and there are other things that it will not do.

Here an important point is how you combine information about individual cases with information with statistical information. I’m going to argue that System 1 has a lot of trouble with statistics. System 1, and here I believe the analogy from perception is very direct, it’s intended, or designed, to deal with individual particular cases, not with ensembles, and it does beautifully when it deals with an individual case. For example, it can accumulate an enormous amount of information about that case. This is what I’m trying to exploit in calling it System 1. It’s coming alive as I’m describing it. You’re accumulating information about it. But combining information of various kinds, information about the case, and information about the statistics, seems to be a lot harder.

Here is an old example.

There are two cab companies in the city. In one, 85 percent of the cabs are blue, and 15 percent of the cabs are green. There was a hit-and-run accident at night, which involved a cab. There was a witness, and the witness says the cab was green, which was the minority. The court tested the witness -we can embellish that a little bit – the court tested the witness and the finding is the witness is 80 percent reliable when the witness says “blue”, and when the witness says “green”, it’s 80 percent reliable. You can make it more precise, there are complexities, but you get the idea. You ask people, what’s your judgment? you’ve had both of these items of information, and people say 80 percent, by and large. That is, they ignore the base rate, and they use the causal information about the case. And it’s causal because there is a causal link between the accident and the witness.

As soon as I started reading that final paragraph, the word problem, I immediately started thinking of it in terms of Bayes’ Theorem. I almost instantly placed the sentence The court tested the witness…and the finding is the witness is 80 percent reliable when the witness says “blue”, and when the witness says “green”, it’s 80 percent reliable as the Likelihood Ratio and determined that it was exactly 1. Which meant that the prior probability (the base rate, as he describes it) does not change due to the witness’ testimony. It was 15% before being introduced to the witness’ testimony, and it’s 15% after.

As he explains, most people say that the probability that the cab that crashed was green is 80 percent, which is false. The witness has a certain reliability, meaning there’s a relationship between the witness’ testimony and whether what he says is actually true. And the witness’ testimony can be determined by the Likelihood Ratio, which isn’t just how many times he’s right but has to be compared to how many times he’s wrong.

The Likelihood Ratio is really just “correct guess divided by incorrect guess”.

Even though it might take significantly more work to get the actual probabilities if he hadn’t used equal numbers in the Likelihood Ratio, dividing one number by another number can be “calculated” by System 1 thinking quickly to determine if the number is bigger than 1 or less than 1. And how much bigger than 1 the Likelihood Ratio is can give you a feel for how much bigger the prior probability (base rate) increases, decreases, or remains the same; scale is something that System 1 does really well (explained in the article’s previous paragraphs).

Just to be thorough, here is the word problem fully computed:

H: Car accident was green cab
E: Witness testimony
P(H): .15
P(E | H): .8
P(E | ~H): .8

P(H | E) = P(E | H) * P(H) / [P(E | H) * P(H)] + [P(E | ~H) * P(~H)]
= .8 * .15 / [.8 * .15] + [.8 * .85]
= .12 / [.12] + [.68]
= .12 / .8
= .15

So all is not lost! Intuitive thinking can be trained to think like System 2 (“rational”) thinking; this is even explained in the concluding paragraphs. But in order to do that, one has to know what “System 1” thinking encapsulates. So, in order to be a good thinker, one must first know how the brain thinks.

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Posted by on January 18, 2012 in Bayes

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