Implications and Reality
In my recent post “Extraordinary Claims”, I touched upon the nature of evidence-collecting. In particular, I discussed how the persuasiveness of the evidence must increase in proportion to the tenacity of the claim. In this follow-up, I will go into more detail on precisely how we (should) use evidence to update beliefs. We’ll need some more basic logic and probability theory in order to do so.
In mathematical logic, statements frequently take the form “If A then B”, or equivalently “A implies B” (written A => B). There are a couple of other ways to say this: we could say A is sufficient for B to be true, or B is necessary for A to be true. An example statement is “If x is an even number, then x is a whole number”. This is obvious, as all even numbers are whole numbers, but it is not true that all whole numbers are even numbers.
Rarely in reality can we make such “true or false” claims. It’s easy to state them using definitions (e.g. “If Jamie has two legs, then Jamie is an animal”), but elsewhere we abandon binary logic and turn to probability by giving implications strengths. The stronger the implication, the more likely it is true. By gaining more information, we improve our knowledge of these strengths. (In mathematical logic, strengths are always 100% or 0%.)
Consider a variation of the famous cancer test problem I posed in “Extraordinary Claims”. If the base rate of some type of cancer is one in a thousand, and you get a positive result on a test for that cancer that is correct 95% of the time, what is the probability that you have cancer?
If we denote the statement “You have cancer” with A, and the statement “You get a positive result on the test” with B, then what we are trying to find out is the probability of A given B, written p(A|B). This is equivalent to the strength of the implication from B to A. The only information we have is the base probability of you having cancer, p(A), and the accuracy of the test.
So why do people wrongly answer 95%, or something close to it? The answer lies in confusing “B => A” with “A => B“. The strength of A => B is indeed 95% – if you have cancer (A), then the probability that you will get a positive result on the test (B) is 95%. However, it is a universal fact that the strength of A => B alone tells you NOTHING about the strength of B => A. This is true in all forms of logic, everywhere, including the true/false logic of mathematics. Yet this is the error almost everyone makes (note that this question was originally posed to real doctors, who performed embarrassingly badly).
In getting from the strength of A => B to knowing the strength of B => A, we need to also consider the strengths of negative cases, i.e. what is the strength of “not A => not B“? If you don’t have cancer, what is the probability of getting a negative result on the test? Again, it’s 95% – the test’s accuracy rating. Also, if you don’t have cancer (not A), the probability of getting a positive result (B) is 5%.
As it turns out, two more pieces of information are vital in getting from A => B to B => A. They are p(A) (the original probability of A) and the strength of not A => B. The latter is needed because we need to know how contingent B is on A. Sure, A may lead to B 95% of the time, but what if B occurred a lot without A as well? The more often B occurs without A also occuring, the smaller the chance that B implies A.
In this problem, we know B occurs without A 5% of the time. That’s a lot less than than the 95% chance that B occurs with A, I hear you saying. It seems B is pretty contingent on A. So why is the final answer still so small?
This is where p(A) comes in. 5% is a smaller number than 95% – but these probabilies assume that, respectively, A has not or has occurred. We have to go back and look at our starting probability that A even occurs at all. In this case, p(A) is 0.1%, while p(not A) is 99.9%. Now it becomes a bit clearer: 95% of that 0.1% of people that have cancer will get a positive result, and 5% of that 99.9% of people that don’t have cancer will also get a positive result. Multiplying appropriately, we find that the probability of a person having cancer AND getting a positive result, i.e. p(A and B), is 0.095%, while the probability of a person NOT having cancer AND getting a positive result, i.e. p(not A and B), is 4.995%.
The proportion of p(A and B) out of the total p(A and B) + p(not A and B) (i.e. everyone who gets a positive result) is 1.87%, and this is our final answer. B implies A 1.87% of the time, i.e. the strength of B => A is 1.87%, i.e. 1.87% of people who get a positive test result will actually have cancer.
What we have just used is called Bayes’ Theorem. It is a relatively simple formula that helps you get from the strength of A => B to the strength of B => A by also considering p(A) (how extraordinary the claim A is) and the strength of not A => B (how contingent B is on A). You can look it up if you want to see it written out, but for now, take away its intuitive meaning: we can completely quantify scientific evidence (B), and use it to update our beliefs (A).
Great job explaining Bayes’ Theorem in simple and relevant terms so it’s easy to understand. My question for this series is, extraordinary claims require extraordinary evidence, but by what criteria do we judge what makes a claim extraordinary?