Understanding Bayes Theorem

I’ve always found Bayes Theorem hard to understand. Even after semesters of studying probability you can can eventually get the hang of the math involved, but developing a solid understanding of the real world implcations can be challenging.

Breakdown

The standard formula you often see is:

$P(A|B) = \frac{P(A)P(B|A)}{P(B)}$

If we break up the formula, there are 4 parts, and 3 quantities that we need to find before we can get the answer.

$P(A)$

This is the probability of A happening not knowing anything else about the problem.

$P(B)$

This is the probability of B happening not knowing anything else about the problem. Sometimes its not possible to know this directly, but we can also find it based on this property:

$P(B) = P(A)P(B|A) + P(\bar{A})P(B|\bar{A})$

We already know $P(A)$ and $P(\bar{A})$ is just $1 - P(A)$, so we just need find $P(B|A)$ and $P(B|\bar{A})$ from the problem statement.

$P(B|A)$

This is the probability of B happening given that A has happened.

$P(A|B)$

This is what we are trying find and in words, it can be described as:

What is the probability of seeing A given that we have seen B?

or maybe:

What is the probability of A happening given what we know about B?

Visualize

It can help to visualize the probability as a large square that can be divided into portions. The square can largely be divided into the probability of hypothesis A happening, $P(\textcolor{#d97706}{A})$ and the probability of A not happening, $P(\textcolor{#059669}{\bar{A}})$. These 2 events are obviously, mutually exclusive, and so the entire probability space is covered by these 2 events.

Lets adjust the probability of A happening, $P(A)$, and see how the probability space changes.

Then, by looking only at the left, we can think about the probability of B happening given A, $P(B|A)$, and the product of these 2 together give us the are of the square in the lower left corner.

We can then look at the right to determine $P(B)P(B|A)$.

We now have everything we need to put these 3 quantities together to get the answer.

Example

You find that a family member tested positive for a genetic defect. What are the odds they actually have the defect?

The doctor tells you that 1% of people have this certain genetic defect. He also tells you that 90% of tests for the gene detect the defect accurately and 9.6% of the tests give false positives.

Take a moment to think how likely it is that your family member has the defect. You probably think that is very high, greater than 50% at least right?

Lets take this information and figure out the odds using Bayes Theorem.

• $P(A)$ = 0.01
• $P(B|A)$ = 0.90
• $P(B|\bar{A})$ = 0.096

$P(A|X)$ = (.9 * .01) / (.9 * .01 + .096 * .99) = 0.0865 (8.65%).

This is only an 8.65% chance that your family member has the defect. This is much lower than you probably thought.