Bayesian Theorem for the Practical Thinkers

I have been reading up on Bayesian Theorem, as you might have noticed based on my past blog posts. What really gets me is how hard it is to think in practical terms for people in general without a degree in Statistics and without resorting to complicated math. Because Bayesian Theorem/Inference is so useful in our daily lives, I would like to share my shortcut so people can calculate the probability using a simple 10-key calculator instead of a computer.

The shortcut is to always think in terms of odds instead of probability. The power of Bayesian Theorem is to take the base rate and after some new evidences provided the modified rate.

The best way to learn this is to use some examples:

Example 1:
From this blog, here is an example:

1% of women have breast cancer (and therefore 99% do not).
80% of mammograms detect breast cancer when it is there (and therefore 20% miss it).
9.6% of mammograms detect breast cancer when itโ€™s not there (and therefore 90.4% correctly return a negative result).
What’s the probability of having the breast cancer once detected positive by mammograms?

The trick is to think of the base odd of getting the breast cancer:
1% to 99% = 1 to 99
Now think of the odd the evidence provided by mammograms:
80% to 9.6% = 8.33 to 1

So the odd of having the breast cancer when tested positive by mammograms are:
1/99 * 8.33/1 = 8.33/99 = 0.084 or “odd of 0.084 to 1”

Now you must convert the odd to probability (if probability is what you’re looking for):
0.084 to 1 odd = 0.084/(1+0.084) = 7.8%

Example 2:
Allen Downey, my favorite Bayesian Statistics author and professor, has this example in his blog:
Elvis Presley had a twin brother who died at birth. What is the probability that Elvis was an identical twin?

You need the following facts:
”Twins are estimated to be approximately 1.9% of the world population, with monozygotic twins making up 0.2% of the total—and 8% of all twins.”

The odd of getting identical twin to fraternal twins are:
8% of twins are identical twins
92% of twins are fraternal twins
So the odd of identical twins is 8% to 92% or 0.087 to 1

Now there is another piece of information we must take into account => It’s a twin brother. The odd of same sex in a twin increases the odd that his brother is an identical twin. What’s the odd? It’s 2:1 ( identical twin brother + fraternal twin brother to fraternal sister).

So the odd of Elvis’s brother being an identical twin is:
0.087:1 x 2:1 = 0.174:1
Converted to probability => 0.174/(1+0.174) = 15%

Example 3:
Let’s do a final example from Example 5 of Allen Downey’s blog:
According to the CDC, “Compared to nonsmokers, men who smoke are about 23 times more likely to develop lung cancer and women who smoke are about 13 times more likely.”
If you learn that a woman has been diagnosed with lung cancer, and you know nothing else about her, what is the probability that she is a smoker?

So the odd of getting cancer as a woman who smoke is 13 to 1 (13:1). Now we need to know the base odd of women who smoke to non-smoking women. From the blog, it’s 17.9% of woman smoke. So the odd is 17.9% to 82.1% or 0.21:1
So the odd that the woman is a smoker is:
13:1 x 0.21:1 = 2.83:1 => 2.83/(1+2.83)= 74%

Now isn’t that more intuitive and practical. Go out and apply the Bayesian Theorem in a party to impress people.
Or maybe you want to calculate your chance of meeting single women if you were a single men. Let’s say you’re going to a friend’s party whose friends are 25% female and 75% male and there’s a probability that 20% of the female are single, 80% is not single or unavailable. There you have base odd of 1:3 (25%:75%)and the odd of meeting single women is 1:4 (20%:80%). Your odd of meeting a single woman in a party is going to be (1/3)*(1/4)=1/12 or 0.083:1 => 7.7%. Suppose you’re highly selective and possess a prince charming quality, your odd of finding your qualified, desirable women is 1:10. Now your odd just drop to 1/120 or 0.83%. Unless the party attendance is going to be > 120 people, then it’s worth a shot. Otherwise, you might as well stay home and watch a sports game at home instead ๐Ÿ˜‰