### Thinking like a Bayesian

*Hi I am Paras Chopra, founder & chairman of VWO. Every fortnight, on this blog and on our email list, I’ll be posting a new idea or a story on experimentation and growth.* Here is my 3rd letter.

I recently finished reading a book on the history of the **Bayes’ theorem** (appropriately called *the theory that would not die*) and thought you may enjoy my notes from it.

**1/ Statistics is all about calculating probabilities, and there are two camps who interpret probability differently.**

- Frequentists = frequency of events over multiple trials
- Bayesians = subjective belief of the outcome of events

**2/ This philosophical divide informs what these two camps usually bother with.**

- Frequentists = probability of data, given a model (of how data could have been generated)
- Bayesians = probability of model, given the data

**3/ Most often we care about the latter question and that is what the Bayesian way of thinking helps with.**

For example, given that the mammography test is positive, we want to know what the probability of having breast cancer is. And given breast cancer, we usually donâ€™t care about the probability of the test being positive.

**4/ These two questions sound similar but have different answers.**

For example, imagine that 80% of mammograms detect breast cancer when itâ€™s there and ~90% come out as negative when itâ€™s not there (which means for 10% times it comes as positive even if itâ€™s not there).

Then if only 1% population has breast cancer, the probability of having it given a positive test is 7.4%.

**5/ Read that again: **

80% times the mammography test works and yet if you get a positive, your chances of having breast cancer are only 7.4%.

How is it possible?

**6/ The math is simple:**

- Chances that the test is positive when a patient has breast cancer = chances of detecting breast cancer when a patient has it * chances of having breast cancer in the first place = 80% * 1% = 0.8%
- Chances that test is positive when a patient does NOT have breast cancer = chances of detecting breast cancer when a patient DOESNâ€™T have it * chances of NOT having breast cancer in the first place = 10% * 99% = 9.9%

Now, the chances of having breast cancer on a positive mammogram are simply:

% times you get a positive mammogram if you have breast cancer / % times you can get a positive mammogram.

We calculated these numbers above, so this becomes

0.8%/(0.8%+9.9%) = 7.4%.

Voila! So even if a test works 80% of the times, it may not be very useful (if population incidence rate is low, which is 1% in this case). This is why doctors recommend taking multiple tests, even after a positive detection.

**7/ When you understand Bayes’ theorem, you realize that it is nothing but arithmetic.**

Itâ€™s perhaps the simplest but most powerful framework I know. If you want to build a better intuition about it, I recommend reading thisÂ visual introductionÂ to Bayes’ theorem (which also contains the breast cancer example we talked about).

**8/ The key idea behind being a Bayesian is that *everything* has a probability. **

So instead of thinking in certainties (yes/no), you start thinking about chances and odds.

**9/ Today, Bayes’ theorem powers many apps we use daily because it helps answer questions like:**

- Given an e-mail, whatâ€™s the probability of it being spam?
- Given an ad, whatâ€™s the probability of it being clicked?
- Given the DNA, is the accused the culprit?
- And, of course, given the data, is variation better than the control in an A/B test? (FYI – we use Bayesian statistics in VWO)

**10/** Thatâ€™s it! Hope you also fall in love with the Bayesian way of looking at the world.

If you enjoyed reading my letter, do send me a note with your thoughts at paras@vwo.com. I read and reply to all emails