Follow us and stay on top of everything CRO
Related content:

Thinking Like a Bayesian

3 Min Read

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.

the theory that would not die book cover image

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

Download Free: A/B Testing Guide

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 don’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 it’s there and ~90% come out as negative when it’s not there (which means for 10% times it comes as positive even if it’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 DOESN’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. 

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

Download Free: A/B Testing Guide

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, what’s the probability of it being spam?
  • Given an ad, what’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/ That’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 🙂

Paras Chopra
Paras Chopra I started Wingify in early 2009 to enable businesses to design and deploy great customer experiences for their websites and apps. I have a background in machine learning and am a gold medalist from Delhi College of Engineering. I have been featured twice in the Forbes 30 under 30 list - India and Asia. I'm an entrepreneur by profession and my curiosity is wide-ranging. Follow me at @paraschopra on Twitter. You can email me at paras@wingify.com
Share
More from VWO on Conversion Rate Optimization
11 Pages on Your Site That Need Optimization to Get More Email Signups

11 Pages on Your Site That Need Optimization to Get More Email Signups

I really couldn’t understand this at first. After all, we live in the modern age…

Read More
Karol

Karol

14 Min Read
20 Conversion Rate Optimization Tips to Increase Your Website Conversion Rate

20 Conversion Rate Optimization Tips to Increase Your Website Conversion Rate

Conversion rate is a function of your website usability, trust in your brand, relevancy of…

Read More
Shubhi Ahluwalia

Shubhi Ahluwalia

15+ Min Read
What Is Click-Through Rate (CTR), How To Calculate It (Formula) and How Does It Compare to Conversion Rate

What Is Click-Through Rate (CTR), How To Calculate It (Formula) and How Does It Compare to Conversion Rate

Note: This is a guest article written by Malaika Nicholas, the content marketing strategist at…

Read More
Malaika Nicholas

Malaika Nicholas

9 Min Read

Scale your A/B testing and experimentation with VWO.

Start Free Trial Request Demo

DOWNLOAD A/B TESTING FREE E-BOOK