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Bayes' Theorem

BayesTheorem.net: A Visual Introduction for Beginners

If you’ve recently searched Google, Bayes’ Theorem was used to display your search results. The same is true for those recommendations on Netflix. Hedge funds? Self-driving cars? Search and rescue? Bayes’ Theorem is used in all of the above and more!

At its core, Bayes’ Theorem is a simple mathematical formula that has revolutionized how we understand and deal with uncertainty. If life is seen as black and white, Bayes’ Theorem helps us think about the gray areas. When new evidence comes our way, it helps us update our beliefs and create a new belief.

Ready to dig in and visually explore the basics? Let’s go!

Welcome to BayesTheorem.net: A Visual Introduction for Beginners. This website is packed with examples and visual aids to help clarify what Bayes’ Theorem is and how it works. At its core, Bayes’ Theorem is very simple and built on elementary mathematics.

Before we dig into different definitions, it needs to be stated that Bayes’ Theorem is often called Bayes’ Rule or Bayes’ Formula. So, don’t be confused – they are the same, and we will be using both theorem and formula throughout this website. Second, we need to make it clear that Bayes’ Theorem is a law of probability theory. It helps us work with, revise, and understand probabilities when we are presented with new evidence. Practically speaking, the theorem helps us quantify or put a number on our skepticism and make more informed rational choices. It helps us answer the following:

When we encounter new evidence, how much should it change our confidence in a belief?

For example:

  • You just had a test for cancer and it came back positive. What is the probability that you have cancer if the test is positive?
  • Your friend has a new dog and when you visit she slobbers all over you, but does that mean the dog likes you? What is the probability that the dog likes you given that she licks you?
  • Your friend claims that stock prices will decrease if interest rates increase. What is the probability stock prices will decrease if interest rates increase?

Bayes’ Theorem Explained: 4 Ways

Here are a few simple ways Bayes’ Theorem can be explained.

  • Bayes’ Theorem helps us update a belief based on new evidence by creating a new belief.
  • Bayes’ Theorem helps us revise a probability when given new evidence.
  • Bayes’ Theorem helps us change our beliefs about a probability based on new evidence.
  • Bayes’ Theorem helps us update a hypothesis based on new evidence.

The only problem? Applying the theorem is not intuitive, at least not for most people. This is where visualizing a problem that entails using Bayes’ Theorem can be a BIG HELP.

Visual Aids
When working with small amounts of data there are a few different visual aids you can use:

  • Venn diagrams
  • Decision trees
  • Letters (e.g., H,T,T,T for head/tail coin flips)
  • Physical Objects (e.g., real coins)

On this website, we’ll be using venn diagrams and decision trees.

Venn diagrams are an excellent way to help us visually understand and solve abstract problems.
Bayes-Theorem-Picture1

Decision trees are a great tool that can help us solve problems where probabilities are not provided and must be discovered.
Decision Trees Bayes' Theorem

Continue on to Chapter 1: Bayes’ Theorem for Dummies.