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Bayes’ Theorem


Bayes’ Theorem sits at the heart of a few well known machine learning algorithms. So a fundamental understanding of the theorem is in order.

Let’s consider the following idea (the following stats are completely made up by the way). Imagine 5% of kids are dyslexic. Now imagine the tests administered for dyslexia at a local school is known to give a false positive 10% of the time. What is the probability a kid has dyslexia given the fact they tested positive?

What we want to know is = P(Dyslexic | Positive Test).

To figure this out, we are going to use Bayes’ Theorem

Let’s start with the equation:

Don’t worry. It is not all that complicated. Let’s break it down into parts:

Let’s take a new look at the formula

So let me put this into English.



First, let’s figure out our probabilities. A tree chart is a great way to start.

Look at the chart below. It branches first between dyslexic and not dyslexic. Then each branch has positive and negative probabilities branching from there.

Now to calculate the probabilities. We do this by multiplying the branches. For example Dyslexic and Positive  0.05 * 0.9 = 0.045

Now, let’s fill in our formula. If you are having trouble seeing where the values come from look at the chart below

So the probability of being dyslexic assuming the kid had a positive test = 0.016 or 1.6%


Another – perhaps more real world use for Bayes’ Theorem is the SPAM filter. Check it out below. See if you can figure your way through it on your own.


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