Bayes' Theorem describes the probability of an event based on the probability of **all** other conditions leading up to that event.

This can often be counter intuitive. Say, for example, you take a test for a disease that exists in 10% of the population, and you test positive. If the test has a **Sensitivity** of 90% (true positive rate) and a **Specificity** of 90% (true negative rate), your intuition may tell you that you have a 90% chance of having the disease.

Whilst this feels correct, we have neglected to take into account the **base probability**; the probability of having the disease in the first place.

In fact, your odds of truly having the disease after testing positive are actually **9%** - far from the expected 90%.

This is worked out by taking the rather slim chance of you having the disease and multiplying it by the chance of the test picking it up (10% x 90%).

# The Test

To get a sense of this we can watch how the modelled test below plays out:

### Key

- Not Infected
- Infected
- False Positive
- True Positive
- False Negative
- True Negative

### Sample Test

**Sample Set**

Positive Candidates:

Negative Candidates:

**Tested Positive**

Percentage Positive:

True Positive Tests:

False Positive Tests:

**Tested Negative**

Percentage Negative:

True Negative Tests:

False Negative Tests:

### Venn Diagram

# Further Reading

If Bayes' Theorem still seems a little unclear, **Kalid Azad** has written a great article which I highly recommend reading: An Intuitive (and Short) Explanation of Bayesâ€™ Theorem. For a more complete explanation of the formula, as well as some history, see the Wiki Page.

If you're ready to see more however, **Andrew Collier** has developed a Shiny application illustrating an intuitive visualization of Bayesian Updates.