While studying through the excellent Udacity Data Analysis Nano Degree , I found myself struggling to answer the Quiz questions on Bayes Theorem. To help myself comprehend it, I did a fair bit of studying other resources too and I came to the conclusion that it might be a. helpful to write an article myself to help reinforce this difficult subject b. but also help others.

In this article I will articulate Bayes Theorem in a simple manner, and guide with some examples.

## What Is Bayes Theorem?

Bayes’ Theorem is a widely used theory in statistics and probability, making it a very important theory in the field of data science and data analysis. For example, Bayesian inference, a particular approach to statistical inference where we can determine and adjust the probability for a hypothesis as more data or information becomes available.

## What Are its Applications?

For example, it can be used to determine the likelihood that a finance transaction is fraud related, or in determining the accuracy of a medical test, or the chances of a particular return on stocks and hundreds of other examples for every industry imaginable from Finance to Sport, Medicine to Engineering, Video Games to Music.

## What does it Do?

So as we mentioned – Bayesian inference gives us the *probability* of an event, given certain *evidence* or *tests.*

We must keep a few things in the back of our mind first

- The
*test* for fraud is separate from the result of it *being* fraud or not. - Tests are not perfect, and so give us
*false positives* (Tell us the transaction is fraud when it isn’t in reality), and * false negatives *(Where the test misses fraud that does exist. - Bayes Theorem turns the results from your tests into the actual probability of the event.
- We start with a
*prior probability* , combine with our evidence which results in out *posterior probability*

## How Does it Work? An Example

Consider the scenario of tests for cancer as an example.

Where we want to ascertain the *probability* of a patient having cancer given a particular *test result.*

- Chances a patient has this type of cancer are 1% , written as P(C) = 1% – the
**prior probability** - Test result is 90% Positive if you have C – written as P(Pos | C) – the
*sensitivity* (we can take 100%-90% = 10% as the remaining Positive percentage where there is no C but the test misdiagnoses it – the *false positives* - Test result is 90% Negative if you do not have C, written as P(Neg | ¬C) – the
*specificity* (we can take 100%-90% = 10% as the percentage of negative results but there is C but the test misses it – * the false negatives*

Lets plot this in a table so it’s a bit more readable.

| Cancer – 1% | Not Cancer – 99% |

Positive Test | 90% | 10% |

Negative Test | 10% | 90% |

- Our
**Posterior ***probability* is what we’re trying to predict – the chances of Cancer actually being present, given a Positive Test – written as P( C | Pos ) – that is, we take account of the chances of *false positives* and *false negatives* -
**Posterior** P( C | Pos ) = P ( Pos | C) x P( C ) = .9 x .001 = 0.009 - While P( ¬C | Pos) = P ( Pos | ¬C) x P(¬C) = .1 x .99 = 0.099

Lets plot this in our table.

| Cancer – 1% | Not Cancer – 99% |

Positive Test | True Pos 90% * 1% = 0.009 | False Pos 10% * 99% = 0.099 |

Negative Test | False Neg 10% * 1% = 0.001 | True Neg 90% * 99% = 0.891 |

But of course that’s not the complete story – We need to account for the number of ways it could happen given all possible outcomes

The chance of getting a real, positive result is .009. The chance of getting any type of positive result is the chance of a true positive plus the chance of a false positive (0.009 + 0.099 = 0.108).

So, our actual *posterior probability* of cancer given a positive test is .009/.108 = 0.0883, or about 8.3%.

In Bayes Theorem terms, this is written as follows, where c is the chance a patent has cancer, and x is the positive result

**P(c|x)** = Chance of having cancer (c) given a positive test (x). This is what we want to know: How likely is it to have cancer with a positive result? In our case it was 8.3%.**P(x|c)** = Chance of a positive test (x) given that you had cancer (c). This is the chance of a true positive, 90% in our case.**P(c) **= Chance of having cancer (1%).**P(¬ c)** = Chance of not having cancer (99%).**P(x|¬ c)** = Chance of a positive test (x) given that you didn’t have cancer (¬ c). This is a false positive, 9.9% in our case.

## Resources