Book Cover

# Review

I read this book mainly to review some concepts I learned in my Statistics and Probability courses. The book was a quick read and it was very colloquial and accessible. The author explained the different parts of the Bayes' theorem with geeky and quirky examples.

# Bayes' Theorem

Bayes' theorem works as follows:

You have a prior hypothesis H and some new evidence E, you want to figure out what the probability of hypothesis H given the new evidence E. The posterior, or probability of hypothesis H occuring given evidence E is written as . The entire theorem is written as follows:

• is the probability of the evidence E occuring given that the hypothesis H is true

• is the probability that hypothesis H is true (the initial degree of belief in H, before we witnessed E), or the prior

• is the probability that the evidence E is true

Baye's theorem is often written in the following form: , where is known as the likelihood ratio.

Let's illustrate this with an example; consider a standard deck of 54 playing cards (including Jokers). Say you want to figure out what the probability is that a card is a King, given that it's a face card (either a Jack, a Queen, or a King). Let's let be the probability that a card is a King, and be the probability that a card is a face card. We are thus looking for . Let's remember our formula:

• is the probability that a card will be a face card if it's a King; this is equal to 1 since all Kings are by definition face cards

• is the probability that a card in the deck is a King; this is equal to since there exists one King for each of the 4 suits and there are 54 cards total

• is the probability that a card in the deck is a face card; this is equal to since there exists three face cards for each suit and there are 13 cards total in each suit

We thus get:

There is thus approximately a 32.1% chance that a card is a King given that it is a face card. This makes intuitive sense since a King makes up of a third of face cards, and even with the two Jokers, the probability of should still be close to .

3Blue1Brown made an amazing visual explanation of Bayes' theorem.

# Conclusion

Bayes' theorem is one of the most powerful theorems in all of statistics and probability and is widely used in techniques such as machine learning.

Unlike the the rigorous proofs I had to do when I first learned the theorem in class, this book had a very intuitive explanation of the theory behind the equation. This book also had some basic exercises in R. It was very enjoyable to read; I believe all math should be taught like this: intuitively in common English. I would highly recommend this book to anyone who is learning about statistics and probability for the first time, or anyone who wants a quick referesher on Bayes'. For experienced statisticians, however, this book is too superficial to add any value.

# Overall rating: 7.4

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