Bayes' rule, named after the English mathematician Thomas Bayes, is a rule for computing conditional probabilities.
A formal statement of Bayes' rule follows.
Proposition Let and be two events. Denote their probabilities by and and suppose that both and . Denote by the conditional probability of given and by the conditional probability of given . Bayes' rule states that
By the conditional probability formula, we have thatandThe second formula can be re-arranged as follows:which is then plugged into the first formula, so as to obtain
The following example shows how Bayes' rule can be applied in a practical situation.
We built a robot that can detect defective items produced in our factory:
if an item is defective, it is spotted with 98% probability by the robot;
when an item is not defective, the robot will not signal any defect with 99% probability.
We draw an item at random from a production lot in which 0,1% of items are defective.
If the robot tells us that the drawn item is defective, what is the probability that the robot is right?
In probabilistic terms, what we know about this problem can be formalized as follows:
Furthermore, the unconditional probability that the robot signals a defective item can be derived using the law of total probability:
Therefore, Bayes' rule gives
Even if the robot is conditionally very accurate, the unconditional probability that the robot is right when he says that an item is defective is less than 10 per cent!
The quantities involved in Bayes' ruleoften take the following names:
is called prior probability or, simply, prior;
is called conditional probability or likelihood;
is called marginal probability;
is called posterior probability or, simply, posterior.
Below you can find some problems with explained solutions.
There are two urns containing colored balls. The first urn contains 50 red balls and 50 blue balls. The second urn contains 30 red balls and 70 blue balls. One of the two urns is randomly chosen (both urns have probability 50% of being chosen) and then a ball is drawn at random from one of the two urns. If a red ball is drawn, what is the probability that it comes from the first urn?
In probabilistic terms, what we know about this problem can be formalized as follows:The unconditional probability of drawing a red ball can be derived using the law of total probability:By using Bayes' rule, we obtain
An economics consulting firm has created a model to predict recessions. The model predicts a recession with probability 80% when a recession is indeed coming and with probability 10% when no recession is coming. The unconditional probability of falling into a recession is 20%. If the model predicts a recession, what is the probability that a recession will indeed come?
What we know about this problem can be formalized as follows:The unconditional probability of predicting a recession can be derived by using the law of total probability:Bayes' rule implies
Alice has two coins in her pocket, a fair coin (head on one side and tail on the other side) and a two-headed coin. She picks one at random from her pocket, tosses it and obtains head. What is the probability that she flipped the fair coin?
What we know about this problem can be formalized as follows:The unconditional probability of obtaining head can be derived by using the law of total probability:With Bayes' rule, we obtain
If you want to know more about Bayes' rule and how it is used, you can read the following pages:
prior probability, where you can find another example and an introduction to the concept of prior distribution;
posterior probability, which also contains an example;
Bayesian inference, which introduces an important branch of statistics entirely based on Bayes' rule.
Please cite as:
Taboga, Marco (2021). "Bayes' rule", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/Bayes-rule.
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