Confident Data Skills: Master the Fundamentals of Working with Data and Supercharge Your Career (Confident Series)

by Kirill Eremenko

Now that we have a taste for Bayesian inference in practice, let’s look at the equation for the Bayes theorem. Here are the notations that will be used in this example: P(Drunk) P(Drunk | Positive) P(Positive | Drunk) P(Positive) where P stands for probability and the vertical bar signifies a conditional probability. Each of the elements above has a mathematical name. P(Drunk) is the probability that a randomly selected driver is drunk. In Bayesian statistics, this probability is called the prior probability. If we recall our initial assumptions, we can calculate the prior probability as P(Drunk) = 1/1,000 = 0.001. P(Drunk | Positive) is the conditional probability that a driver is actually drunk, given that the breathalyser test was positive (detected him or her as being drunk). This probability is called the posterior probability and this is what we are interested in calculating. P(Positive | Drunk) is the conditional probability that the breathalyser will test positive, given that a driver is actually drunk. The name for this probability is likelihood. In our case, any truly drunk driver is detected by the device without failure, and therefore P(Positive | Drunk) = 1. P(Positive) is the probability that any randomly selected driver will test positive on the breathalyser. This probability is called the marginal likelihood and in our example is calculated as P(Positive) = 50.95 / 1,000 = 0.05095.