1.6 Probabilities, uncertainty, and logic

Probabilities can help us to quantify uncertainty. If we do not have information about a problem, it is reasonable to state that every possible event is equally likely. This is equivalent to assigning the same probability to every possible event. In the absence of information, our uncertainty is maximum, and I am not saying this colloquially; this is something we can compute using probabilities. If we know instead that some events are more likely, then this can be formally represented by assigning a higher probability to those events and less to the others. Notice that when we talk about events in stats-speak, we are not restricting ourselves to things that can happen, such as an asteroid crashing into Earth or my auntie’s 60^th birthday party. An event is just any of the possible values (or a subset of values) a variable can take, such as the event that you are older than 30, the price of a Sachertorte, or the number of bikes that will be sold next year around the world.

The concept of probability is also related to the subject of logic. Under classical logic, we can only have statements that take the values of true or false. Under the Bayesian definition of probability, certainty is just a special case: a true statement has a probability of 1, and a false statement has a probability of 0. We would assign a probability of 1 to the statement that there is Martian life only after having conclusive data indicating something is growing, reproducing, and doing other activities we associate with living organisms.

Notice, however, that assigning a probability of 0 is harder because we could always think that there is some Martian spot that is unexplored, or that we have made mistakes with some experiments, or there are several other reasons that could lead us to falsely believe life is absent on Mars even if it is not. This is related to Cromwell’s rule, which states that we should reserve the probabilities of 0 or 1 to logically true or false statements. Interestingly enough, it can be shown that if we want to extend the logic to include uncertainty, we must use probabilities and probability theory.

As we will soon see, Bayes’ theorem is just a logical consequence of the rules of probability. Thus, we can think of Bayesian statistics as an extension of logic that is useful whenever we are dealing with uncertainty. Thus, one way to justify using the Bayesian method is to recognize that uncertainty is commonplace. We generally have to deal with incomplete and or noisy data, we are intrinsically limited by our evolution-sculpted primate brain, and so on.