2.5 Logic of argumentation
The Frequentist approach is based on deductive logic, this means that it starts from a statement about the true state of nature (null hypothesis), and predicts what should be seen if this statement were true. On the other hand, the Bayesian approach is based on inductive logic, this means that it defines what hypothesis is more consistent with what is seen. The former inferential approach establishes that the true of the premises implies the true of the conclusion, that is why we reject or not reject hypothesis. The latter establishes that the premises supply some evidence, but not full assurance, of the true of the conclusion, that is why we get probabilistic statements.
Here, there is a difference between effects of causes (forward causal inference) and causes of effects (reverse causal inference) ((Andrew Gelman and Imbens 2013), (Dawid, Musio, and Fienberg 2016)). To illustrate this point, imagine that a firm increases the price of a specific good, then economic theory would say that its demand decreases. The premise (null hypothesis) is a price increase, and the consequence is a demand reduction. Another view would be to observe a demand reduction, and try to identify which cause is more tenable. For instance, demand reduction can be caused by any positive supply shocks or any negative demand shocks. The Frequentist logic sees the first view, and the Bayesian reasoning gives the probability associated with possible causes.