Laplace's rule of thumb

Laplace's rule of succession is only a rule of thumb. The idea is that since probabilities cannot be zero or one, if one has seen a hundred white swans and no black swans, one should not assign a probability of 100/100=100%. Instead, one assigns 100/101 = 99.01%. n/n+1, not n/n.

This is more sensible than assigning 100% because it is open to the possibility of error, and makes it possible to change your mind. If your prior probability is zero or one, no amount of evidence can change your mind according to Bayes' theorem.

However, in the real world, events are rarely exclusive. There are many alternatives to a hypothesis. We should not assign equal probability to the alternatives. For example, if you had only seen five white swans, there might exist red, green, blue, orange or black swans. According to a naive version of Laplace's rule, we would expect a random swan to be white with only 50% probability, assigning 10% each to red, green, blue, orange and black.

Clearly, a more sensible prediction for the colour of a random adult swan in Europe is more like 99% white, 1% black and close to zero for red, green, blue and orange, even if you have only seen fifty swans in your lifetime.

Popper versus Bayes

Popper thought he could turn inductive science into a deductive process. We only see white swans, so we form a theory that all swans are white, but we might turn out to be wrong. The theory can never be proved. Aha, says Popper, but it can be disproved. If we see a single black swan, that disproves the theory.

The problem is that one can never be sure of evidence. Perhaps the evidence is wrong and the theory is correct. One man's modus ponens is another man's modus tollens. If I receive reports of black swans from Australia, maybe the reports are wrong, for a variety of reasons. Maybe all swans I have ever seen are in disguise. Maybe I am in the Matrix.

Another criticism of Popper's method is that his deductive claims are not very useful. If it is true that "not all swans are white", this does not help me make predictions. Such a statement says nothing about the probability that a given swan is black of white.

David Stove showed that criticisms of inductivism, that it doesn't live up to deductive standards, are misguided. They beg the question by assuming "deductivism", that only deductive arguments are valid. In fact, inductive arguments can be useful, with evidence supporting a conclusion despite not logically entailing it. Perhaps inductivism should be renamed "probabilism", or "Bayesianism".

The correct epistemology is Bayesian: knowledge is probabilistic. A correct epistemology covers everything: science is a subset of it. You don't have different epistemologies for different things. Therefore the practice of science should be probabilistic. What makes science special, then? It involves techniques of "epistemic hygiene" to increase our confidence in our beliefs. By carefully designing experiments, and using good statistics, we can increase the probability of a hypothesis by reducing the probability of alternative hypotheses.

Bayesianism makes one hyper-aware of the infinite number of hypotheses.