It is profoundly important. It is a formalization of induction which is what humans (and other living things) do all the time as they sense the state of the physical world and, in turn, reactively manipulate it.
Clayton -
Clayton,
Bayesian probability is not the formalisation of induction. The increased probability of a hypothesis (of universal form) does not transmit equally to each of its logical consequences. Those predictions beyond the evidence do not change in probability. It is a fallacy of division to claim otherwise. In any case, an equation cannot be inductive, by definition, because it asserts an equality relation. Equality is a subset of deduction.
The formalisation of induction has already been achieved, it’s just invalid by the standards of deductive logic. And nobody “does” induction per se, our thoughts may or may not correspond to deductive or inductive forms of inference, but that correspondence is not an identity.
Not to be pedantic but I said “a formalization” not “the formalization”. The human brain is the ultimate formalization of human induction. Bayes’s theorem (and Bayesian analysis generally) gives us a very high-level outline of what human brains - and the other inductive/predictive systems in the biosphere - are doing when they make inferences about the future state of the world based on its present state.
But of course they do, else you must discard Ockham’s razor which is, essentially, to discard all of rationality and science. The essence of reason is to choose the fewest axioms necessary to construct your theorems. The essence of science is to choose the “simplest” hypothesis that explains all the data. Quantifying the change in probability associated with choosing more axioms over fewer or more complex hypotheses over simpler is a very involved subject (google “minimum description length” and “Solomonoff induction”) but the basic principle involved is relatively straightforward and intuitive.
Well, yeah, Bayes’ theorem per se is not induction but it expresses the basic tradeoff between the complexity of a hypothesis and how well it fits the data, permitting the comparison (in principle) of competing hypotheses for a given dataset with one another.
Well, induction is really just a synonym for prediction. It is a stylized way of inferring the future from the present. All living things perform prediction, in some sense, that is, they perform a non-stylized version of what we humans call induction.
Clayton -
Bayesian networks (which are based on Bayesian reasoning) are a huge topic in artificial inteligence. Some AI techniques make use of Bayesian networks, including automated troubleshooting systems.
So to some people, yes. They are important.
Strictly speaking it is Solomonoff induction that is the generally accepted formalization of induction, within which Bayes theorem is instrumental. I think it may be one of the most important concepts to understand in the course of ones intellectual development.
edit: Clayton, just read your post. Excellent explanation
Thanks. I’m astounded you know about Solomonoff induction… nobody knows about that stuff. It’s more obscure than AE. By the way, computability theory and algorithmic information theory have profound implications for central planning. I’ve batted around the idea of trying to write a paper deriving a calculation argument starting from computability theory. If you’ve ever listened to any of Chaitin’s lectures or read his material, you’ll know that AIT is an “ignored” field of mathematics. I suspect that its “ignored” status has something to do with its implications regarding the planned society.
Clayton -