What is a rule of inductive inference? I'm not looking for any examples, but for definitions - what makes the logical form of an inductive argument a rule of inductive inference?
Inductive reasoning, in the broad sense of the term, is usually understood to differ from deductive reasoning in that the premises of a cogent argument support the conclusion without entailing it. In practice, it is difficult to spell out just what this involves.
Some accounts see an inductively strong argument as one of partial entailment. The problem here is to explain what a degree of entailment amounts to. Many take a probabilistic turn and claim that the truth of the premises of an inductively strong argument renders the conclusion probable, or at least more probable than it would be without them. This makes sense in the context of an epistemic understanding of probability as a degree of rational belief.
But we are still left with the problem of whether there are any general formal rules that would count as rules of inductive inference. Rudolf Carnap attempted to set out a general formal theory of inductive logic, but it has been widely criticised. Many modern approaches treat inductive reasoning as something that can only proceed piecemeal for particular subject areas by reference to background facts and assumptions. Others appeal to some general heuristics but do not attempt to codify these as formal rules.
Bayesian conditioning is often used in practice as a way to express the probability of a hypothesis given some data. But it also has its issues. Strictly speaking, Bayesianism does not tell us the probability of a proposition given the evidence, but only how to update a prior probability to a posterior one. It is a separate question as to where our priors come from. Also, often we do not know the probability of the evidence term, so the Bayesian updating formula is frequently used in its ratio form and tells us only whether some given evidence tends to support one hypothesis relative to another rival one. This means it is not telling us in any absolute sense how probable a hypothesis is given the evidence.
There are some algorithms for generating probabilities and rules from data: everything from Laplace's Rule of Succession to decision tree algorithms such as C4.5 to approximations to Solomonoff induction. But these are often quite specific in terms of their range of applicability and are sometimes not much more than ways to adjust parameters in a model. Beyond that, there are artificial intelligence applications that use machine learning techniques to generate hypotheses and test them against data. These can be described as inductive, but they do not simplify down into straightforward rules of inductive inference.