Inference

Inference is the act or process of drawing a conclusion based solely on what one already knows. Suppose you see rain on your window  you can infer from that, quite trivially, that the sky is grey. Looking out the window would have yielded the same fact, but through a process of perception, not inference (note however that perception itself can be viewed as an inferential process).
Inference is studied within several different fields. Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of cognitive psychology. The rules and process of inference are some of philosophy's oldest subject matters. Logic studies the laws of valid inference. Statisticians have developed formal rules for inference from quantitative data. Artificial intelligence researchers develop automated inference systems.
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Valid inferences
Inferences can be correct or incorrect. Philosophical logic has attempted to define the rules of proper inference, i.e. the rules that when applied to true premisses lead to true conclusions. Aristotle has given one of the most famous statements of those rules in his Organon. Modern mathematical logic, beginning in the 19th century, has built numerous formal systems that embody Aristotelian logic (or variants thereof).
An example: the classic syllogism
Greek philosophers defined a number of syllogisms, correct threepart inferences, that can be used as building blocks for more complex reasoning. We'll begin with the most famous of them all:
All men are mortal Socrates is a man  Therefore Socrates is mortal.
The reader can check that the premises and conclusion are true. The inference is valid  it strongly appeals to our common sense (whether that suffices to make an inference valid is an open question).
In predicate logic (a simple but useful formalisation of Aristotelician logic), this syllogism can be stated as follows:
∀ X, man(X) → mortal(X) man(Socrates)  ∴mortal(Socrates)
Or in its general form:
∀ X, A(X) → B(X) A(x)  ∴B(x)
∀, the universal quantifier, is pronounced "for all". It allows us to state a general property. Here it is used to say that "if any X is a man, X is also mortal". Socrates is a man, and the conclusion follows.
Compare to the following inference:
All dogs are mortal Socrates is mortal  Therefore Socrates is a dog
Formally :
∀ X, dog(X) → mortal(X) mortal(Socrates)  ∴dog(Socrates)
This inference is obviously incorrect, because it led us from true premises to the wrong conclusion. Note that while an inference that leads from true premisses to a wrong conclusion is necessarily incorrect, true premises and a true conclusion do not necessarily imply that the inference was correct.
Although in this particular example the error in reasoning is particularly easy to spot, it is not always so, especially if the conclusion appears true. An incorrect inference is known as a fallacy. Philosophers who study informal logic have compiled large lists of them, and cognitive psychologists have documented many biases in human reasoning that favour incorrect reasoning.
Automatic logical inference
Although now somewhat past their heyday, AI systems for automated logical inference once were extremely popular research topics, and have known industrial applications under the form of expert systems.
An inference system's job is to extend a knowledge base automatically. The knowledge base (KB) is a set of propositions that represent what the system knows about the world. Several techniques can be used by that system to extend KB by means of valid inferences. An additional requirement is that the conclusions the system arrives at are relevant to its task.
An example: inference using Prolog
Prolog (Programming in Logic) is a programming language based on a subset of predicate calculus. Its main job is to check whether a certain proposition can be inferred from the KB using an algorithm called backward chaining.
Let us return to our Socrates syllogism. We enter into our Knowledge Base the following piece of code:
mortal(X) : man(X).
man(socrates).
This states that all men are mortal and that Socrates is a man. Now we can ask Prolog about Socrates.
? mortal(socrates).
Yes
On the other hand :
? mortal(plato).
No
This is because Prolog does not know anything about Plato, and hence defaults to any property about Plato being false (the socalled closed world hypothesis). Prolog can be used for vastly more complicated inference tasks. See the corresponding article for further examples.
Inference and uncertainty
Traditional logic is only concerned with certainty  one progresses from certain premises to certain conclusions. There are several motivations for extending logic to deal with uncertain propositions and weaker modes of reasoning.
 Philosophical motivations
 A large part of our everyday reasoning does not follow the strict rules of logic, but is nevertheless effective in many cases
 Science itself is not deductive, but largely inductive, and its process cannot be captured by standard logic (see problem of induction).
 Technical motivations
 Statisticians and scientists wish to be able to infer parameters or test hypothesis on statistical data in a rigorous, quantified way.
 Artificial intelligence systems need to reason efficiently about uncertain quantities.
Common sense and uncertain reasoning
The reason most examples of applying deductive logic, such as the one above, seem artificial is because they are rarely encountered outside fields such as mathematics. Most of our everyday reasoning is of a less "pure" nature.
To take an example: suppose you live in a flat. Late at night, you are awoken by creaking sounds in the ceiling. You infer from these sounds that your neighbour upstairs is having another bout of insomnia and is pacing in his room, sleepless.
Although that reasoning seems sound, it does not fit in the logical framework described above. First, the reasoning is based on uncertain facts: what you heard were creaks, not necessarily footsteps. But even if those facts were certain, the inference is of an inductive nature: perhaps you have often heard your neigbour at night, and the best explanation you have found is that he or she is an insomniac. Hence tonight's footsteps.
It is easy to see that this line of reasoning does not necessarily lead to true conclusions: perhaps your neighbour had a very early plane to catch, which would explain the footsteps just as well. Uncertain reasoning can only find the best explanation among many alternatives.
Bayesian statistics and probability logic
Philosophers and scientists who follow the Bayesian framework for inference use the mathematical rules of probability to find this best explanation. The Bayesian view has a number of desirable features  one of them is that it embeds deductive (certain) logic as a subset (this prompts some writers to call Bayesian probability "probability logic", following E. T. Jaynes).
Bayesianists identify probabilities with degrees of beliefs, with certainly true propositions having probability 1, and certainly false propositions having probability 0. To say that "it's going to rain tomorrow" has a 0.9 probability is to say that you consider the possibility of rain tomorrow as extremely likely.
Through the rules of probability, the probability of a conclusion and of alternatives can be calculated. The best explanation is most often identified with the most probable (see Bayesian decision theory). A central rule of Bayesian inference is Bayes' theorem, which gave its name to the field.
See Bayesian inference for examples.
Frequentist statistical inference
(to be written)Fuzzy logic
(to be written)See also
 Logic
 Fuzzy logic
 Bayesian inference
 Deduction
 Induction
 Rule
 Expert system
 Inference engine
 Inference procedure
 Reasoning
 Logical assertion
 Rules of inference
 List of rules of inference
References
 Ian Hacking. An Introduction to Probability and Inductive Logic. Cambridge University Press, (2000).
 Edwin Thompson Jaynes. Probability Theory: The Logic of Science. (http://titles.cambridge.org/catalogue.asp?isbn=0521592712) Cambridge University Press, (2003). ISBN 0521592712.
 David J.C. McKay. Information Theory, Inference, and Learning Algorithms. Cambridge University Press, (2003).
 Stuart Russell, Peter Norvig. Artificial Intelligence: A Modern Approach. Prentice Hall, (2002).