For the 1992 album by pianist Marilyn Crispell and saxophonist Tim Berne, seeInference (album). For the process in statistics and machine learning, seeStatistical inference.
Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, andcognitive psychology;artificial intelligence researchers develop automated inference systems to emulate human inference.Statistical inference uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case. Statistical inference uses quantitative or qualitative (categorical) data which may be subject to random variations.
The process by which a conclusion is inferred from multipleobservations is calledinductive reasoning. The conclusion may be correct or incorrect, or correct to within a certain degree of accuracy, or correct in certain situations. Conclusions inferred from multiple observations may be tested by additional observations.
This definition is disputable (due to its lack of clarity. Ref: Oxford English dictionary: "induction ... 3. Logic the inference of a general law from particular instances."[clarification needed]) The definition given thus applies only when the "conclusion" is general.
Two possible definitions of "inference" are:
A conclusion reached on the basis of evidence and reasoning.
Ancient Greek philosophers defined a number ofsyllogisms, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with a famous example:
All humans are mortal.
All Greeks are humans.
All Greeks are mortal.
The reader can check that the premises and conclusion are true, but logic is concerned with inference: does the truth of the conclusion follow from that of the premises?
The validity of an inference depends on the form of the inference. That is, the word "valid" does not refer to the truth of the premises or the conclusion, but rather to the form of the inference. An inference can be valid even if the parts are false, and can be invalid even if some parts are true. But a valid form with true premises will always have a true conclusion.
For example, consider the form of the followingsymbological track:
All meat comes from animals.
All beef is meat.
Therefore, all beef comes from animals.
If the premises are true, then the conclusion is necessarily true, too.
Now we turn to an invalid form.
All A are B.
All C are B.
Therefore, all C are A.
To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion.
All apples are fruit. (True)
All bananas are fruit. (True)
Therefore, all bananas are apples. (False)
A valid argument with a false premise may lead to a false conclusion, (this and the following examples do not follow the Greek syllogism):
All tall people are French. (False)
John Lennon was tall. (True)
Therefore, John Lennon was French. (False)
When a valid argument is used to derive a false conclusion from a false premise, the inference is valid because it follows the form of a correct inference.
A valid argument can also be used to derive a true conclusion from a false premise:
All tall people are musicians. (Valid, False)
John Lennon was tall. (Valid, True)
Therefore, John Lennon was a musician. (Valid, True)
In this case we have one false premise and one true premise where a true conclusion has been inferred.
Evidence: It is the early 1950s and you are an American stationed in theSoviet Union. You read in theMoscow newspaper that asoccer team from a small city inSiberia starts winning game after game. The team even defeats the Moscow team. Inference: The small city in Siberia is not a small city anymore. The Soviets are working on their own nuclear or high-value secret weapons program.
Knowns: The Soviet Union is acommand economy: people and material are told where to go and what to do. The small city was remote and historically had never distinguished itself; its soccer season was typically short because of the weather.
Explanation: In acommand economy, people and material are moved where they are needed. Large cities might field good teams due to the greater availability of high quality players; and teams that can practice longer (possibly due to sunnier weather and better facilities) can reasonably be expected to be better. In addition, you put your best and brightest in places where they can do the most good—such as on high-value weapons programs. It is an anomaly for a small city to field such a good team. The anomaly indirectly described a condition by which the observer inferred a new meaningful pattern—that the small city was no longer small. Why would you put a large city of your best and brightest in the middle of nowhere? To hide them, of course.
An incorrect inference is known as afallacy. Philosophers who studyinformal logic have compiled large lists of them, and cognitive psychologists have documented manybiases in human reasoning that favor incorrect reasoning.
AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form ofexpert systems and laterbusiness rule engines. More recent work onautomated theorem proving has had a strongerbasis in formal logic.
An inference system's job is to extend a knowledge base automatically. Theknowledge 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 arerelevant to its task.
Additionally, the term 'inference' has also been applied to the process of generating predictions from trainedneural networks. In this context, an 'inference engine' refers to the system or hardware performing these operations. This type of inference is widely used in applications ranging fromimage recognition tonatural language processing.
Let us return to ourSocratessyllogism. We enter into our Knowledge Base the following piece of code:
mortal(X) :- man(X).man(socrates).
( Here:- can be read as "if". Generally, ifP Q (if P then Q) then in Prolog we would codeQ:-P (Q if P).) This states that all men are mortal and that Socrates is a man. Now we can ask the Prolog system about Socrates:
?- mortal(socrates).
(where?- signifies a query: Canmortal(socrates). be deduced from the KB using the rules)gives the answer "Yes".
On the other hand, asking the Prolog system the following:
?- mortal(plato).
gives the answer "No".
This is becauseProlog does not know anything aboutPlato, and hence defaults to any property about Plato being false (the so-calledclosed world assumption). Finally?- mortal(X) (Is anything mortal) would result in "Yes" (and in some implementations: "Yes": X=socrates) Prolog can be used for vastly more complicated inference tasks. See the corresponding article for further examples.
Recently automatic reasoners found insemantic web a new field of application. Being based upondescription logic, knowledge expressed using one variant ofOWL can be logically processed, i.e., inferences can be made upon it.
Philosophers and scientists who follow theBayesian framework for inference use the mathematical rules ofprobability 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", followingE. T. Jaynes).
Bayesians 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 (seeBayesian decision theory). A central rule of Bayesian inference isBayes' theorem.
A relation of inference ismonotonic if the addition of premises does not undermine previously reached conclusions; otherwise the relation isnon-monotonic.Deductive inference is monotonic: if a conclusion is reached on the basis of a certain set of premises, then that conclusion still holds if more premises are added.
By contrast, everyday reasoning is mostly non-monotonic because it involves risk: we jump to conclusions from deductively insufficient premises.We know when it is worth or even necessary (e.g. in medical diagnosis) to take the risk. Yet we are also aware that such inference is defeasible—that new information may undermine old conclusions. Various kinds of defeasible but remarkably successful inference have traditionally captured the attention of philosophers (theories of induction, Peirce's theory ofabduction, inference to the best explanation, etc.). More recently logicians have begun to approach the phenomenon from a formal point of view. The result is a large body of theories at the interface of philosophy, logic and artificial intelligence.
Angluin, Dana (1976).An Application of the Theory of Computational Complexity to the Study of Inductive Inference (Ph.D.). University of California at Berkeley.
Gabbay, Dov M.; Hartmann, Stephan; Woods, John, eds. (2009).Inductive Logic. Handbook of the History of Logic. Vol. 10. Elsevier.ISBN978-0-444-52936-7.
Burns, B. D. (1996). "Meta-Analogical Transfer: Transfer Between Episodes of Analogical Reasoning".Journal of Experimental Psychology: Learning, Memory, and Cognition.22 (4):1032–1048.doi:10.1037/0278-7393.22.4.1032.
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