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Deduction is inference deriving logical conclusions from premises known or assumed to be true , with the laws of valid inference being studied in logic.

### Likelihoods & Bayesian Statistics

Induction is inference from particular premises to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce , distinguishing abduction from induction, where abduction is inference to the best explanation. Various fields study how inference is done in practice.

Human inference i. 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.

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The process by which a conclusion is inferred from multiple observations is called inductive 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.

## Improving your statistical inferences

Ref: Oxford English dictionary: "induction Logic the inference of a general law from particular instances. Ancient Greek philosophers defined a number of syllogisms , correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with a famous example:.

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 following symbological track:. To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion. A valid argument with a false premise may lead to a false conclusion, this and the following examples do not follow the Greek syllogism :. 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.

In this case we have one false premise and one true premise where a true conclusion has been inferred. Evidence: It is the early s and you are an American stationed in the Soviet Union. You read in the Moscow newspaper that a soccer team from a small city in Siberia 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 a command 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 a command 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 weather, 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 i. Why would you put a large city of your best and brightest in the middle of nowhere?

Knowns: The Soviet Union is a command economy : people and material are told where to go and what to do.

## Skills and Strategies | Making Inferences

The small city was remote and historically had never distinguished itself; its soccer season was typically short because of the weather. Explanation: In a command 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 weather, 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 i.

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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 a fallacy. Philosophers who study informal logic have compiled large lists of them, and cognitive psychologists have documented many biases 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 of expert systems and later business rule engines.

More recent work on automated theorem proving has had a stronger basis in formal logic. 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. Prolog for "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 a KB knowledge base using an algorithm called backward chaining. Let us return to our Socrates syllogism. We enter into our Knowledge Base the following piece of code:. Here :- can be read as "if".

## A Practical Comparison of Qualitative Inferences with Preferred Ranking Models - Semantic Scholar

This states that all men are mortal and that Socrates is a man. Now we can ask the Prolog system about Socrates:. This is because Prolog does not know anything about Plato , and hence defaults to any property about Plato being false the so-called closed world assumption. See the corresponding article for further examples. Recently automatic reasoners found in semantic web a new field of application.

Being based upon description logic , knowledge expressed using one variant of OWL can be logically processed, i. 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.

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. 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.

A relation of inference is monotonic if the addition of premises does not undermine previously reached conclusions; otherwise the relation is non-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.

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We know when it is worth or even necessary e. Citations Publications citing this paper.

A boundary-optimized rejection region test for the two-sample binomial problem. References Publications referenced by this paper. Proving non-inferiority or equivalence of two treatments with dichotomous endpoints using exact methods. Ivan S. Test-based exact confidence intervals for the difference of two binomial proportions.

Chan , Zhengjun Zhang. Combining one-sample confidence procedures for inference in the two-sample case.