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Luce's Choice Axiom
In probability theory, Luce's choice axiom, formulated by R. Duncan Luce (1959), states that the probability of selecting one item over another from a pool of many items is not affected by the presence or absence of other items in the pool. Selection of this kind is said to have "independence of irrelevant alternatives, independence from irrelevant alternatives" (IIA). Overview Consider a set X of possible outcomes, and consider a selection rule P, such that for any a\in A \subset X with A a finite set, the selector selects a from A with probability P(a , A). Luce proposed two choice axioms. The second one is usually meant by "Luce's choice axiom", as the first one is usually called "independence of irrelevant alternatives, independence from irrelevant alternatives" (IIA). Luce's choice axiom 1 (IIA): if P(a, A) = 0, P(b, A) > 0, then for any a, b \in B \subset \textA, we still have P(a, B) = 0. Luce's choice axiom 2 ("path independence"): P(a , A) = P(a, B)\sum_P(b, A)for any a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Of Irrelevant Alternatives
The independence of irrelevant alternatives (IIA), also known as binary independence or the independence axiom, is an axiom of decision theory and various social sciences. The term is used in different connotation in several contexts. Although it always attempts to provide an account of rational individual behavior or aggregation of individual preferences, the exact formulation differs widely in both language and exact content. Perhaps the easiest way to understand the axiom is how it pertains to casting a ballot. There the axiom says that if Charlie (the irrelevant alternative) enters a race between Alice and Bob, with Alice (leader) liked better than Bob (runner-up), then the individual voter who likes Charlie less than Alice will not switch his vote from Alice to Bob. Because of this, a violation of IIA is commonly referred to as the "spoiler effect": support for Charlie "spoils" the election for Alice, while it "logically" should not have. After all, Alice ''was'' liked better t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching Law
In operant conditioning, the matching law is a quantitative relationship that holds between the relative rates of response and the relative rates of reinforcement in concurrent schedules of reinforcement. For example, if two response alternatives A and B are offered to an organism, the ratio of response rates to A and B equals the ratio of reinforcements yielded by each response.Poling, A., Edwards, T. L., Weeden, M., & Foster, T. (2011). The matching law. ''Psychological Record'', 61(2), 313-322. This law applies fairly well when non-human subjects are exposed to concurrent variable interval schedules (but see below); its applicability in other situations is less clear, depending on the assumptions made and the details of the experimental situation. The generality of applicability of the matching law is subject of current debate. The matching law can be applied to situations involving a single response maintained by a single schedule of reinforcement if one assumes that alternativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Softmax Function
The softmax function, also known as softargmax or normalized exponential function, converts a vector of real numbers into a probability distribution of possible outcomes. It is a generalization of the logistic function to multiple dimensions, and used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes, based on Luce's choice axiom. Definition The softmax function takes as input a vector of real numbers, and normalizes it into a probability distribution consisting of probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some vector components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval (0, 1), and the components will add up to 1, so that they can be interpreted as probab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto e^ where is the probability of the system being in state , is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (see for the proportionality constant). The term ''system'' here has a very wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore the Boltzmann distribution can be used to solve a very wide variety of problems. The distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interactions of Agent (economics), economic agents and how economy, economies work. Microeconomics analyzes what's viewed as basic elements in the economy, including individual agents and market (economics), markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyzes the economy as a system where production, consumption, saving, and investment interact, and factors affecting it: employment of the resources of labour, capital, and land, currency inflation, economic growth, and public policies that have impact on glossary of economics, these elements. Other broad distinctions within economics include those between positive economics, desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consumer
A consumer is a person or a group who intends to order, or uses purchased goods, products, or services primarily for personal, social, family, household and similar needs, who is not directly related to entrepreneurial or business activities. The term most commonly refers to a person who purchases goods and services for personal use. Consumer rights “Consumers, by definition, include us all," said President John F. Kennedy, offering his definition to the United States Congress on March 15, 1962. This speech became the basis for the creation of World Consumer Rights Day, now celebrated on March 15. In his speech : John Fitzgerald Kennedy outlined the integral responsibility to consumers from their respective governments to help exercise consumers' rights, including: *The right to safety: To be protected against the marketing of goods that are hazardous to health or life. *The right to be informed: To be protected against fraudulent, deceitful, or grossly misleading informatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Behaviorism
Behaviorism is a systematic approach to understanding the behavior of humans and animals. It assumes that behavior is either a reflex evoked by the pairing of certain antecedent (behavioral psychology), antecedent stimuli in the environment, or a consequence of that individual's history, including especially reinforcement (psychology), reinforcement and punishment (psychology), punishment three-term contingency, contingencies, together with the individual's current motivating operation, motivational state and Stimulus control, controlling stimuli. Although behaviorists generally accept the important role of heredity in determining behavior, they focus primarily on environmental events. Behaviorism emerged in the early 1900s as a reaction to depth psychology and other traditional forms of psychology, which often had difficulty making predictions that could be tested experimentally, but derived from earlier research in the late nineteenth century, such as when Edward Thorndike p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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