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Bayesian Inference
Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian inference uses a prior distribution to estimate posterior probabilities. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability". Introduction to Bayes' rule Formal explanation Bayesian inference derives the posterior probability as a consequence of two antecedents: a prior probability and a "likelihood function" derive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Statistical Inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population. Inferential statistics can be contrasted with descriptive statistics. Descriptive statistics is solely concerned with properties of the observed data, and it does not rest on the assumption that the data come from a larger population. In machine learning, the term ''inference'' is sometimes used instead to mean "make a prediction, by evaluating an already trained model"; in this context inferring properties of the model is referred to as ''training'' or ''learning'' (rather than ''inference''), and using a model for prediction is referred to as ''inference'' (instead of ''prediction''); se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bas Van Fraassen
Bastiaan Cornelis "Bas" van Fraassen (; ; born 5 April 1941) is a Dutch-American philosopher noted for his contributions to philosophy of science, epistemology and formal logic. He is a Distinguished Professor of Philosophy at San Francisco State University and the McCosh Professor of Philosophy Emeritus at Princeton University. Biography and career Van Fraassen was born in the German-occupied Netherlands on 5 April 1941. His father, a steam fitter, was forced by the Nazis to work in a factory in Hamburg. After the war, the family reunited and, in 1956, emigrated to Edmonton, in western Canada. Van Fraassen earned his B.A. (1963) from the University of Alberta and his M.A. (1964) and Ph.D. (1966, under the direction of Adolf Grünbaum) from the University of Pittsburgh. He previously taught at Yale University, the University of Southern California, the University of Toronto and, from 1982 to 2008, at Princeton University, where he is now emeritus. Since 2008, Van Fraasse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Conditional Probability
In probability theory, conditional probability is a measure of the probability of an Event (probability theory), event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This particular method relies on event A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as or occasionally . This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac. For example, the probability that any given person has a cough on any given day ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logical Negation
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. For example, if P is "Spot runs", then "not P" is "Spot does not run". An operand of a negation is called a ''negand'' or ''negatum''. Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the refutations of P. Definition ''Classical negation'' is an operation on one logical value, typically th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Marginal Likelihood
A marginal likelihood is a likelihood function that has been integrated over the parameter space. In Bayesian statistics, it represents the probability of generating the observed sample for all possible values of the parameters; it can be understood as the probability of the model itself and is therefore often referred to as model evidence or simply evidence. Due to the integration over the parameter space, the marginal likelihood does not directly depend upon the parameters. If the focus is not on model comparison, the marginal likelihood is simply the normalizing constant that ensures that the posterior is a proper probability. It is related to the partition function in statistical mechanics. Concept Given a set of independent identically distributed data points \mathbf=(x_1,\ldots,x_n), where x_i \sim p(x, \theta) according to some probability distribution parameterized by \theta, where \theta itself is a random variable described by a distribution, i.e. \theta \sim p(\t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Likelihood Function
A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the joint probability distribution of the random variable that (presumably) generated the observations. When evaluated on the actual data points, it becomes a function solely of the model parameters. In maximum likelihood estimation, the argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often approximated by the likelihood's Hessian matrix at the maximum) gives an indication of the estimate's precision. In contrast, in Bayesian statistics, the estimate of interest is the ''converse'' of the likelihood, the so-called posterior probability of the parameter given the observed data, which is calculated via Bayes' rule. Definition The likelihood function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Experimental Data
Experimental data in science and engineering is data produced by a measurement, test method, experimental design or quasi-experimental design. In clinical research any data produced are the result of a clinical trial. Experimental data may be qualitative or quantitative, each being appropriate for different investigations. Generally speaking, qualitative data are considered more descriptive and can be subjective in comparison to having a continuous measurement scale that produces numbers. Whereas quantitative data are gathered in a manner that is normally experimentally repeatable, qualitative information is usually more closely related to phenomenal meaning and is, therefore, subject to interpretation by individual observers. Experimental data can be reproduced by a variety of different investigators and mathematical analysis may be performed on these data. See also * Accuracy and precision * Computer science * Data analysis * Empiricism * Epistemology * Informatic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Statistical Model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model represents, often in considerably idealized form, the Data generating process, data-generating process. When referring specifically to probability, probabilities, the corresponding term is probabilistic model. All Statistical hypothesis testing, statistical hypothesis tests and all Estimator, statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman J. Adèr, Herman Adèr quoting Kenneth A. Bollen, Kenneth Bollen). Introduction Informally, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Likelihood Function
A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the joint probability distribution of the random variable that (presumably) generated the observations. When evaluated on the actual data points, it becomes a function solely of the model parameters. In maximum likelihood estimation, the argument that maximizes the likelihood function serves as a point estimate for the unknown parameter, while the Fisher information (often approximated by the likelihood's Hessian matrix at the maximum) gives an indication of the estimate's precision. In contrast, in Bayesian statistics, the estimate of interest is the ''converse'' of the likelihood, the so-called posterior probability of the parameter given the observed data, which is calculated via Bayes' rule. Definition The likelihood function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Prior Probability
A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable. In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern. There are many ways to const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Antecedent (logic)
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the ''protasis''. Examples: * If P, then Q. This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q. In the implication "\phi implies \psi", \phi is called the antecedent and \psi is called the consequent.Sets, Functions and Logic - An Introduction to Abstract Mathematics, Keith Devlin, Chapman & Hall/CRC Mathematics, 3rd ed., 2004 Antecedent and consequent are connected via logical connective to form a proposition. * If X is a man, then X is mortal. "X is a man" is the antecedent for this proposition while "X is mortal" is the consequent of the proposition. * If men have walked on the Moon, then I am the king of France. Here, "men have walked on the Moon" is the antecedent and "I am the king of France" is the consequent. Let y=x+1. * If x=1 then y=2,. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Consequence Relation
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises?Beall, JC and Restall, Greg, Logical Consequence' The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.). All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |