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Bayesian Statistical Analysis
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. 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" derived from a statistical model for the observed data. Bayesian inference computes the posterior probability according to Bayes' theorem: P(H\mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of a Statistical population, population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is Sampling (statistics), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contingency Table
In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables. They are heavily used in survey research, business intelligence, engineering, and scientific research. They provide a basic picture of the interrelation between two variables and can help find interactions between them. The term ''contingency table'' was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation", part of the ''Drapers' Company Research Memoirs Biometric Series I'' published in 1904. A crucial problem of multivariate statistics is finding the (direct-)dependence structure underlying the variables contained in high-dimensional contingency tables. If some of the conditional independences are revealed, then even the storage of the data can be done in a smarter way (see Lauritzen (2002)). In order to do this one can use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard C
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick", "Dickon", " Dickie", "Rich", "Rick", "Rico", "Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (other) * Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Kinematics
Radical probabilism is a hypothesis in philosophy, in particular epistemology, and probability theory that holds that no facts are known for certain. That view holds profound implications for statistical inference. The philosophy is particularly associated with Richard Jeffrey who wittily characterised it with the ''dictum'' "It's probabilities all the way down." Background Bayes' theorem states a rule for updating a probability conditioned on other information. In 1967, Ian Hacking argued that in a static form, Bayes' theorem only connects probabilities that are held simultaneously; it does not tell the learner how to update probabilities when new evidence becomes available over time, contrary to what contemporary Bayesians suggested. According to Hacking, adopting Bayes' theorem is a temptation. Suppose that a learner forms probabilities ''P''old(''A'' & ''B'') = ''p'' and ''P''old(''B'') = ''q''. If the learner subsequently learns that ''B'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dutch Book
In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which the gamblers bet. It is associated with probabilities implied by the odds not being coherent. In economics, the term usually refers to a sequence of trades that would leave one party strictly worse off and another strictly better off. Typical assumptions in consumer choice theory rule out the possibility that anyone can be Dutch-booked. In philosophy it is used to explore degrees of certainty of beliefs. There is no agreement on the etymology of the term.Hajek, Alan ''Dutch Book Arguments'', Chapter 7 in The Oxford Handbook of Rational and Social Choice, ed. Paul Anand, Prasanta Pattanaik, and Clemens Puppe, 173-195, Oxford University Press Gambling The main point of the Dutch book argument is to show that rational people must have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, 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 the value of a proposition, that produces a value of ''true'' when its operand is false, and a value of ''false'' when its operand is true. Thus if statement is true, then \neg P (pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 from a prior and is therefore often referred to as model evidence or simply evidence. 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(\theta\mid\alpha), the marginal likelihood in general asks what the probability p(\mathbf\mid\alpha) is, where \theta has been marginalized out (integrated out): :p(\mathbf\mid\alpha) = \int_\theta p(\mathbf\mid\theta) \, p(\theta\mid\alpha)\ \operatorname\!\theta The above definition is phrased in the context of Bayesian statistics in which case p(\theta\mid\alpha) is called prior density and p(\mathbf\mid\theta) is the likelihood. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood Function
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood function indicates which parameter values are more ''likely'' than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as \mathcal(\theta\mid X) instead of P(X \mid \theta), to emphasize that it is to be understood as a function of the parameters \theta instead of the random variable X. In maximum likelihood estimation, the arg max of the likelihood function serves as a point estimate for \theta, while local curvature (approximated by the likelihood's Hessian matrix) indicates the estimate's precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 * Informatics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. 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). 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. Introduction Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood Function
The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood function indicates which parameter values are more ''likely'' than others, in the sense that they would have made the observed data more probable. Consequently, the likelihood is often written as \mathcal(\theta\mid X) instead of P(X \mid \theta), to emphasize that it is to be understood as a function of the parameters \theta instead of the random variable X. In maximum likelihood estimation, the arg max of the likelihood function serves as a point estimate for \theta, while local curvature (approximated by the likelihood's Hessian matrix) indicates the estimate's precision. Meanwhile in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior Probability
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity 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. Bayes' theorem calculates the renormalized pointwise product of the prior and the likelihood function, to produce the ''posterior probability distribution'', which is the conditional distribution of the uncertain quantity given the data. Similarly, the prior probability of a random event or an uncertain proposition is the unconditional probability that is assigned before any relevant evidence is taken into account. Priors can be created using a num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
