|
Relative Frequency
In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, i.e. by means not of a theoretical sample space but of an actual experiment. More generally, empirical probability estimates probabilities from experience and observation. Given an event in a sample space, the relative frequency of is the ratio being the number of outcomes in which the event occurs, and being the total number of outcomes of the experiment. In statistical terms, the empirical probability is an ''estimator'' or ''estimate'' of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modelling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likelihood estimate. It is the Bayesian estimate for the same case if certain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory or probability calculus 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 of probability, axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure (mathematics), 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 (probability theory), event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of determinism, non-deterministic or uncertain processes or measured Quantity, quantities that may either be single occurrences or evolve over time in a random fashion). Although it is no ... [...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, 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] |
Applied Probability
Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. Scope Much research involving probability is done under the auspices of applied probability. However, while such research is motivated (to some degree) by applied problems, it is usually the mathematical aspects of the problems that are of most interest to researchers (as is typical of applied mathematics in general). Applied probabilists are particularly concerned with the application of stochastic processes, and probability more generally, to the natural, applied and social sciences, including biology, physics (including astronomy), chemistry, medicine, computer science and information technology, and economics. Another area of interest is in engineering: particularly in areas of uncertainty, risk management, probabilistic design, and Quality assurance. History Having initially been defined at a symposium of the American Mathematical Society in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Probability
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability (the ''long-run probability'') as the limit of a sequence, limit of its Empirical probability, relative frequency in infinitely many Experiment (probability theory), trials. Probabilities can be found (in principle) by a repeatable objective process, as in repeated sampling (statistics), sampling from the same population (statistics), population, and are thus ideally devoid of subjectivity. The continued use of frequentist methods in scientific inference, however, has been called into question. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the Classical definition of probability, classical interpretation. In the classical interpretation, probability was defined in terms of the principle of indifference, based on the natural symmetry of a problem, so, for example, the probabilities of dice game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as '' quartiles'' (four groups), '' deciles'' (ten groups), and '' percentiles'' (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points. -quantiles are values that partition a finite set of values into subsets of (nearly) equal sizes. There are partitions of the -quantiles, one for each integer satisfying . In some cases the value of a quantile may not be uniquely determined, as can be the case for the median (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to continuous di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Measure
In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empirical measures are relevant to mathematical statistics. The motivation for studying empirical measures is that it is often impossible to know the true underlying probability measure P. We collect observations X_1, X_2, \dots , X_n and compute relative frequencies. We can estimate P, or a related distribution function F by means of the empirical measure or empirical distribution function, respectively. These are uniformly good estimates under certain conditions. Theorems in the area of empirical processes provide rates of this convergence. Definition Let X_1, X_2, \dots be a sequence of independent identically distributed random variables with values in the state space ''S'' with probability distribution ''P''. Definition :The ''empirical measure'' ''P''''n'' is defined for m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Distribution Function
In statistics, an empirical distribution function ( an empirical cumulative distribution function, eCDF) is the Cumulative distribution function, distribution function associated with the empirical measure of a Sampling (statistics), sample. This cumulative distribution function is a step function that jumps up by at each of the data points. Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value. The empirical distribution function is an Estimator, estimate of the cumulative distribution function that generated the points in the sample. It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem. A number of results exist to quantify the rate of Convergence of random variables#Convergence in distribution, convergence of the empirical distribution function to the underlying cumulative distribution function. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Reasoning
Deductive reasoning is the process of drawing valid inferences. An inference is valid if its conclusion follows logically from its premises, meaning that it is impossible for the premises to be true and the conclusion to be false. For example, the inference from the premises "all men are mortal" and " Socrates is a man" to the conclusion "Socrates is mortal" is deductively valid. An argument is ''sound'' if it is valid ''and'' all its premises are true. One approach defines deduction in terms of the intentions of the author: they have to intend for the premises to offer deductive support to the conclusion. With the help of this modification, it is possible to distinguish valid from invalid deductive reasoning: it is invalid if the author's belief about the deductive support is false, but even invalid deductive reasoning is a form of deductive reasoning. Deductive logic studies under what conditions an argument is valid. According to the semantic approach, an argument is valid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Priori 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 constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posterior Probability
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
A Posteriori Probability
In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, i.e. by means not of a theoretical sample space but of an actual experiment. More generally, empirical probability estimates probabilities from experience and observation. Given an event in a sample space, the relative frequency of is the ratio being the number of outcomes in which the event occurs, and being the total number of outcomes of the experiment. In statistical terms, the empirical probability is an ''estimator'' or ''estimate'' of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modelling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likelihood estimate. It is the Bayesian estimate for the same case if certain a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
