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Exchangeability
In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence ''X''1, ''X''2, ''X''3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. In other words, the joint distribution is invariant to finite permutation. Thus, for example the sequences : X_1, X_2, X_3, X_4, X_5, X_6 \quad \text \quad X_3, X_6, X_1, X_5, X_2, X_4 both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models. Exchangeable sequences of random variables arise in cases of simple random sampling. Definition Formally, an exchangeable sequence of random variables is a finite or infinite sequence ''X''1, ''X''2, ''X''3, ... of random variables such that for any finite permutation σ of the indices 1, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Finetti's Theorem
In probability theory, de Finetti's theorem states that exchangeable random variables, exchangeable observations are conditionally independent relative to some latent variable. An epistemic probability probability distribution, distribution could then be assigned to this variable. It is named in honor of Bruno de Finetti, and one of its uses is in providing a pragmatic approach to de Finetti's well-known dictum "Probability does not exist". For the special case of an exchangeable sequence of Bernoulli distribution, Bernoulli random variables it states that such a sequence is a "mixture distribution, mixture" of sequences of independent and identically distributed (i.i.d.) Bernoulli random variables. A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of a finite set of indices. In general, while the variables of the exchangeable sequence are not ''themselves'' independent, only exchangeable, there is an '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruno De Finetti
Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 , which discussed probability founded on the coherence of betting odds and the consequences of exchangeability. Life De Finetti was born in Innsbruck, Austria, and studied mathematics at Politecnico di Milano. He graduated in 1927, writing his thesis under the supervision of Giulio Vivanti. After graduation, he worked as an actuary and a statistician at ( National Institute of Statistics) in Rome and, from 1931, the Trieste insurance company Assicurazioni Generali. In 1936 he won a competition for Chair of Financial Mathematics and Statistics, but was not nominated due to a fascist law barring access to unmarried candidates; he was appointed as ordinary professor at the University of Trieste only in 1950. He published extensively (17 papers in 193 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predictive 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] |
Statistical Control
Statistical process control (SPC) or statistical quality control (SQC) is the application of statistical methods to monitor and control the quality of a production process. This helps to ensure that the process operates efficiently, producing more specification-conforming products with less waste scrap. SPC can be applied to any process where the "conforming product" (product meeting specifications) output can be measured. Key tools used in SPC include run charts, control charts, a focus on continuous improvement, and the design of experiments. An example of a process where SPC is applied is manufacturing lines. SPC must be practiced in two phases: the first phase is the initial establishment of the process, and the second phase is the regular production use of the process. In the second phase, a decision of the period to be examined must be made, depending upon the change in 5M&E conditions (Man, Machine, Material, Method, Movement, Environment) and wear rate of parts used in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Large Numbers
In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law of large numbers states that given a sample of independent and identically distributed values, the sample mean converges to the true mean. The law of large numbers is important because it guarantees stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a ''large number'' of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a stre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Random Sampling
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample (statistics), sample) chosen from a larger Set (mathematics), set (a statistical population, population) in which a subset of individuals are chosen randomization, randomly, all with the same probability. It is a process of selecting a sample in a random way. In SRS, each subset of ''k'' individuals has the same probability of being chosen for the sample as any other subset of ''k'' individuals. Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods. Introduction The principle of simple random sampling is that every set with the same number of items has the same probability of being chosen. For example, suppose ''N'' college students want to get a ticket for a basketball game, but there are only ''X'' < ''N'' tickets for them, so they decide to have a fair way to see who gets to go. Then, everybody is given a number in the range from 0 to ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Ernest Johnson
William Ernest Johnson, Fellow of the British Academy, FBA (23 June 1858 – 14 January 1931), usually cited as W. E. Johnson, was a British philosopher, logician and economic theorist.Zabell, S.L. (2008"Johnson, William Ernest (1858–1931)"In: Durlauf S.N., Blume L.E. (eds) ''The New Palgrave Dictionary of Economics.''(2nd ed, 2008.) Palgrave Macmillan, Londoalso online He is mainly remembered for his 3 volume ''Logic'' which introduced the concept of Exchangeable random variables, exchangeability. Life and career Johnson was born in Cambridge on 23 June 1858 to William Henry Farthing Johnson and his wife, Harriet (''née'' Brimley). He was their fifth child. The family were Baptists and political liberals. He attended the Llandaff House School, Cambridge where his father was the proprietor and headteacher, then the The Perse School, Perse School, Cambridge, and the Liverpool Royal Institution School. At the age of around eight he became seriously ill and developed severe as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequentist Statistics
Frequentist inference is a type of statistical inference based in frequentist probability, which treats “probability” in equivalent terms to “frequency” and draws conclusions from sample-data by means of emphasizing the frequency or proportion of findings in the data. Frequentist inference underlies frequentist statistics, in which the well-established methodologies of statistical hypothesis testing and confidence intervals are founded. History of frequentist statistics Frequentism is based on the presumption that statistics represent probabilistic frequencies. This view was primarily developed by Ronald Fisher and the team of Jerzy Neyman and Egon Pearson. Ronald Fisher contributed to frequentist statistics by developing the frequentist concept of "significance testing", which is the study of the significance of a measure of a statistic when compared to the hypothesis. Neyman-Pearson extended Fisher's ideas to apply to multiple hypotheses. They posed that the ratio ... [...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] |
Banach Limit
In mathematical analysis, a Banach limit is a continuous linear functional \phi: \ell^\infty \to \mathbb defined on the Banach space \ell^\infty of all bounded complex-valued sequences such that for all sequences x = (x_n), y = (y_n) in \ell^\infty, and complex numbers \alpha: # \phi(\alpha x+y) = \alpha\phi(x) + \phi(y) (linearity); # if x_n\geq 0 for all n \in \mathbb, then \phi(x) \geq 0 (positivity); # \phi(x) = \phi(Sx), where S is the shift operator defined by (Sx)_n=x_ (shift-invariance); # if x is a convergent sequence, then \phi(x) = \lim x . Hence, \phi is an extension of the continuous functional \lim: c \to \mathbb C where c \subset\ell^\infty is the complex vector space of all sequences which converge to a (usual) limit in \mathbb C. In other words, a Banach limit extends the usual limits, is linear, shift-invariant and positive. However, there exist sequences for which the values of two Banach limits do not agree. We say that the Banach limit is not uniquely d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
