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Inverse Probability
In probability theory, inverse probability is an old term for the probability distribution of an unobserved variable. Today, the problem of determining an unobserved variable (by whatever method) is called inferential statistics. The method of inverse probability (assigning a probability distribution to an unobserved variable) is called Bayesian probability, the distribution of data given the unobserved variable is the likelihood function (which does not by itself give a probability distribution for the parameter), and the distribution of an unobserved variable, given both data and a prior distribution, is the posterior distribution. The development of the field and terminology from "inverse probability" to "Bayesian probability" is described by . The term "inverse probability" appears in an 1837 paper of De Morgan, in reference to Laplace's method of probability (developed in a 1774 paper, which independently discovered and popularized Bayesian methods, and a 1812 book), th ... [...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] |
Jerzy Neyman
Jerzy Spława-Neyman (April 16, 1894 – August 5, 1981; ) was a Polish mathematician and statistician who first introduced the modern concept of a confidence interval into statistical hypothesis testing and, with Egon Pearson, revised Ronald Fisher's null hypothesis testing. Neyman allocation, an optimal strategy for choosing sample sizes in stratified sampling, is named for him. Spława-Neyman spent the first part of his professional career at various institutions in Warsaw, Poland, and then at University College London; and the second part, at the University of California, Berkeley. Life and career He was born into a Polish people, Polish family in Bendery, in the Bessarabia Governorate of the Russian Empire, the fourth of four children of Czesław Spława-Neyman and Kazimiera Lutosławska. His family was Roman Catholic, and Neyman served as an Altar server, altar boy during his early childhood. Later, Neyman would become an agnostic. Neyman's family descended from a long line ... [...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.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] |
Bayes' Theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole. Based on Bayes' law, both the prevalence of a disease in a given population and the error rate of an infectious disease test must be taken into account to evaluate the meaning of a positive test result and avoid the ''base-rate fallacy''. One of Bayes' theorem's many applications is Bayesian inference, an approach to statistical inference, where it is used to invert the probability of Realization (probability), observations given a model configuration (i.e., th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, marine navigation, air navigation, aeronautic navigation, and space navigation. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks. All navigational techniques involve locating the navigator's Position (geometry), position compared to known locations or patterns. Navigation, in a broader sense, can refer to any skill or study that involves the determination of position and Relative direction, direction. In this sense, navigation includes orienteering and pedestrian navigation. For marine navigation, this involves the safe movement of ships, boats and other nautical craft either on or underneath the water using positions from navigation equipment with appropriate nautical char ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biology
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and distribution of life. Central to biology are five fundamental themes: the cell (biology), cell as the basic unit of life, genes and heredity as the basis of inheritance, evolution as the driver of biological diversity, energy transformation for sustaining life processes, and the maintenance of internal stability (homeostasis). Biology examines life across multiple biological organisation, levels of organization, from molecules and cells to organisms, populations, and ecosystems. Subdisciplines include molecular biology, physiology, ecology, evolutionary biology, developmental biology, and systematics, among others. Each of these fields applies a range of methods to investigate biological phenomena, including scientific method, observation, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include planets, natural satellite, moons, stars, nebulae, galaxy, galaxies, meteoroids, asteroids, and comets. Relevant phenomena include supernova explosions, gamma ray bursts, quasars, blazars, pulsars, and cosmic microwave background radiation. More generally, astronomy studies everything that originates beyond atmosphere of Earth, Earth's atmosphere. Cosmology is a branch of astronomy that studies the universe as a whole. Astronomy is one of the oldest natural sciences. The early civilizations in recorded history made methodical observations of the night sky. These include the Egyptian astronomy, Egyptians, Babylonian astronomy, Babylonians, Greek astronomy, Greeks, Indian astronomy, Indians, Chinese astronomy, Chinese, Maya civilization, M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Statistics
Bayesian statistics ( or ) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation, which views probability as the limit of the relative frequency of an event after many trials. More concretely, analysis in Bayesian methods codifies prior knowledge in the form of a prior distribution. Bayesian statistical methods use Bayes' theorem to compute and update probabilities after obtaining new data. Bayes' theorem describes the conditional probability of an event based on data as well as prior information or beliefs about the event or conditions related to the event. For example, in Bayesian inference, Bayes' theorem can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequentist Inference
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 rati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egon Pearson
Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson and Maria, née Sharpe, and, like his father, a British statistician. Career Pearson was educated at Winchester College and Trinity College, Cambridge, and succeeded his father as professor of statistics at University College London and as editor of the journal '' Biometrika''. He is best known for development of the Neyman–Pearson lemma of statistical hypothesis testing. He was elected a Fellow of the Econometric Society in 1948. Pearson was President of the Royal Statistical Society in 1955–56, and was awarded its Guy Medal in gold in 1955. He was appointed a CBE in 1946. Pearson was elected a Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequentist Probability
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability (the ''long-run probability'') as the limit of its relative frequency in infinitely many trials. Probabilities can be found (in principle) by a repeatable objective process, as in repeated sampling from the same 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 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 games arise from the natural symmetric 6-sidedness of the cube. This classical interpretation stumbled at any statistical problem that has no natural symmetry f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical description of a Randomness, random phenomenon in terms of its sample space and the Probability, probabilities of Event (probability theory), events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that fair coin, the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values. Probability distributions can be defined in different ways and for discrete or for continuous variables. Distributions with special properties or for especially important applications are given specific names. Introduction A prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
