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Principle Of Insufficient Reason
The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or "degrees of belief") equally among all the possible outcomes under consideration. In Bayesian probability, this is the simplest non-informative prior. Examples The textbook examples for the application of the principle of indifference are coins, dice, and cards. In a macroscopic system, at least, it must be assumed that the physical laws that govern the system are not known well enough to predict the outcome. As observed some centuries ago by John Arbuthnot (in the preface of ''Of the Laws of Chance'', 1692), :It is impossible for a Die, with such determin'd force and direction, not to fall on such determin'd side, only I don't know the force and direction which makes it fall on such determin'd side, and therefore I call it Ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epistemic Probability
Uncertainty quantification (UQ) is the science of quantitative characterization and estimation of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc., will lead to different results that can only be predicted in a statistical sense. Many problems in the natural sciences and engineering are also rife with sources of uncertainty. Computer experiments on computer simulations are the most common approach to study problems in uncertainty quantification. Sources Uncertainty can enter mathematical models and experimental measurements in various contexts. One way to categorize the sources of uncertainty is to consider: ; Parame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Transformation Groups
The principle of transformation groups is a methodology for assigning prior probabilities in statistical inference issues, initially proposed by physicist E. T. Jaynes. It is regarded as an extension of the principle of indifference. Prior probabilities determined by this principle are objective in that they rely solely on the inherent characteristics of the problem, ensuring that any two individuals applying the principle to the same issue would assign identical prior probabilities. Thus, this principle is integral to the objective Bayesian interpretation of probability. Motivation and Method Description The principle is motivated by the following normative principle, or desideratum: ''In scenarios where the prior information is identical, individuals should assign the same prior probabilities.'' This rule is implemented by identifying symmetries, defined by transformation groups, that allow a problem to converted into an equivalent one, and utilizing these symmetries to c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Sufficient Reason
The principle of sufficient reason states that everything must have a Reason (argument), reason or a cause. The principle was articulated and made prominent by Gottfried Wilhelm Leibniz, with many antecedents, and was further used and developed by Arthur Schopenhauer and Sir William Hamilton, 9th Baronet, William Hamilton. History The modern formulation of the principle is usually ascribed to the early Age of Enlightenment, Enlightenment philosopher Gottfried Wilhelm Leibniz, Gottfried Leibniz, who formulated it, but was not its originator.See chapter on Leibniz and Spinoza in A. O. Lovejoy, ''The Great Chain of Being''. The idea was conceived of and utilized by various philosophers who preceded him, including Anaximander, Parmenides, Archimedes, Plato, Aristotle,Sir William Hamilton, 9th Baronet, Hamilton 1860:66. Cicero, Avicenna, Thomas Aquinas, and Baruch Spinoza. One often pointed to is in Anselm of Canterbury: his phrase ''quia Deus nihil sine ratione facit'' (because God d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gottfried Leibniz
Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Isaac Newton, Sir Isaac Newton, with the creation of calculus in addition to many other branches of mathematics, such as binary arithmetic and statistics. Leibniz has been called the "last universal genius" due to his vast expertise across fields, which became a rarity after his lifetime with the coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics. He wrote works on philosophy, theology, ethics, politics, law, history, philology, games, music, and other studies. Leibniz also made major contributions to physics and technology, and anticipated notions that surfaced much later in probability theory, biology, medicine, geology, psychology, linguistics and computer science. Leibniz contributed to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Von Kries
Johannes Adolf von Kries (6 October 1853 – 30 December 1928) was a German physiological psychologist who formulated the modern “duplicity” or “duplexity” theory of vision mediated by rod cells at low light levels and three types of cone cells at higher light levels. He made important contributions in the field of haemodynamics. In addition, von Kries was a significant theorist of the foundations of probability. Biography When von Kries was at Freiburg (1880–1924), he was called to succeed Professor Emil Du Bois-Reymond as chair of physiology at the University of Berlin, but he declined. Von Kries has been called Helmholtz's "greatest German disciple". Works “Über den Druck in den Blutcapillaren der menschlichen Haut” ''Arbeiten aus der Physiologischen Anstalt zu Leipzig'' p 69-80 (1875). “Die Zeitdauer einfachster psychischer Vorgänge”with Felix Auerbach. ''Archiv für Physiologie'' p 297-378 (1877). * “Über die Bestimmung des Mitteldruckes durch das Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Simon Laplace
Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summarized and extended the work of his predecessors in his five-volume ''Mécanique céleste'' (''Celestial Mechanics'') (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. Laplace also popularized and further confirmed Sir Isaac Newton's work. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Bernoulli
Jacob Bernoulli (also known as James in English or Jacques in French; – 16 August 1705) was a Swiss mathematician. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibnizian calculus, to which he made numerous contributions. A member of the Bernoulli family, he, along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant . However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work '' Ars Conjectandi''.Jacob (Jacques) Bernoulli [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucretius
Titus Lucretius Carus ( ; ; – October 15, 55 BC) was a Roman poet and philosopher. His only known work is the philosophical poem '' De rerum natura'', a didactic work about the tenets and philosophy of Epicureanism, which usually is translated into English as ''On the Nature of Things''—and somewhat less often as ''On the Nature of the Universe''. Very little is known about Lucretius's life; the only certainty is that he was either a friend or client of Gaius Memmius, to whom the poem was addressed and dedicated. ''De rerum natura'' was a considerable influence on the Augustan poets, particularly Virgil (in his ''Aeneid'' and ''Georgics'', and to a lesser extent on the '' Eclogues'') and Horace. The work was almost lost during the Middle Ages, but was rediscovered in 1417 in a monastery in Germany by Poggio Bracciolini and it played an important role both in the development of atomism (Lucretius was an important influence on Pierre Gassendi) and the efforts of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epicurus
Epicurus (, ; ; 341–270 BC) was an Greek philosophy, ancient Greek philosopher who founded Epicureanism, a highly influential school of philosophy that asserted that philosophy's purpose is to attain as well as to help others attain tranquil lives, characterized by freedom from fear and the absence of pain. Epicurus advocated that people were best able to pursue philosophy by living a self-sufficient life surrounded by friends; he and his followers were known for eating simple meals and discussing a wide range of philosophical subjects at "the Garden", the school he established in Athens. Epicurus taught that although the gods exist, they have no involvement in human affairs. Like the earlier philosopher Democritus, Epicurus claimed that all occurrences in the natural world are ultimately the result of tiny, invisible particles known as ''Atomism, atoms'' moving and interacting in empty space, though Epicurus also deviated from Democritus by proposing the idea of Clinamen, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wine/water Paradox
The wine/water paradox is an apparent paradox in probability theory. It is stated by Michael Deakin as follows: The core of the paradox is in finding consistent and justifiable simultaneous prior distributions for x and \frac. Calculation This calculation is the demonstration of the paradoxical conclusion when making use of the principle of indifference The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their cre .... To recapitulate, We do not know x, the wine to water ratio. When considering the numbers above, it is only known that it lies in an interval between the minimum of one quarter wine over three quarters water on one end (i.e. 25% wine), to the maximum of three quarters wine over one quarter water on the other (i.e. 75% wine). In term of ratios, x_\mathrm=\frac = \frac resp. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Variable
Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation—in physics called the Heisenberg uncertainty principle—between them. In mathematical terms, conjugate variables are part of a symplectic basis, and the uncertainty relation corresponds to the symplectic form. Also, conjugate variables are related by Noether's theorem, which states that if the laws of physics are invariant with respect to a change in one of the conjugate variables, then the other conjugate variable will not change with time (i.e. it will be conserved). Conjugate variables in thermodynamics are widely used. Examples There are many types of conjugate variables, depending on the type of work a certain system is doing (or is being subjected to). Examples of canonically conjugate variables include the following: * Time and fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution function is constant along the Trajectory, trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. Liouville's theorem applies to conservative systems, that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
