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A Priori Probability
An ''a priori'' probability is a probability that is derived purely by deductive reasoning. One way of deriving ''a priori'' probabilities is the principle of indifference, which has the character of saying that, if there are ''N'' mutually exclusive and collectively exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/''N''. Similarly the probability of one of a given collection of ''K'' events is ''K'' / ''N''. One disadvantage of defining probabilities in the above way is that it applies only to finite collections of events. In Bayesian inference, " uninformative priors" or "objective priors" are particular choices of ''a priori'' probabilities. Note that "prior probability" is a broader concept. Similar to the distinction in philosophy between a priori and a posteriori, in Bayesian inference ''a priori'' denotes general knowledge about the data distribution before making an inference, while ''a posteriori'' denotes knowledge t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deductive Reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if 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. Some theorists define 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. Psychology is interested in deductive reasoning as a psychological process, i.e. how people ''actually'' draw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the context of QFT, the ''S''-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the ''in-states'' and the ''out-states'') in the Hilbert space of physical states. A multi-particle state is said to be ''free'' (non-interacting) if it transforms under Lorentz transformations as a tensor product, or ''direct product'' in physics parlance, of ''one-particle states'' as prescribed by equation below. ''Asymptotically free'' then means that the state has this appearance in either the distant past or the distant future. While the ''S''-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simple form in the case of the Minkowsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Statistics
Bayesian statistics 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 that views probability as the limit of the relative frequency of an event after many trials. 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 be used to estimate the parameters of a probability distribution or statistical model. Since Bayesian statistics treats probabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Assessment
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Transport Equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of Thermodynamic equilibrium, equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), Rita G. Lerner, R. G. Lerner, G. L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number. The equation arises not by analyzing the individual position vector, positions and momentum, momenta of each particle in the fluid but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated by Austrian physicist Wolfgang Pauli in 1925 for electrons, and later extended to all fermions with his spin–statistics theorem of 1940. In the case of electrons in atoms, it can be stated as follows: it is impossible for two electrons of a poly-electron atom to have the same values of the four quantum numbers: ''n'', the principal quantum number; ', the azimuthal quantum number; ''m'', the magnetic quantum number; and ''ms'', the spin quantum number. For example, if two electrons reside in the same orbital, then their ''n'', ', and ''m'' values are the same; therefore their ''ms'' must be different, and thus the electrons must have opposite half-integer spin projections of 1/2 and −1/2. Particles with an integer spin, or bosons, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bose–Einstein Statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles can be distributed in this way. The idea was later adopted and extended by Albert Einstein in collaboration with Bose. The Bose–Einstein statistics applies only to the particles not limited to single occupancy of the same state – that is, particles that do not obey the Pauli exclusion principle restrictions. Such particles have integer values of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi–Dirac Statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of particles over energy states. It is named after Enrico Fermi and Paul Dirac, each of whom derived the distribution independently in 1926 (although Fermi derived it before Dirac). Fermi–Dirac statistics is a part of the field of statistical mechanics and uses the principles of quantum mechanics. F–D statistics applies to identical and indistinguishable particles with half-integer spin (1/2, 3/2, etc.), called fermions, in thermodynamic equilibrium. For the case of negligible interaction between particles, the system can be described in terms of single-particle energy states. A result is the F–D distribution of particles over these states where no two particles can occupy the same state, which has a considerable effec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution Function (physics)
:''This article describes the ''distribution function'' as used in physics. You may be looking for the related mathematical concepts of cumulative distribution function or probability density function.'' In molecular kinetic theory in physics, a system's distribution function is a function of seven variables, f(x,y,z,t;v_x,v_y,v_z), which gives the number of particles per unit volume in single-particle phase space. It is the number of particles per unit volume having approximately the velocity \mathbf=(v_x,v_y,v_z) near the position \mathbf=(x,y,z) and time t. The usual normalization of the distribution function is :n(x,y,z,t) = \int f \,dv_x \,dv_y \,dv_z, :N(t) = \int n \,dx \,dy \,dz, where, ''N'' is the total number of particles, and ''n'' is the number density of particles – the number of particles per unit volume, or the density divided by the mass of individual particles. A distribution function may be specialised with respect to a particular set of dimensions. E.g. t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies. In the case of triangles, one has a ''degenerate triangle'' if at least one side length or angle is zero. Equivalently, it becomes a "line segment". Often, the degenerate cases are the exceptional cases where changes to the usual dimension or the cardinality of the object (or of some part of it) occur. For example, a triangle is an object of dimension two, and a degenerate triangle is contained in a line, which makes its dimension one. This is similar ... [...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 and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the 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. There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems. There are extensions of Liouville's theorem to stochastic systems. Liouville equations The Liouville equation describes the time evolution of the ''phase space distribution function''. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the impor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 credence (or 'degrees of belief') equally among all the possible outcomes under consideration. In Bayesian probability, this is the simplest non-informative prior. The principle of indifference is meaningless under the frequency interpretation of probability, in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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