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Equiprobable
Equiprobability is a property for a collection of events that each have the same probability of occurring. In statistics and probability theory it is applied in the discrete uniform distribution and the equidistribution theorem for rational numbers. If there are n events under consideration, the probability of each occurring is \frac. In philosophy it corresponds to a concept that allows one to assign equal probabilities to outcomes when they are judged to be equipossible or to be "equally likely" in some sense. The best-known formulation of the rule is Laplace's principle of indifference (or ''principle of insufficient reason''), which states that, when "we have no other information than" that exactly N mutually exclusive events can occur, we are justified in assigning each the probability \frac. This subjective assignment of probabilities is especially justified for situations such as rolling dice and lotteries since these experiments carry a symmetry structure, and one's st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Content
In information theory, the information content, self-information, surprisal, or Shannon information is a basic quantity derived from the probability of a particular event occurring from a random variable. It can be thought of as an alternative way of expressing probability, much like odds or log-odds, but which has particular mathematical advantages in the setting of information theory. The Shannon information can be interpreted as quantifying the level of "surprise" of a particular outcome. As it is such a basic quantity, it also appears in several other settings, such as the length of a message needed to transmit the event given an optimal source coding of the random variable. The Shannon information is closely related to ''entropy'', which is the expected value of the self-information of a random variable, quantifying how surprising the random variable is "on average". This is the average amount of self-information an observer would expect to gain about a random variable when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory 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. Typically these axioms formalise probability in terms of a probability space, which assigns a 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. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability
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] |
Scientific Modelling
Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted knowledge. It requires selecting and identifying relevant aspects of a situation in the real world and then developing a model to replicate a system with those features. Different types of models may be used for different purposes, such as conceptual models to better understand, operational models to operationalize, mathematical models to quantify, computational models to simulate, and graphical models to visualize the subject. Modelling is an essential and inseparable part of many scientific disciplines, each of which has its own ideas about specific types of modelling. The following was said by John von Neumann. There is also an increasing attention to scientific modelling in fields such as science education, philosophy of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Uniform Distribution
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair dice. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aequiprobabilism
Aequiprobabilism, also spelled æquiprobabilism or equiprobabilism, is one of several doctrines in moral theology opposed to probabilism. Teaching #If the opinions for and against the current existence of a law have equal or nearly equal probabilities, it is permissible to act on the less safe opinion. #If the opinions for and against the cessation of a previously existing law have equal or nearly equal probabilities, then it is not permissible to act on the less safe opinion. #If the safe opinion is certainly more probable than the less safe opinion, then it is unlawful to follow the less safe opinion. With the first of these propositions Probabilists agree—but they deny the truth of the second and third propositions (cf. Marc, "Institutiones Morales", I, nn. 91-103). Arguments for Æquiprobabilism # In proof of their first proposition, Æquiprobabilists quote, among other things, the axiom: in dubio melior est condition possidentis. When the doubt regards the existence, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 knowled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uninformative Prior
In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity 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. Bayes' theorem calculates the renormalized pointwise product of the prior and the likelihood function, to produce the ''posterior probability distribution'', which is the conditional distribution of the uncertain quantity given the data. Similarly, the prior probability of a random event or an uncertain proposition is the unconditional probability that is assigned before any relevant evidence is taken into account. Priors can be created using a numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian Smoothing
Laplacian smoothing is an algorithm to smooth a polygonal mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex polyg .... For each vertex in a mesh, a new position is chosen based on local information (such as the position of neighbours) and the vertex is moved there. In the case that a mesh is topologically a rectangular grid (that is, each internal vertex is connected to four neighbours) then this operation produces the Laplacian of the mesh. More formally, the smoothing operation may be described per-vertex as: :\bar_= \frac \sum_^\bar_j Where N is the number of adjacent vertices to node i, \bar_ is the position of the j-th adjacent vertex and \bar_ is the new position for node i. See also * Tutte embedding, an embedding of a planar mesh in which each vertex is already at the aver ... [...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] |
Principle Of Transformation Groups
The principle of transformation groups is a rule for assigning ''epistemic'' probabilities in a statistical inference problem. It was first suggested by Edwin T. Jaynes and can be seen as a generalisation of the principle of indifference. This can be seen as a method to create ''objective ignorance probabilities'' in the sense that two people who apply the principle and are confronted with the same information will assign the same probabilities. Motivation and description of the method The method is motivated by the following normative principle, or desideratum: ''In two problems where we have the same prior information we should assign the same prior probabilities'' The method then comes about from "transforming" a given problem into an equivalent one. This method has close connections with group theory, and to a large extent is about finding symmetry in a given problem, and then exploiting this symmetry to assign prior probabilities. In problems with discrete variables (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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