**outcomes**- One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same. (wikipedia.org)
- A probability space consists of three parts: A sample space, Ω {\displaystyle \Omega } , which is the set of all possible outcomes. (wikipedia.org)
- this contrasts with global methods such as minimax, which considers worst-case analysis over the entire space of outcomes, and probabilistic decision theory, which considers all possible outcomes, and assigns some probability to them. (wikipedia.org)
- In contrast to probabilistic decision theory, info-gap analysis does not use probability distributions: it measures the deviation of errors (differences between the parameter and the estimate), but not the probability of outcomes - in particular, the estimate u ~ {\displaystyle {\tilde {u}}} is in no sense more or less likely than other points, as info-gap does not use probability. (wikipedia.org)
- Info-gap, by not using probability distributions, is robust in that it is not sensitive to assumptions on probabilities of outcomes. (wikipedia.org)
- 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. (wikipedia.org)
- Examples: Throwing dice, experiments with decks of cards, random walk, and tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. (wikipedia.org)
- The probability associated with each outcome is 1/ The toss of two cois: The four possible outcomes are H,H, H,T, T,H ad TT.The probability of each is 1/4. (docplayer.net)
- If there is a ifiite umber of possible outcomes, the probability of a outcome is ot defied i the classical sese. (docplayer.net)
- Nothig has to be observed i terms of outcomes to deduce the probabilities. (docplayer.net)
- If we just take QM seriously as a theory that predicts the probability of different measurement outcomes, all is well. (discovermagazine.com)
- Numerical "point" probabilities are possible when the relevant outcomes involved are finite, exclusive and equiprobable (Gillies 2000: 35). (blogspot.it)
- Laplace's formula for calculating probabilities is the familiar one where the probability P( E ) of any event E in a finite sample space S , where all outcomes are equally likely, is the number of outcomes for E divided by the total number of outcomes in S . (blogspot.it)

**discrete probability**- Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. (wikipedia.org)
- Discrete probability theory deals with events that occur in countable sample spaces. (wikipedia.org)

**Treatise on Probability**- (1) Dow (1995) is an explicit recognition of the idea that Keynesian uncertainty comes in degrees, on the basis of the Treatise on Probability and Keynes's concept of the weight of evidence. (blogspot.pt)
- Keynes had finished the proofs of his Treatise on Probability in 1913, but it was not published until 1921. (blogspot.it)
- Keynes After Ramsey: In Defence of A Treatise on Probability ," Studies in History and Philosophy of Science 25.1: 97-121. (blogspot.it)

**distributions**- Wald's paper renewed and synthesized many concepts of statistical theory, including loss functions , risk functions , admissible decision rules , antecedent distributions , Bayesian procedures , and minimax procedures. (like2do.com)
- Prior, likelihood and posterior probability distributions. (intechopen.com)
- 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. (wikipedia.org)
- The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have identical cumulants as well, and similarly the cumulants determine the moments. (wikipedia.org)
- 0. The Bernoulli distributions, (number of successes in one trial with probability p of success). (wikipedia.org)
- The geometric distributions, (number of failures before one success with probability p of success on each trial). (wikipedia.org)
- The binomial distributions, (number of successes in n independent trials with probability p of success on each trial). (wikipedia.org)

**Uncertainty**- Gillies on Probability and Uncertainty in Keynes' Economic Thinking," July 6, 2014. (blogspot.com)
- The area of choice under uncertainty represents the heart of decision theory. (like2do.com)
- Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. (wikipedia.org)
- Possibility may refer to: Probability, the measure of the likelihood that an event will occur Epistemic possibility, a topic in philosophy and modal logic Possibility theory, a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory Subjunctive possibility, (also called alethic possibility) is a form of modality studied in modal logic. (wikipedia.org)
- Info-gap decision theory is a non-probabilistic decision theory that seeks to optimize robustness to failure - or opportuneness for windfall - under severe uncertainty, in particular applying sensitivity analysis of the stability radius type to perturbations in the value of a given estimate of the parameter of interest. (wikipedia.org)
- Info-gap is a decision theory: it seeks to assist in decision-making under uncertainty. (wikipedia.org)
- Info-gap theory models uncertainty α {\displaystyle \alpha } (the horizon of uncertainty) as nested subsets U ( α , u ~ ) {\displaystyle {\mathcal {U}}(\alpha ,{\tilde {u}})} around a point estimate u ~ {\displaystyle {\tilde {u}}} of a parameter: with no uncertainty, the estimate is correct, and as uncertainty increases, the subset grows, in general without bound. (wikipedia.org)
- The probabilistic approach to modelling uses probability theory to express all aspects of uncertainty in the model. (biomedsearch.com)
- Dow (1995: 119-120) notes that the weight of evidence "allows understanding of uncertainty as a relative concept," and that this idea of relative uncertainties was an important, but neglected, aspect of Keynes's famous 1937 article "The General Theory of Employment" (Keynes 1937). (blogspot.pt)
- We assign a probability distribution to p to express our uncertainty, not to attribute randomness to p. (wikipedia.org)

**Interpretations**- The article probability interpretations outlines several alternative views of what "probability" means and how it should be interpreted. (wikipedia.org)
- 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. (wikipedia.org)
- It is currently considered a mainstream interpretation along with the other decoherence interpretations, collapse theories (including the historical Copenhagen interpretation), and hidden variable theories such as the Bohmian mechanics. (wikipedia.org)

**axioms**- The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function P {\displaystyle P} . The Russian mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. (wikipedia.org)
- Axiomatic Approach to Probability Put simply, the axiomatic approach build up probability theory from a umber of assumptios (axioms). (docplayer.net)

**results in probability theory**- The conformal prediction framework comes from deep results in probability theory and is inspired by Kolmogorov and Martin-Lof's ideas on algorithmic complexity theory. (wordpress.com)
- Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem. (wikipedia.org)

**1994**- Roussas, George G. (2001) [1994], "Contiguity of probability measures", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 van der Vaart, A. W. (1998). (wikipedia.org)

**mathematical theory**- This implicit assumption is completely unnecessary and the mathematical theory of probability makes use of it only through one crucial assumption, which turns out to be wrong in principle but right in practice for many actual events in the real world. (discovermagazine.com)

**2000**- Chapter 8 of Donald Gillies' Philosophical Theories of Probability (2000) deals with intersubjective and pluralist views of probability. (blogspot.com)
- Gillies (2000: 169) argues that there is an intermediate class of probability called intersubjective probability that lies between fully subjective and objective probability. (blogspot.com)
- Furthermore, Gillies makes the case for a "pluralist" view of probability, a development of the two-concept view of probability by Ramsey and Carnap (Gillies 2000: 180-181). (blogspot.com)
- A fully objective probability is one that is found in natural phenomena or the material world and is independent of human beings, such as, for example, the probability of radioactive decay of atoms (Gillies 2000: 180). (blogspot.com)
- An artefactual probability is objective in the sense that it is part of the material world, but is at the same time the result of human interaction with nature (Gillies 2000: 179). (blogspot.com)
- If such probabilities can be established though stable long-run observed relative frequencies in trials or experiments, then they can be said to be objective (Gillies 2000: 180). (blogspot.com)
- Between artefactual and intersubjective probabilities, Gillies sees some borderline cases such as in medicine or population studies (Gillies 2000: 180, 194). (blogspot.com)
- An intersubjective probability is a measure of the degree of belief of a social group where a consensus has formed (Gillies 2000: 179). (blogspot.com)
- If a group of people, though shared beliefs, can agree on a common betting quotient about some probability, then this can be seen as an intersubjective or consensus probability (Gillies 2000: 171). (blogspot.com)
- A fully subjective probability is one of an individual and is a mere measure of a degree of belief, which has a non-objective status (Gillies 2000: 179). (blogspot.com)
- Fully objective and artefactual probabilities are objective and properly described by the propensity theory of probability, but intersubjective and fully subjective probabilities are epistemological, and often such subjective probabilities have no real nor quantifiable objective "propensity" probabilities at all (Gillies 2000: 180-185). (blogspot.com)
- Contiguity Asymptopia: 17 October 2000, David Pollard Asymptotic normality under contiguity in a dependence case A Central Limit Theorem under Contiguous Alternatives Superefficiency, Contiguity, LAN, Regularity, Convolution Theorems Testing statistical hypotheses Necessary and sufficient conditions for contiguity and entire asymptotic separation of probability measures R Sh Liptser et al 1982 Russ. (wikipedia.org)
- Chapter 3 of Donald Gillies' Philosophical Theories of Probability (2000) deals with Keynes' logical theory of probability, which was taken up by the Vienna circle and logical positivists like Carnap (Gillies 2000: 25). (blogspot.it)
- Keynes' logical theory was partly inspired by the lectures of W. E. Johnson at Cambridge university, which were also attended by Harold Jeffreys who later formulated his own logical theory of probability (Gillies 2000: 25). (blogspot.it)
- Keynes' views on probability were influenced by the intellectual climate at Cambridge university, particularly the ethical work of G. E. Moore and Bertrand Russell's logicist work on mathematics (Gillies 2000: 27). (blogspot.it)
- Gillies (2000: 27) sees Keynes' attempts to provide a "logical" foundation for probability, and particularly inductive reasoning, as inspired by Russell and Whitehead's attempts to found mathematics on logic. (blogspot.it)
- Keynes thought that we have knowledge of this probability relation by logical intuition or direct acquaintance, but this is a problematic part of Keynes' theory (Gillies 2000: 31-32). (blogspot.it)
- Gillies (2000: 32-33) argues that underlying Keynes' idea that probability is objective and known by logical intuition is a problematic concept derived from Platonic ontology: that the probability relation is objective and belongs to some Platonic realm. (blogspot.it)
- he thought that not every probability had a numerical value, and also that some probabilities can only be arranged in an ordinal ranking, while others cannot even be compared at all (Gillies 2000: 33-34). (blogspot.it)
- Chapter 2 of Donald Gillies' Philosophical Theories of Probability (2000) deals with the Classical interpretation of probability theory. (blogspot.it)
- It is only human ignorance that prevents perfect forecasting, and leads us to calculate probabilities (Gillies 2000: 17). (blogspot.it)
- Thus probability, according to Laplace, is a measure of human ignorance (Gillies 2000: 21). (blogspot.it)
- Law of total expectation Law of total variance Law of total cumulance Marginal distribution Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. (wikipedia.org)

**assigns**- Then a probability distribution assigns a number P(A) between zero and one to each possible outcome. (discovermagazine.com)

**inductive**- citation needed] In the 1940s, Rudolf Carnap investigated a probability-based theory of inductive reasoning, and developed measures of degree of confirmation, which he considered as alternatives to Laplace's rule of succession. (wikipedia.org)

**Laplace**- As Gillies notes, the "Classical" interpretation was the earliest theory of probability and its most important statement was by Pierre-Simon Laplace (1749-1827) in his Essai Philosophique sur les Probabilités [ A Philosophical Essay on Probabilities ] (1814). (blogspot.it)
- In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. (wikipedia.org)
- Laplace used the rule of succession to calculate the probability that the sun will rise tomorrow, given that it has risen every day for the past 5000 years. (wikipedia.org)

**axiom**- Axiom 3 corresponds to the additivity axiom in probabilities. (wikipedia.org)
- Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. (wikipedia.org)
- 1 3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio ecompasses the Classical ad Frequecy defiitios of probability Note that i a umber of places i these otes I use axiom as a syomom for assumptio. (docplayer.net)

**Bayesian**- His dissertation and his paper Stochastic Independence, Causal Independence, and Shieldability are precursors of the theory of Bayesian networks and their causal interpretation. (wikipedia.org)
- Therefore, the aim of this chapter is to introduce a Bayesian approach to hypothesis testing that may represent a useful complement, or even an alternative, to the current NHST. (intechopen.com)
- The probabilistic approach is synonymous with Bayesian modelling, which simply uses the rules of probability theory in order to make predictions, compare alternative models, and learn model parameters and structure from data. (biomedsearch.com)
- The other interpretation, often called Bayesian, is that probability gives a best guess at what the answer will be in any given trial. (discovermagazine.com)

**objective**- In Gillies' "pluralist" theory of probability, probabilities can be conceptualised on a spectrum or continuum from the fully objective to fully subjective, and intermediate types of probability, as in the diagram below (which needs to be opened in a new window to be seen properly). (blogspot.com)
- An intersubjective probability may be utterly subjective, or subjective but based on an underlying objective probability. (blogspot.com)
- We provide a formally rigorous framework for integrating singular causation, as understood by Nuel Belnap's theory of causae causantes, and objective single case probabilities. (pitt.edu)
- Very often, when I try to discuss my theory of consciousness with people, the discussion falls apart because the people I'm talking to want to assume that objective reality is primary, or else that subjective experiential reality is primary. (goertzel.org)
- The outcome in a (fair) game of roulette can be calculated in an objective sense though a priori probabilities, so it is clear Keynes is excluding mathematical or physical probabilities here. (blogspot.pt)

**estimate**- Oe s estimate of the probability of a head is.52. (docplayer.net)
- Frequecy probability allows to estimate probabilities whe Classical probability provides o isight. (docplayer.net)
- The formula is still used, particularly to estimate underlying probabilities when there are few observations, or for events that have not been observed to occur at all in (finite) sample data. (wikipedia.org)

**thus**- Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. (wikipedia.org)

**occur**- In probability theory, a probability space or a probability triple ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} is a mathematical construct that models a real-world process (or "experiment") consisting of states that occur randomly. (wikipedia.org)
- Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. (wikipedia.org)
- Phenomena with very small probabilities do not occur' (p. 3). (answersingenesis.org)
- This argument was not fully developed since events of low probabilities actually could occur by chance. (answersingenesis.org)
- Dembski's key insight, which will be elaborated on below, is stated succinctly as, ' Specified events of small probability do not occur by chance ' (p. 5). (answersingenesis.org)
- The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. (wikipedia.org)
- mu it-\sigma ^{2}{\frac {t^{2}}{2}}+\cdots } An advantage of H(t)-in some sense the function K(t) evaluated for purely imaginary arguments-is that E(eitX) is well defined for all real values of t even when E(etX) is not well defined for all real values of t, such as can occur when there is "too much" probability that X has a large magnitude. (wikipedia.org)

**displaystyle**- that is, a function P {\displaystyle P} from events to probabilities. (wikipedia.org)
- A probability space is a mathematical triplet ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} that presents a model for a particular class of real-world situations. (wikipedia.org)
- The probability measure P {\displaystyle P} is a function returning an event's probability. (wikipedia.org)
- Pr ( A ∣ B Y ) = 95 100 {\displaystyle \Pr(A\mid B_{Y})={95 \over 100}} is the probability that a bulb manufactured by Y will work for over 5000 hours. (wikipedia.org)

**probabilistic**- He is known especially for his contributions to the defense and development of alternatives to the classical calculus for probabilistic modeling and decision-making. (wikipedia.org)
- Probability and Probabilistic Reasoning for Electrical Engineering, Pearson/Prentice-Hall, 2006. (wikipedia.org)
- Probability and Probabilistic Causality. (wikipedia.org)
- To Keynes expectations are a question of weighing probabilities by 'degrees of belief,' beliefs that often have preciously little to do with the kind of stochastic probabilistic calculations made by the rational agents as modeled by 'modern' social sciences. (blogspot.pt)

**epistemic**- His paper How to Make Sense of Game Theory is a forerunner of epistemic game theory. (wikipedia.org)

**ontology**- Spohn's research extends to philosophy of science, the theory of causation, metaphysics and ontology, philosophy of language and mind, two-dimensional semantics, philosophical logic, and decision and game theory (see the collection of papers). (wikipedia.org)

**mutually**- The probability measure function must satisfy two simple requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these events. (wikipedia.org)
- To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs is given by the sum of the probabilities of the events. (wikipedia.org)
- Terms to ote i the defiitio of classical probability are radom,, mutually exclusive, ad equally likely. (docplayer.net)

**fuzzy logic**- Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. (wikipedia.org)

**expresses**- The prior probability density function that expresses total ignorance of p except for the certain knowledge that it is neither 1 nor 0 (i.e., that we know that the experiment can in fact succeed or fail) is equal to 1 for 0 (wikipedia.org)
- It expresses the total probability of an outcome which can be realized via several distinct events-hence the name. (wikipedia.org)

**calculate**- By assuming some probability distribution for the value (women's average lifespan-men's average lifespan), one could calculate the probability of obtaining just by chance the difference in average age found, with the variation that is found within each group, and, for the number of people in each group. (answersingenesis.org)

**finite**- but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti. (wikipedia.org)
- You do a few experiments and see how the finite distribution of results compares to the probabilities, and then assign a confidence level to the conclusion that a particular theory of the data is correct. (discovermagazine.com)

**likelihood**- It has also been referred to as modeling fields, modeling fields theory (MFT), Maximum likelihood artificial neural networks (MLANS). (wikipedia.org)

**consists**- The central notion is that of a causal probability space whose sample space consists of causal alternatives. (pitt.edu)
- So, in many cases the unique alternative consists of working with interpolated, outdated, or even absolutely unknown values. (hindawi.com)

**density**- On the histogram as a density estimator: L2 theory" (PDF). (wikipedia.org)

**posterior**- xn is the number of "successes" and n is of course the number of trials, and then normalizes, to get the "posterior" (i.e., conditional on the data) probability distribution of p. (wikipedia.org)

**mathematics**- This article is concerned with the mathematics of manipulating probabilities. (wikipedia.org)
- Probability theory is the branch of mathematics concerned with probability. (wikipedia.org)
- The theory of non-well-founded sets shows that this kind of circularity is perfectly consistent in terms of logic and mathematics. (goertzel.org)
- In order theory, a branch of mathematics, the 1/3-2/3 conjecture states that, if one is comparison sorting a set of items then, no matter what comparisons may have already been performed, it is always possible to choose the next comparison in such a way that it will reduce the number of possible sorted orders by a factor of 2/3 or better. (wikipedia.org)

**stochastic**- PDF version : http://www.uni-konstanz.de/FuF/Philo/Philosophie/files/grundlagen_der_entscheidungstheorie.pdf „Stochastic Independence, Causal Independence, and Shieldability", Journal of Philosophical Logic 9 (1980) 73-99 „How to Make Sense of Game Theory", in: W. Stegmüller, W. Balzer, W. Spohn (Hg. (wikipedia.org)
- Because queuing theory is highly complex, only the (stochastic) arrival process will be discussed in detail. (egms.de)

**Empirical**- Empirical applications of this rich theory are usually done with the help of statistical and econometric methods, especially via the so-called choice models, such as probit and logit models. (like2do.com)
- From almost one and a half centuries, scientific research mostly relies on empirical findings to provide support to their hypotheses, models, or theories. (intechopen.com)
- For a set empirical measurements sampled from some probability distribution, the Freedman-Diaconis rule is designed to minimize the difference between the area under the empirical probability distribution and the area under the theoretical probability distribution. (wikipedia.org)

**distribution**- In the 20th century, interest was reignited by Abraham Wald's 1939 paper pointing out that the two central procedures of sampling-distribution-based statistical-theory, namely hypothesis testing and parameter estimation , are special cases of the general decision problem. (like2do.com)
- One may define a uniform probability distribution on the linear extensions in which each possible linear extension is equally likely to be chosen. (wikipedia.org)
- The 1/3-2/3 conjecture states that, under this probability distribution, there exists a pair of elements x and y such that the probability that x is earlier than y in a random linear extension is between 1/3 and 2/3. (wikipedia.org)
- Physical theories are often couched in the form of equations for the time evolution of the probability distribution, even in classical physics. (discovermagazine.com)
- In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. (wikipedia.org)
- In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. (wikipedia.org)

**causation**- Probability theory and causation. (pitt.edu)
- Why is the transference theory of causation insuffcient? (pitt.edu)

**Decoherence**- The decoherence approaches to interpreting quantum theory have been further explored and developed, becoming quite popular. (wikipedia.org)

**BIBLIOGRAPHY**- Bibliography on Keynes's Theory of Probability (Updated)," July 6, 2014. (blogspot.it)

**framework**- The core of this framework is evaluated according to the probability theory principles. (hindawi.com)
- A study of mathematical and interpretive alternatives to the standard framework for mathematical probability. (wikipedia.org)

**Bernoulli**- In 1738, Daniel Bernoulli published an influential paper entitled Exposition of a New Theory on the Measurement of Risk , in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. (like2do.com)
- If the probablity a particular even side comes up is 1/9 and the probability a particular odd side comes up is 2/9, can you consider this a Bernoulli trial in determining the probability of rolling five 4's in 10 rolls? (brainmass.com)

**statistical**- These include probabilities of games of chance, probabilities of repeatable scientific experiments, and certain statistical probabilities. (blogspot.com)
- Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. (wikipedia.org)

**marginal**- In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. (wikipedia.org)

**whereas**- Closely related to the field of game theory , decision theory is concerned with the choices of individual agents whereas game theory is concerned with interactions of agents whose decisions affect each other. (like2do.com)

**Algebra**- Some of the terminology in von Neumann algebra theory can be confusing, and the terms often have different meanings outside the subject. (wikipedia.org)
- The similar-looking perpendicular symbol (⟂, \perp in LaTeX, U+27C2 in Unicode) is a binary relation symbol used to represent: Perpendicularity of lines in geometry Orthogonality in linear algebra Independence of random variables in probability theory Comparability in order theory Coprimality in number theory Tee (symbol) (⊤) Table of mathematical symbols Alternative plus sign List of mathematical symbols "Mathematical Operators - Unicode" (PDF). (wikipedia.org)

**approaches**- It has been criticized as unsuited for this purpose, and alternatives proposed, including such classical approaches as robust optimization. (wikipedia.org)
- Of the application of such theories to quantum mechanics, Bill Jefferys has said: "Such approaches are also not necessary and in my opinion they confuse more than they illuminate. (wikipedia.org)

**concepts**- Concepts of probability theory. (wikipedia.org)

**Furthermore**- Furthermore the impact of different treatment alternatives on the waiting list can be tested. (egms.de)

**foundations**- This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. (wikipedia.org)
- Theories of Probability: An Examination of Foundations, Academic Press, 1973. (wikipedia.org)

**successes**- If we repeat an experiment that we know can result in a success or failure, n times independently, and get s successes, then what is the probability that the next repetition will succeed? (wikipedia.org)

**measure**- Glenn Shafer is also well known for his co-development of Dempster-Shafer theory, which is a brewing alternative to standard measure-theoretic probability theory which is quite useful in sensor fusion, and I think some machine learning frameworks. (wordpress.com)
- The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more. (wikipedia.org)
- Learning is an essential part of perception and cognition, and in NMF theory it is driven by the dynamics that increase a similarity measure between the sets of models and signals, L({X},{M}). The similarity measure is a function of model parameters and associations between the input bottom-up signals and top-down, concept-model signals. (wikipedia.org)

**Proportion**- The table shows, for some particular year, a listing of several income levels and, for each level, the proportion of the population in the level and the probability that a person in that level bought a new car during the year. (brainmass.com)

**measures**- In probability theory, two sequences of probability measures are said to be contiguous if asymptotically they share the same support. (wikipedia.org)
- Econometrics Asymptotic theory (statistics) Contiguity Probability space Wolfowitz J.(1974) Review of the book: "Contiguity of Probability Measures: Some Applications in Statistics. (wikipedia.org)
- Roussas, George G. (1972), Contiguity of Probability Measures: Some Applications in Statistics, CUP, ISBN 978-0-521-09095-7. (wikipedia.org)
- Scott, D.J. (1982) Contiguity of Probability Measures, Australian & New Zealand Journal of Statistics, 24 (1), 80-88. (wikipedia.org)

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