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Topic: Likelihood principle

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Likelihood function

Maximum likelihood

Statistical inference

Bayesian inference

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Testing statistical hypotheses

Bayes factor

Probability density function

Bernoulli process

Lebesgue integration

Equivalence class

Bayesian

Random variable

Bernoulli trial

Normal distribution

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	Likelihood principle - Wikipedia, the free encyclopedia
		In statistics, the likelihood principle is a controversial principle of statistical inference which asserts that all of the information in a sample is contained in the likelihood function.
		Combining the likelihood principle with the law of likelihood yields the consequence that the parameter value which maximizes the likelihood function is the value which is most strongly supported by the evidence.
		The likelihood principle was first identified by that name in print in 1962 (Barnard et al., Birnbaum, and Savage et al.), but arguments for the same principle, unnamed, and the use of the principle in applications goes back to the works of R.A. Fisher in the 1920s.
	en.wikipedia.org /wiki/Likelihood_principle (1810 words)

	Likelihood function - Wikipedia, the free encyclopedia
		For a likelihood function of more than one parameter, it is sometimes possible to write some parameters as functions of other parameters, thereby reducing the number of independent parameters.
		Attempting to interpret the likelihood of a hypothesis given observed evidence as the probability of the hypothesis is a common error, with potentially disastrous real-world consequences in medicine, engineering or jurisprudence.
		The likelihood function is not a probability density function -- for example, the integral of a likelihood function is not in general 1.
	en.wikipedia.org /wiki/Likelihood (787 words)

	likelihood principle 1 (Site not responding. Last check: 2007-10-21)
		Two likelihood functions are considered equivalent if either is a scalar multiple of the other; the likelihood principle says that all information relevant to inferences about the value of θ is found in the equivalence class.
		From a Bayesian point of view, the likelihood principle is a consequence that falls out of Bayes' theorem.
		The likelihood principle implies that any event that did not happen has no effect on an inference, since if an unrealized event does affect an inference then there is some information not contained in the likelihood function.
	www.yourencyclopedia.net /likelihood_principle_1.html (997 words)

	Likelihood principle (Site not responding. Last check: 2007-10-21)
		In statistics, the likelihood principle is a controversialprinciple of statistical inference which asserts that allof the information in a sample is contained in the likelihoodfunction.
		Combining the likelihood principle with the law of likelihood yields the consequence that the parameter value which maximizesthe likelihood function is the value which is most strongly supported by the evidence.
		The likelihood principle was first identified by that name in print in 1962 (Barnard et al., Birnbaum, and Savage et al.), butarguments for the same principle, unnamed, and the use of the principle in applications goes back to the works of R.A. Fisher in the 1920s.
	www.therfcc.org /likelihood-principle-127716.html (830 words)

	Likelihood principle (Site not responding. Last check: 2007-10-21)
		Two likelihood functions are considered equivalent either is a scalar multiple of the the likelihood principle says that all information to inferences about the value of θ found in the equivalence class.
		Combining the likelihood principle with the law likelihood yields the consequence that the parameter which maximizes the likelihood function is the which is most strongly supported by the This is the basis for the widely-used method of maximum likelihood.
		The likelihood principle was first identified by name in print in 1962 (Barnard et Birnbaum and Savage et al.) but arguments the same principle unnamed and the use the principle in applications goes back to works of R.A. Fisher in the 1920s.
	www.freeglossary.com /Likelihood_principle (1077 words)

	likelihood principle (Site not responding. Last check: 2007-10-21)
		The likelihood principle is a principle of inference which asserts that all of the information in a sample is contained in the likelihood function.
		A widely-used application of the likelihood principle is the method of maximum likelihood.
		Some widely-used methods of conventional statistics, for example significance tests, are not consistent with the likelihood principle.
	www.yourencyclopedia.net /Likelihood_principle.html (421 words)

	Likelihood principle: Definition and Links by Encyclopedian.com - All about Likelihood principle
		In statistical theory, the likelihood principle asserts that the information in any sample can be found, if at all, from the likelihood function, that function of unknown parameters[?] which specifies the probability of the sample observed on the basis of a known model, in terms of the model's parameters.
		The Likelihood principle says the same statistical inference about the value of p should be drawn in each case.) In particular, this principle suggests that it does not matter whether you started out planning to observe N trials or you just decided to stop on a whim.
		The issue of the likelihood principle is still controversial.
	www.encyclopedian.com /li/Likelihood-principle.html (313 words)

	Likelihood principle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A (Click link for more info and facts about likelihood function) likelihood function is a (Click link for more info and facts about conditional probability distribution) conditional probability distribution considered as a function of its second argument, holding the first fixed.
		The likelihood principle has been applied to the (Click link for more info and facts about philosophy of science) philosophy of science by R. Royall.
		From a Bayesian point of view, the likelihood principle is a direct consequence of (Click link for more info and facts about Bayes' theorem) Bayes' theorem.
	www.absoluteastronomy.com /encyclopedia/L/Li/Likelihood_principle.htm (1562 words)

	Talk:Likelihood principle - Wikipedia, the free encyclopedia
		The value of "high enough" is a matter of choice; one approach might be to use 1 as the critical value, but for a Bayesian looking at point hypotheses the figure would best be a combination of the (inverse) ratio of the priors and the relative costs of Type I errors and Type II errors.
		I'm willing to consider a compromise of the form "The conventional likelihood-ratio test is not consistent with the likelihood principle, although there is an unconventional LR test which is".
		Consequently, contrary to a widely held opinion, the likelihood principle is not a direct consequence of Bayes theorem.
	en.wikipedia.org /wiki/Talk:Likelihood_principle (1363 words)

	likelihood (Site not responding. Last check: 2007-10-21)
		Thus a likelihood ratio of 1 corresponds to indifference between the hypotheses on the basis of the evidence in the data, whilst the maximum-likelihood value of a parameter is regarded as the best-supported value, other values being ranked by their lesser likelihoods accordingly.
		Most importantly, the likelihood approach is compatible with Bayesian statistical inference in the sense that the posterior Bayes distribution for a parameter is, by Bayes’s Theorem, found by multiplying the prior distribution by the likelihood function.
		Likelihood ratio tests are based on the distribution under repeated-sampling of the likelihood ratio and are therefore not part of likelihood theory.
	www.cimat.mx /reportes/enlinea/D-99-10.html (2128 words)

	Likelihood principle (Site not responding. Last check: 2007-10-21)
		By contrast, a likelihood-ratio test is based on the principle.
		The likelihood principle also yields results which seems to some people to be apparently paradoxical results.
		Such apparently-paradoxical results of this kind are considered evidence against the likelihood principle.
	www.sciencedaily.com /encyclopedia/likelihood_principle_1 (947 words)

	Likelihood (Site not responding. Last check: 2007-10-21)
		In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first agument held fixed, thus:
		This mode of reasoning is formalized in Bayes' theorem; note the appearance of a likelihood function for B given A in the numerator:
		In the colloquial language, "likelihood" is one of several informal synomyms for "probability", but throughout this article we use only the technical definition.
	www.portaljuice.com /likelihood.html (648 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The likelihood function is a measure of probability or uncertainty applicable to the estimation of unknown parameters EMBED Equation.3 on the basis of a given data set.
		Likelihood, on the other hand, is predicted on fixed data, and for varying hypotheses may be regarded as a function of the hypotheses or of the parameters.
		On the basis of the argument presented, the likelihood is proportional to the probability of obtaining the data as a function of the parameter.
	www.ent.orst.edu /info2000/Week3/Chapter2Likelihood.doc (2804 words)

	Read about Likelihood principle at WorldVillage Encyclopedia. Research Likelihood principle and learn about Likelihood ... (Site not responding. Last check: 2007-10-21)
		In statistics, the likelihood principle is a controversial principle of statistical inference which asserts that all of the information in a
		the likelihood principle was first identified by that name in print in 1962 (barnard et al., birnbaum, and savage et al.), but arguments for the same principle, unnamed, and the use of the principle in applications goes back to the works of
		jeffreys prior, can fail the likelihood principle as it may depend on the design of the experiment.
	encyclopedia.worldvillage.com /s/b/Likelihood_principle (1368 words)

	Integrated likelihood methods for eliminating nuisance parameters, James O. Berger, Brunero Liseo, Robert L. Wolpert
		Bjørnstad, J. On the generalization of the likelihood function and the likelihood principle.
		The problems associated with the use of profile likelihood methods appear to be due to the relatively large number of nuisance parameters.
		We are somewhat concerned with the apparent double use of the data in the definitions of Lest and L est (using the data once in the likelihood function and again in the weight function for integration).
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.ss/1009211804 (1839 words)

	Peter van der Helm: Occam versus Bayes (Site not responding. Last check: 2007-10-21)
		The likelihood principle, on the one hand, reflects the assumption that the visual system is adapted to its environment, that is, it yields optimal veridicality in this specific world.
		Hence, the likelihood principle assumes that vision is guided by external veridicality, and that stimuli are interpreted on the basis of knowledge about the world.
		Then, the likelihood principle can be said to favor interpretations that maximize certainty by way of a Bayesian multiplication of prior and conditional probabilities, whereas the simplicity principle can be said to favor interpretations that minimize information by way of an Occamian summation of prior and conditional complexities.
	www.nici.kun.nl /~peterh/doc/occambayes.html (523 words)

	Likelihood principle - Wikipedia
		The likelihood principle asserts that the information in any sample can be found, if at all, from the likelihood function, that function of unknown parameters which specifies the probability of the sample observed.
		In particular, this principle suggests that it does not matter whether you started out planning to observe N trials or you just decided to stop on a whim.
		A deeper discussion of the topic is available in the article about the maximum likelihood principle.
	nostalgia.wikipedia.org /wiki/Likelihood_principle (207 words)

	Citations: Cambridge University Press - Edwards (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		A maximum likelihood method of inference chooses the hypothesis H maximising the likelihood function for....
		As a principle, ML is backed by n s asymptotic efficiency in the repeated sampling paradigm (under some regularity conditions) and its attainment of the Cramer Rao information lower bound in many exponential family....
		As a principle, ML is backed by # n s asymptotic e#ciency in the repeated sampling paradigm #under some regularity conditions# and its attainment of the Cramer Rao information lower bound in many exponential family....
	citeseer.ist.psu.edu /context/309791/0 (1676 words)

	How not to Detect Design, Review of "The Design Inference": Fitelson, Stephens, and Sober
		The likelihood of a hypothesis is the probability it confers on the observations; it is not the probability that the observations confer on the hypothesis.
		The likelihood of H relative to E is Pr(EH), not Pr(HE).
		As noted earlier, that principle states that if two hypotheses confer different probabilities on the same observations, the one that entails the higher probability is the one that is better supported by those observations.
	www.arn.org /docs/dembski/wd_wisconsinureview.htm (5989 words)

	Likelihood principle and inference (Site not responding. Last check: 2007-10-21)
		For a given prior distribution, if the likelihood functions p1(x1?),p2(x2
		Inference procedures should lead to identical inferences when applied to proportional likelihoods.
		Once the data has been observed, the likelihood function contains all the relevant information about the unknown parameter.
	www.stat.duke.edu /~dalene/talks/ncssm/tsld059.htm (53 words)

	Likelihood and Probability in R
		On the Mathematical Foundations of Theoretical Statistics (1922) was not a likelihood text in the same sense.
		Although likelihood inference was proclaimed as a fundamental form of inference in all editions of Statistical Methods for Research Workers, it was not developed any further for more than 30 years.
		Likelihood inference re-appears in Statistical Methods and Scientific Inference (1956) and the discussion there picks up from where the 1921 paper and Statistical Methods for Research Workers had left off, “The likelihood supplies a natural order of preference among the possibilities under consideration” (chapter III, §6).
	www.economics.soton.ac.uk /staff/aldrich/fisherguide/prob+lik.htm (859 words)

	Likelihood Ratio Test (Site not responding. Last check: 2007-10-21)
		variables, was selected in preference to logistic regression for the analyses, because it has the advantage of calculating the likelihood ratio test for each...
		A likelihood-ratio test is a statistical test relying on a test statistic computed by taking the ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum with that constraint relaxed.
		If that ratio is Λ and the null hypothesis holds, then for commonly occurring families of probability distributions, −2 log Λ has a particularly handy asymptotic distribution.
	www.wikiverse.org /likelihood-ratio-test (638 words)

	Peter van der Helm: Psychological Bulletin (2000) Simplicity versus Likelihood (Site not responding. Last check: 2007-10-21)
		The likelihood principle states that the visual system prefers the most likely interpretation of a stimulus, whereas the simplicity principle states that it prefers the most simple interpretation.
		It is argued that, in visual perception, the two principles are perhaps very different with respect to the viewpoint-independent aspects of perception but probably very close with respect to the viewpoint-dependent aspects which, moreover, seem decisive in everyday perception.
		This implies that either principle may have guided the evolution of visual systems and that the simplicity paradigm may provide perception models with the necessary quantitative specifications of the often plausible but also intuitive ideas provided by the likelihood paradigm.
	www.nici.kun.nl /~peterh/abstracts/psybull_precisal.html (199 words)

	Profile likelihood (Site not responding. Last check: 2007-10-21)
		In the section on likelihood based confidence region we used the tangent parabola, rather than the likelihood ratio itself.
		The profile likelihood is then the skyline of this landscape in one dimension.
		The practical problem in the application of profile likelihoods is that for each value of the parameter under consideration, the MLE's for the other parameters have to be obtained, which might be computationally intensive.
	www.bio.vu.nl /thb/course/tb/tb/node92.html (282 words)

	Maximum likelihood estimation
		The maximum likelihood principle is a very popular approach to obtain practical estimators.
		It has been proved in [14] that the maximum likelihood estimator (MLE) has the asymptotic properties (for large data records) of being unbiased and achieving the Cramér-Rao lower bound .
		The maximum likelihood method is based on the principle that I should be estimated by its most likely value given the observations S, i.e.
	dutnsic.tn.tudelft.nl:8080 /main/node17.html (1093 words)

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