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Topic: Prior probability distribution

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	Prior probability - Biocrawler (Site not responding. Last check: 2007-09-06)
		A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence.
		As prior and posterior are not terms used in frequentist analyses, this article uses the vocabulary of Bayesian probability and Bayesian inference.
		And in the continuous case, the maximum entropy prior given that the density is normalized with mean zero and variance unity is the standard normal distribution.
	www.biocrawler.com /encyclopedia/Prior_distribution (1325 words)

	NationMaster - Encyclopedia: Prior probability distribution
		Bayes' rule is the recipe for computing the new probability distribution for the proportion, called the posterior, based on knowledge of the prior probability distribution and the sample survey data.
		The notion of a sampling distribution is perhaps the most difficult concept, since the student is asked to think about the variation in samples other than the one that he or she observed.
		The prior probability distribution can be used to state initial beliefs about the population of interest, relevant sample data is collected, and the posterior probability distribution reflects one's new beliefs about the population in light of the data that were collected.
	www.nationmaster.com /encyclopedia/Prior-probability-distribution (952 words)

	Evidence: Victim's Research Paper
		Essentially, the choice between probability levels of Type I and Type II errors is the basis on which the relative importance of two alternative types of mistakes are assessed in hypothesis testing in classical statistical analysis8.
		In Bayesian decision analysis, this type of probability distribution is often subjective, in that the data upon which it is based is itself based on the judgments of individuals.
		The prior probability distribution is descriptive of the uncertainty which is associated with the decision maker's estimate of the probability of occurrence of a random event.
	www.crimescene.com /stadium/evidence.term.paper.php (2036 words)

	NationMaster - Encyclopedia: Jeffreys prior
		Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement.
		In Bayesian probability theory, a conjugate prior is a prior distribution which has the property that the posterior distribution is the same type of distribution.
		In Bayesian probability, the Jeffreys prior is a noninformative prior distribution proportional to the square root of the Fisher information: and is invariant under reparameterization of.
	www.nationmaster.com /encyclopedia/Jeffreys-prior (348 words)

	UCLA Soc. 210A, Topic 5, Conditional Probability
		The reason for choosing beta distributions is that they are "conjugate" to the process of estimating a population proportion, which means that a beta prior yields a beta posterior as well, only shifting the parameter values (Leamer pp 40-51; Winkler and Hays pp 498-506).
		Notice that the numbers in the prior probability column are all non-negative, and sum to 1.0, as required of a probability distribution.
		To complete the calculation of posterior probabilities, each of the likelihoods is multiplied by the corresponding prior, in the 4th column, and finally each of those products is divided by their sum, yielding the posterior probabilities in the 5th column.
	www.sscnet.ucla.edu /soc/faculty/mcfarland/soc210a/prob2.htm (2605 words)

	Prior probability - Wikipedia, the free encyclopedia
		A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence.
		And in the continuous case, the maximum entropy prior given that the density is normalized with mean zero and variance unity is the standard normal distribution.
		We could specify, say, a normal distribution as the prior for his speed, but alternatively we could specify a normal prior for the time he takes to complete 100 metres, which is proportional to the reciprocal of the first prior.
	en.wikipedia.org /wiki/Prior_probability_distribution (1485 words)

	Bayesian sequential indicator simulation of lithology from seismic data - Patent 5416750
		In an aspect of this invention, the likelihood function is derived from the lithoclass-conditional probability distributions of the seismic attributes.
		3 is the prior sand/shale probability distribution for pixel x.sub.21 ;
		The prior sand/shale probability distribution as determined from Indicator Kriging at pixel i, is multiplied by the corresponding likelihood function to define the posterior lithoclass probability distribution p (x.sub.i.vertline.z.sub.i, x.sub.l,.
	www.freepatentsonline.com /5416750.html (2857 words)

	CDC - Estimating Time and Size of Bioterror Attack
		The incubation time distribution for anthrax has been estimated by Brookmeyer and colleagues on the basis of the Swerdlovsk outbreak (4); data describing the incubation distribution for smallpox are summarized by Fenner et al.
		At the moment the attack is detected, consistent with Bayesian principles (7), we presume a prior probability distribution (henceforth, prior) governing the size of the attack.
		The joint prior distribution of N and A under the stated assumptions is shown in Appendix Figure 1; Appendix Figure 2 displays the joint posterior distribution for N and A after a total of 23 cases have been observed by the end of 5 days after the first case was reported (Figure 1).
	www.cdc.gov /ncidod/Eid/vol10no7/03-0632.htm (2411 words)

	Re: Bayesian & Frequentist Probability Theory
		The MAP parameter estimate is the set of parameter values that maximizes the conditional probability distribution (discrete case) or conditional probability density (continuous case) of the parameters, _given_ the observed data.
		It requires that one supply a prior probability distribution for the parameters from which to calculate the a posteriori distribution of parameter values given observed values from the conditional distribution of observed values given parameter values provided by the model.
		The ML parameter estimate is the set of parameter value that maximizes the probability distribution or density of observed values given the parameter values provided by the model, when the latter is evaluated at the values observed.
	www.lns.cornell.edu /spr/2002-03/msg0040334.html (526 words)

	DeBovis's PhD Abstract (Site not responding. Last check: 2007-09-06)
		That is, prior information is used to estimate the expected gain in mean survival (terminal utility) obtained by sampling with n patient, n greater-than and/or equal to 1, and to estimate the expected risk (cost) to those sampled.
		The prior distribution assigned to the parameters of the survival distribution is used to estimate the expected value of both perfect and sample information.
		The expected risk is based on a prior difference in mean survival of interest and is determined by the prior probability distributions assigned to the parameters of the survival distribution.
	www.sph.emory.edu /bios/news/library/debovis.html (353 words)

	Free Data Mining Source Code - Prior Probability
		A prior probability distribution, often called simply the prior, of an uncertain quantity p is the probability distribution that would express one's uncertainty about p before the "data" are taken into account.
		When we multiply the prior by the likelihood function and then do normalizing, we get the posterior probability distribution, which is the conditional distribution of the uncertain quantity given the data.
		Prior probability is assessed in the absence of empirical data, or which may incorporate pre-existing data and information.
	www.kdkeys.net /forums/2899/ShowPost.aspx (251 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-09-06)
		Multiplying the prior probability of the hypothesis by this factor would result in a large posterior probability of the hypothesis given the evidence.
		Either the defendant is guilty (with prior probability 0.3) and thus his DNA is present with probability 1, or he is innocent (with prior probability 0.7) and he is unlucky enough to be one of the 1 in a million matching people.
		A conjugate prior is a prior distribution, such as the beta distribution in the above example, which has the property that the posterior is the same type of distribution.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Bayesian_decision_theory (3501 words)

	Field mice: Extracting hand geometry from electric field measurements
		Figure 2 shows two distributions that, by construction, give the same loading measurements: the four small dark objects that comprise the second distribution can be moved in from infinity until the signals are the same as those from the first distribution.
		Assuming a Gaussian approximation to the posterior probability distribution, the inverse curvature of a peak in a particular direction gives the uncertainty of the estimate of the parameter value (or linear combination of parameter values) corresponding to that direction.
		To invert the signals we maximize the log probability, which corresponds to minimizing a prior term plus the sum of squared error between the measured value and that predicted by the current estimate of the hand position.
	www.research.ibm.com /journal/sj/353/sectione/smith.html (7390 words)

	A General Objective for Inductive Inference
		Further, we show how the specification language for theories defines a prior probability distribution over families of models and a framework in which prior expectations about simple components of the model may be combined.
		Part of the specification of a PTM is the specification of the probabilities of moves.
		Given a random sample from an unknown multivariate distribution, numerical taxonomy or classification techniques attempt to construct and test hypotheses that treat the population as comprising several distinct classes.
	www.csse.monash.edu.au /~lloyd/tildeMML/Structured/TR32 (4263 words)

	[No title] (Site not responding. Last check: 2007-09-06)
		When a prior probability distribution on this nuisance parameter is given, the marginal distribution is the classical tool to account for it.
		If the prior distribution is not given, but we have partial knowledge such as a fixed number of moments, we can use the maximum entropy principle to assign a prior law and thus go back to the previous case.
		It is obtained by first remarking that the marginal distribution can be considered as the mean value of the original distribution with respect to the prior probability law of the nuisance parameter, and then, by using the median in place of the mean.
	www.mdpi.org /entropy/htm/e8020067.htm (195 words)

	Workshop Statistics: Discovery with Data, A Bayesian Approach, Sample Survey Report
		After the prior distribution for the proportion has been constructed and the data are taken, we use the methodology of Topic 16 to compute the posterior probability distribution.
		You enter the prior distribution into two columns of the spreadsheet, input the number of yes's and no's, and the program computes the posterior distribution.
		You use this probability distribution to construct a probability distribution for p as described in Topic 16.
	www.keycollege.com /ws/Bayesian/project_assignment.htm (1026 words)

	Bayesian inference
		O’Hagan 1994 p 74 et seq.) that whatever prior probability is used at the outset, Bayes’s theorem ensures that everyone is driven towards the same conclusion as the data accumulate.
		where Prob(Null) is the prior probability for the null hypothesis that there is no real difference in the toxin level in the children, and BF is the so-called Bayes Factor, which measures how much we should alter our prior belief about the null hypothesis in the light of the new data, as captured by z.
		The prior information was captured through a probability distribution which peaked at an RR of 0.83 while giving low probabilities to RRs greater than 1.0 (no benefit) or less than 0.6 (dramatic improvement).
	ourworld.compuserve.com /homepages/rajm/twooesef.htm (7413 words)

	What is Bayesian Inference? (Site not responding. Last check: 2007-09-06)
		Of course very little may be known and in such a case the prior is chosen so that it does not discriminate sharply amongst the possible values.
		In any case once we have determined the prior probability distribution for the unknowns we can combine the prior for w, b and g together with the conditional distribution for the data, as given by the likelihood above, to get the joint distribution of the data and the unknowns.
		This is known as a particular case of a Dirichlet distribution and algorithms are available for the calculation of various probabilities and expectations associated with this class of distributions.
	www.bayesian.org /bayesian/whatis.html (1358 words)

	MrBayes
		The prior distribution represents your prior beliefs about the parameter before observation of the data.
		Symdirihyperpr specifies the distribution on the variance parameter of the dirichlet.
		You must specify a variable rate prior for at least two partitions, otherwise the option is not activated when calculating likelihoods.
	mrbayes.csit.fsu.edu /Help/prset.html (2195 words)

	For Debate: The statistical basis of public policy: a paradigm shift is overdue -- Lilford and Braunholtz 313 (7057): ...
		Thus the probability P of a fair that it provides probability distributions for coin landing heads up is 0.5 because in a long series of parameters--which is exactly what is needed to inform tosses it lands heads up half the time.
		A Bayesian might judge the value of P changes in prior probability distributions, or indeed to to be close to 0.5, without the need for any previous changes in the model used to create the likelihood.
		In this case the model we have in conflict with this--for instance, when calculating P assumed specifies that the probability distribution for the values, which take into account the probability of obser- "observed" log relative risk will be normal with a mean of vations more extreme than the actual observations.
	bmj.bmjjournals.com /cgi/content/full/313/7057/603 (5228 words)

	Worksheet to Construct Prior Probability Distribution for Project (Site not responding. Last check: 2007-09-06)
		Worksheet to Construct Prior Probability Distribution for Project
		We will assume in defining your prior that p can only be one of the values 0,.1,.2,....
		Assign the number 10 to the value of p that you think is most likely.
	www-math.bgsu.edu /~albert/make_prior.htm (208 words)

	For Debate: The statistical basis of public policy: a paradigm shift is overdue -- Lilford and Braunholtz 313 (7057): ...
		Bayesian statistics The key difference between Bayesian and conven- tribution is formed by weighting the prior probability tional (or frequentist) statistics is the view of what prob- distribution by the likelihood.
		These probabilities cial interest, which in some way summarise the can be plotted on a graph, and, when thought of as pro- interesting facets of the data.
		The controversy relates (1731) and Introduction to the Doctrine of Fluxions (1736) to deriving the probability of statistical hypotheses (prior are the only works known to have been published during his probability distributions), especially before any data of lifetime.
	www.bmj.com /cgi/content/full/313/7057/603 (5186 words)

	A Dirichlet process model for detecting positive selection in protein-coding DNA sequences -- Huelsenbeck et al. 103 ...
		(0, 1); a gamma distribution, truncated on the interval (0.
		probability of all of the parameters of the model is
		a stationary distribution that is the posterior probability
	www.pnas.org /cgi/content/full/103/16/6263 (3393 words)

	Reply by Guthrie Miller et al.
		For example, the U.S. Nuclear Regulatory Commission has stated that "it is generally believed by most probability risk assessment (PRA) analysts that, for PRA application to complex systems, such as nuclear power plants, the advantages of Bayesian methods outweigh the disadvantages." Thus, Bayesian methods are widely used in PRA.
		Note that the Bayesian method, unlike the classical method, gives us exactly what we are interested in, namely the probability distribution of true counts given the measurement result.
		A generalized question we have frequently encountered in litigation cases, from both plaintiff and defendant counsel, is what is the probability that the dose is less than some value of interest.
	www.pnl.gov /bayesian/recent/reply.htm (781 words)

	Bayes
		Ultimately, one has to face the fact that probability cannot be usefully defined in terms of the frequency of occurence of some event over a large (or infinite) number of trials.
		In the jargon of probability theory, the frequentist interpretation of probability is wrong.
		Given the prior one can >compute probabilities, but without a prior one can't compute the >probability that someone else's predictions are right or not, so one >can't "judge" a prior without reference to some other prior.
	math.ucr.edu /home/baez/bayes.html (8656 words)

	The accumulation of evidence (Site not responding. Last check: 2007-09-06)
		Starting with a uniform prior probability distribution for object identity, evidence is gathered around the viewsphere of an unknown object.
		Given the first image, the probability of observing a particular object is determined by Bayes rule: it is the product of the prior probability of that object with the likelihood distribution, determined by inverting the distribution found during training.
		This is achieved by using the posterior distribution found at the previous iteration as the new prior distribution.
	www.cim.mcgill.ca /~cathy/report526/node31.html (230 words)

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