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Topic: Conjugate prior

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	Conjugate prior - Wikipedia, the free encyclopedia
		The concept, as well as the term "conjugate prior", was introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian decision theory.
		A conjugate prior is an algebraic convenience: otherwise a difficult numerical integration may be necessary.
		All members of the exponential family have conjugate priors.
	en.wikipedia.org /wiki/Conjugate_prior (429 words)

	Meaning and role of the prior: many data limit versus frontier type measurements
		The beta distribution is the conjugate prior of the binomial distribution, i.e.
		Anyway, instead of playing blindly with mathematics, looking around for `objective' priors, or priors that come from abstract arguments, it is important to understand at once the role of prior and likelihood.
		Priors are logically important to make a `probably inversion' via the Bayes formula, and it is a matter of fact that no other route to probabilistic inference exists.
	www.roma1.infn.it /~dagos/proportions/node4.html (1086 words)

	Conjugation - Wikipedia, the free encyclopedia
		A conjugate in algebra is similar to a complex conjugate, but is used to rationalize the denominator of a fraction.
		Conjugate prior, in probability theory, a family of prior probability distributions.
		This includes bacterial conjugation, which takes place without fusion; conjugation in ciliates, which involves fusion of nuclei but not cells; and conjugation in certain algae and fungi, which involves fusion of cells but not filaments.
	en.wikipedia.org /wiki/Conjugate (281 words)

	Implement Bayesian inference using PHP: Part 2
		The concept of conjugate priors is attractive from the point of view of implementing Bayes networks and imagining how you might propagate information from parent nodes to child nodes.
		Another attractive feature of the conjugate prior updating rule is that it is recursive and, in the limit, can be used to tell you how to update your posterior probabilities on the basis of a single new observation (another exercise for you to think about).
		The use of conjugate priors is not, however, without its critics who argue that the mindless use of conjugate priors abrogates a Bayesian's responsibility to use all information at his disposal to represent prior knowledge about a parameter.
	www-128.ibm.com /developerworks/web/library/wa-bayes2 (5017 words)

	eMedicine - Hyperbilirubinemia, Conjugated : Article Excerpt by: Richard A Weisiger, MD
		Accumulation of bilirubin or its conjugates in body tissues produces jaundice (ie, icterus), which is characterized by high plasma bilirubin levels and deposition of yellow bilirubin pigments in skin, sclerae, mucous membranes, and other less visible tissues.
		Diseases that reduce the rate of secretion of conjugated bilirubin into the bile or the flow of bile into the intestine produce a mixed or predominantly conjugated hyperbilirubinemia due to reflux of conjugates back into the plasma.
		Pathophysiology: Conjugated hyperbilirubinemia results from reduced secretion of conjugated bilirubin into the bile, such as occurs in patients with hepatitis, or it results from impaired flow of bile into the intestine, such as occurs in patients with biliary obstruction.
	www.emedicine.com /med/byname/hyperbilirubinemia-conjugated.htm (601 words)

	PyMML 0.5
		The form of prior used for this estimator is a "conjugate prior".
		The conjugate form leads to a fairly natural choice of prior, and this allows more confident comparison of regression estimates based on different sets of data -- which is the main reason you would want to use MML regression.
		Individual priors are given by a tuple containing the class of estimator to use for that field followed by the parameters of the prior to be passed to that estimator.
	www.logarithmic.net /pymml-doc/doc.html (2329 words)

	8.1.10. How can Bayesian methodology be used for reliability evaluation?
		Prior knowledge is not used except to suggest the choice of a particular population model to "fit" to the data, and this choice is later checked against the data for reasonableness.
		Bayes formula is a useful equation from probability theory that expresses the conditional probability of an event A occurring, given that the event B has occurred (written P(AB)), in terms of unconditional probabilities and the probability the event B has occurred, given that A has occurred.
		For example, the Beta distribution model is a conjugate prior for the proportion of successes p when samples have a binomial distribution.
	www.itl.nist.gov /div898/handbook/apr/section1/apr1a.htm (1039 words)

	How to put error bars on histograms
		The conjugate prior/posterior to the multinomial distribution is the Dirichlet distribution.
		To study the effect of the smooting parameter s on the credibility band, the experiment of Figure 21 was repeated for a range of s-values and the average width of the credibility band was calculated for each of them.
		No smoothing prior was used for the construction of Figure 23 because it is not our goal to constrain the distribution of the underlying population, but only to see if the different observations are compatible with each other.
	pangea.stanford.edu /research/noble/epdu/paper (8238 words)

	ALK (Alkaline Phosphatase) Conjugate Stabilizers from Research Diagnostics Inc
		Conjugates prepared in this manner will maintain AP activity for up to 2 years at 4 DEG C. (actual time must be determined with your particular conjugate and assay characteristics).
		Ideally, sterile filter your conjugate prior to addition to the stabilizer (or after dilution for maximum stabilizing effect).
		NOTE: Stability of HRP conjugates is prolonged by keeping conjugates cool and reducing exposure to light and oxygen.
	www.researchd.com /rdioem/alkstab.htm (936 words)

	Statistical Inference (Site not responding. Last check: 2007-09-10)
		It is convenenient if, starting from a prior xi(theta) in a particular family and given an observation x, we end up with posterior distribution xi(thetax) from the same family.
		This is the defining property for a family of conjugate prior distributions.
		In cases in which the experimenter feels uncomfortable assigning a prior xi(theta), a maximum likelihood estimator is often used instead of a Bayes estimator.
	www.cs.brown.edu /research/ai/dynamics/tutorial/Documents/StatisticalInference.html (792 words)

	Conjugate priors
		Because of computational problems, modelling priors has been traditionally a compromise between a realistic assessment of beliefs and choosing a mathematical function that simplifies the analytic calculations.
		A well-known strategy is to choose a prior with a suitable form so the posterior belongs to the same functional family as the prior.
		For instance, given a Gaussian likelihood and choosing a Gaussian prior, the posterior is still Gaussian, as we have seen in Eqs.
	www.roma1.infn.it /~dagos/rpp/node31.html (323 words)

	Neal Vs. ME
		The Entropic Prior for the parameters of a Gaussian mixture turns out to be very similar to the popular conjugate prior except that the uncertainty on the parameters of each component depends on the weight assigned to that component.
		The proper prior p(params) that minimizes I(P:Q), when data consists of alpha independent observations of the model, is the Entropic Prior with parameters h and alpha.
		The only prior information that we assume that we have about the beatles is the one in the likelihood and the parameters of the entropic prior (h and alpha).
	omega.albany.edu:8008 /about-0201016.html (877 words)

	Whatman - Leadership in separations technology for the life sciences
		Conjugate release materials are critical to lateral flow assays.
		To ensure strong assay performance, the conjugate must dry without damage or aggregation and release rapidly when the sample is applied.
		Do not require treatment prior to conjugate application, decreasing time required for diagnostic kit production.
	www.whatman.com /products/?pageID=7.32.40 (250 words)

	Lectures
		This is not usually how priors are built though because it seems quite an exhaustive process to build up a whole density prior, instead we are going to use families of priors who have easy updating processes with regards to the specific likelihoods at hand.
		Sometimes a prior distribution can be approximated by one that is in a convenient family of distributions, which combines with the likelihood to produce a posterior that is manageable.
		We see that an ``objective'' way of building priors for the binomial parameter was to use the `conjugate family' distribution that has the property that the updated distribution is in the same family.
	www-stat.stanford.edu /~susan/courses/s166/node2.html (1942 words)

	Gaussian node
		The conjugate prior distribution of the inverse of the prior variance of a Gaussian random variable is the gamma distribution Gelman95.
		Using such gamma prior pdf causes the posterior distribution to be gamma, too, which is mathematically convenient.
		However, the conjugate prior pdf of the second parameter of the gamma distribution is something quite intractable.
	www.cis.hut.fi /praiko/papers/bayes_blocks/node8.html (205 words)

	Bayesian Parameter Estimation
		Not surprisingly, the probability of heads is estimated as the empirical frequency of heads in the data sample.
		The Beta distribution is conjugate to the binomial distribution which gives the likelihood of iid Bernoulli trials.
		As we will see, a conjugate prior perfectly captures the results of past experiments.
	www-ccrma.stanford.edu /~jos/bayes/Bayesian_Parameter_Estimation.html (413 words)

	The model
		These should be written as conditional distributions conditional to the parameters of the hyperprior but the conditioning variables have been dropped out to simplify the notation.
		Their values should be chosen to reflect true prior knowledge on the possible initial states and transition probabilities of the chain.
		The conjugate prior for variance of a Gaussian is the inverse gamma distribution.
	www.cis.hut.fi /ahonkela/dippa/node49.html (314 words)

	HRP Conjugate Stabilizers from RDI Division of Fitzgerald Industries Intl/RDI Divison of Fitzgerald Industries Intl
		Conjugates prepared in this manner will maintain HRP activity for up to 2 years at 4 DEG C. (actual time must be determined with your particular conjugate and assay characteristics).
		If you have the capabilities and you have a very unstable conjugate, you may also wish to add the components, mix in a teflon, glass or plastic (not metal) lined container under a nitrogen atmosphere.
		The varied stability/purity of the HRP conjugate itself will have a great effect on the stability of any stabilizer to be of help.
	www.researchd.com /rdioem/hrpstab.htm (1441 words)

	Prior distribution assessment for a multivariate normal distribution: an experimental study
		Prior distribution assessment for a multivariate normal distribution: an experimental study
		A variety of methods of eliciting a prior distribution for a multivariate normal (MVN) distribution have recently been proposed.
		Our results compare prior models and show, in particular, that it can be better to assume the mean and variance of an MVN distribution are independent a priori, rather than to model opinion by the conjugate prior distribution.
	ideas.repec.org /a/taf/japsta/v28y2001i1p5-23.html (358 words)

	Conjugate Analysis of Multivariate Normal Data with Incomplete Observations (ResearchIndex)
		Abstract: In this article we discuss families of prior distributions that are conjugate to the multivariate normal likelihood when some of the observations are incomplete.
		We present a general class of priors, modifying a proposal of Kadane and Trader, to allow incorporation of information about unidentified parameters in the covariance matrix within a conjugate setting.
		We analyze the important special case of monotone patterns of missing data, providing an explicit recursive form for the posterior...
	citeseer.ist.psu.edu /98833.html (498 words)

	bayes04lab3
		R-codes for Normal sample with conjugate priors for the mean
		An expert has a prior belief that the median of the new system should be close to.8.
		Triplot of the prior, likelihood, and the posterior distribution in one figure using the function 'nn.trip':
	cc.oulu.fi /~hyon/bayes04lab3.html (1056 words)

	A Bayesian Characterization of Hardy-Weinberg Disequilibrium -- Shoemaker et al. 149 (4): 2079 -- Genetics (Site not responding. Last check: 2007-09-10)
		based on prior A. For the posterior distribution, the data were from the LDLR locus in the FBI Caucasian data base.
		based on prior C. For the posterior distribution, the data were from the LDLR locus in the FBI Caucasian data base.
		The posterior probabilities for prior A and priors B or C were
	www.genetics.org /cgi/content/full/149/4/2079 (3735 words)

	Bayesian inference - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Bayesian_inference (3653 words)

	A Bayesian Factor Analysis Model With Generalized Prior Information - Rowe (ResearchIndex) (Site not responding. Last check: 2007-09-10)
		Abstract: In the Bayesian approach to factor analysis, available prior knowledge regarding the model parameters is quantified in the form of prior distributions and incorporated into the inferences.
		The incorporation of prior knowledge has the added consequence of eliminating the ambiguity of rotation found in the traditional factor analysis model.
		Incorporating Prior Knowledge Regarding the Mean in Bayesian..
	citeseer.ist.psu.edu /345257.html (517 words)

	CiteULike: Bayesian Multioutput Feedforward Neural Networks Comparison: A Conjugate Prior Approach (Site not responding. Last check: 2007-09-10)
		As opposed to classic point-prediction-based cross-validation methods, this expected utility is defined from the logarithmic score of the neural model predictive probability density.
		It is shown how the advocated choice of a conjugate probability distribution as prior for the parameters of a competing network, allows a consistent approximation of the network posterior predictive density.
		A comparison of the performances of the proposed method with the performances of usual selection procedures based on classic cross-validation and information-theoretic criteria, is performed first on a simulated case study, and then on a well known food analysis dataset.
	www.citeulike.org /user/drmabuse/article/507847 (293 words)

	8.1.10. How can Bayesian methodology be used for reliability evaluation?
		Old information, or subjective judgment, is used to come up with a prior distribution for these population parameters.
		It makes a great deal of practical sense to use all the information available, old or new, objective or subjective, when making decisions under uncertainty.
		Main stream statistical analysis, however, seeks objectivity by generally restricting the information used in an analysis to that obtained from a current set of clearly relevant data.
	www.itl.nist.gov /div898/handbook/apr/section2/apr1a.htm (950 words)

	Dialnet: A Common Conjugate Prior Structure For Several Randomized Response Models
		Suitable truncated beta distributions are used throughout in a common conjugate prior structure to obtain the Bayes estimates for the proportion of a "sensitive" attribute in the population of interest.
		The results of this common conjugate prior approach are contrasted with those of Winkler and Franklin's (1979), in which non-conjugate priors have been used in the context of Warner's model.
		The results are illustrated numerically in several cases and exemplified further with data reported in Liu and Chow (1976) concerning incidents of induced abortions.
	dialnet.unirioja.es /servlet/oaiart?codigo=752268 (152 words)

	[No title]
		Note: The prior N(0,1) converts the posterior pdf of (x to another Normal pdf.
		We say N(0,1) is a conjugate prior for N((,(2) when (2 known.
		In this case, N(a,b2) is a conjugate prior for N((,(2) when (2 is known.
	www.sci.wsu.edu /math/faculty/jpascual/math556/handouts/ch9-2.doc (327 words)

	How to use the Bayes Net Toolbox
		This prior does not satisfy the likelihood equivalence principle, which says that Markov equivalent models should have the same marginal likelihood.
		When you specify a prior, you should set row i of the CPT to the normalized version of row i of the pseudo-count matrix, i.e., to the expected values of the parameters.
		B Models 2 and 3 are Markov equivalent, and therefore indistinguishable from observational data alone, so we expect their posteriors to be the same (assuming a prior which satisfies likelihood equivalence).
	bnt.sourceforge.net /usage.html (11383 words)

	[No title]
		The use of conjugate families for prior distributions is often criticized as being too restrictive.
		However, we can increase the flexibility of these families by considering mixtures of conjugate prior distributions.
		(See also Problem 5.6 in BDA) (a) Show that the posterior distribution is also a mixture of distributions from the conjugate family.
	www.sph.umich.edu /~qin/biostat682/homework3.doc (226 words)

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