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# Topic: Posterior probability distribution

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 Posterior probability - Wikipedia, the free encyclopedia In Bayesian probability theory, the posterior probability of an uncertain proposition is its conditional probability given empirical data into account. Compare with prior probability, which may be assessed in the absence of empirical data, or which may incorporate pre-existing data and information. Similarly a posterior probability distribution is the conditional probability distribution of the uncertain quantity given the data. en.wikipedia.org /wiki/Posterior_probability   (176 words)

 Prior probability - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-14) The posterior probability is computed from the prior and the likelihood function via Bayes' theorem. As prior and posterior are not terms used in frequentist analyses, this article uses the vocabulary of Bayesian probability and Bayesian inference. 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   (1355 words)

 Prior probability - Wikipédia A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. The posterior probability is then the conditional probability of the variable taking the evidence into account. Some attempts have been made at finding probability distributions in some sense logically required by the nature of one's state of uncertainty; these are a subject of philosophical controversy. su.wikipedia.org /wiki/Prior_probability   (545 words)

 Téoréma Bayes - Wikipédia Teorema Bayes ngarupakeun hasil dina tiori probabiliti, which gives the conditional probability distribution of a variabel acak A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. In the context of Bayesian probability theory and statistical inference, the marginal probability distribution of A alone is usually called the prior probability distribution or simply the prior. The conditional distribution of A given the "data" B is called the posterior probability distribution or just the posterior. su.wikipedia.org /wiki/Bayes'_theorem   (1668 words)

 The Posterior Probability Distribution of Alignments   (Site not responding. Last check: 2007-10-14) The Posterior Probability Distribution of Alignments and its Application to Parameter Estimation of Evolutionary Trees and to Optimisation of Multiple Alignments. P(H) is the prior probability of the hypothesis H. P(HD) is the posterior probability of H. The message length (ML) of an event E is the minimal length, in bits, of a message to transmit E using an optimal code. The probability of a particular value `j' is proportional to the sum of the probabilities of all alignments that involve its choice ie. www.csse.monash.edu.au /~lloyd/tildeStrings/Multiple/94.JME   (7580 words)

 Probability Distribution for Exact Nonlinear Estimation A prior knowledge of the error of linearization is considered to estimate a posterior probability distribution of the estimated parameters properly. With such a distribution the computation of the posterior probability distribution is almost as easy as in the linear and normal case. Probability density functions of the both distributions are compared in Figure 1. www.utia.cas.cz /user_data/scientific/ZOI_dept/nonlin.html   (281 words)

 Bayes' theorem   (Site not responding. Last check: 2007-10-14) Bayes' theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. The likelihood function for such a problem is just the probability of 7 successes in 10 trials for a binomial distribution. The prior probability that more than half the voters will vote "yes" is 1/2, by the symmetry of the uniform distribution. www.bidprobe.com /en/wikipedia/b/ba/bayes__theorem.html   (1600 words)

 Expected A Priori (EAP) Estimation of Latent Trait Scores In this context it refers to a posterior probability distribution of latent trait scores--specifically, the predicted distribution of scores for a given case given (a) the response pattern of that case, and (b) the estimated model parameters. Distribution of the posterior probability of latent trait level T given estimated model parameters pi and a case with response pattern X. Recall that in Eq. The prior distribution Pr[T(s)] is the density of the multivariate normal distribution corresponding to T(s). ourworld.compuserve.com /homepages/jsuebersax/irt2.htm   (2332 words)

 Belief networks The basic task for a BBN is to compute the posterior probability distribution for a set of query variables, given exact values for some evidence variables. For each simulation trial, the probability of the evidence given the sampled state values is used to increment the count of each event of interest. The estimated probability distribution is obtained by normalizing after all the simulation trials are completed. www.cs.ualberta.ca /~jonathan/Grad/pena/node50.html   (366 words)

 Viewing results   (Site not responding. Last check: 2007-10-14) Posterior probability distribution of any node in a Bayesian network can be viewed by bringing the cursor over the Updated [ ] status icon. An alternative way of viewing the posterior probability distribution is to choose the Value tab from the Node Property sheet. In cases when a node's probability distribution is affected by a decision or by a node that precedes a decision node, the posterior probability distribution is indexed by the outcomes of these nodes. www.sis.pitt.edu /~genie/GeNIeHelp/ViewingResults.htm   (329 words)

 Workshop Statistics: Discovery with Data, A Bayesian Approach, An Introduction to Bayesian Thinking The value of the proportion p is unknown, and a person expresses his or her opinion about the uncertainty in the proportion by means of a probability distribution placed on a set of possible values of p. From the set of posterior probabilities, one finds that the probability the proportion value is less than or equal to one-half is.032. Note that the posterior probability of the hypothesis H is approximately equal to the classical p-value. www.keycollege.com /ws/Bayesian/tutorial/a_brief_tutorial.htm   (1171 words)

 paper   (Site not responding. Last check: 2007-10-14) Psychometric function of the probability of a correct response in a two-alternative method, where the mean probability of correct at zero stimulus intensity is 0.5. The posterior probability distribution is the distribution of probability correct at each stimulus level that has been used so far. However, the variance of the posterior probability distribution, which sets this goal for the Minimum Variance Method, cannot be readily expanded to two dimensions because the threshold and slope dimensions are incommensurate with each other. www.ski.org /cwt/CWTyler/Pubtopics/PsiMethod/Psi.html   (3777 words)

 BEAST: Bayesian Evolutionary Analysis Sampling Trees Bayesian inference - Bayesian inference is a branch of statistical inference that permits the use of prior knowledge in assessing the probability of model parameters in the presence of new data. The posterior probability distribution - The posterior (or posterior probability density) is the entity that an MCMC analysis attempts to obtain an estimate of. The posterior is the probability distribution over the parameter state space, given the data under the chosen model of evolution. evolve.zoo.ox.ac.uk /beast/glossary.html   (822 words)

 Markov Chain Monte Carlo   (Site not responding. Last check: 2007-10-14) The standard situation is where it is difficult to normalize a posterior probability distribution. The distribution of the final state will be approximately that of one chosen randomly from the posterior distribution. In practice, if you are interested in a parameter value such as the proportion of contingency tables with the same row and column sums as a given table with a chi-square statistic greater than that of the given table, the long-run proportion of tables in the sequence will converge to this number. www.mathcs.duq.edu /larget/math496/mcmc.html   (351 words)

 BioMed Central | Full text | Molecular phylogeny of Subtribe Artemisiinae (Asteraceae), including Artemisia and its ... Posterior probabilities are indicated on the ML tree (Fig. 1), i.e., 67% posterior probability as estimated by MrBayes and unsupported as measured by parsimony bootstrap analysis (<50%). Bayesian analysis uses Markov Chain Monte Carlo (MCMC) methods to approximate posterior probability distributions that are a direct estimation of branch support because they are the true probabilities of the resulting clades under the assumed models, unlike bootstrap values [87,88]. www.biomedcentral.com /1471-2148/2/17   (5919 words)

 Bayesian Phylogenetic Analysis Instead of using a gamma distribution and learning which sites have what rates from the data, we are instead using our prior knowledge about the structure of the genetic code to specify that all 1st positions have the same rate, all 2nd positions have the same rate, and all 3rd positions have the same rate. The Dirichlet distribution is a handy way of specifying the prior probability distribution of nucleotide (or amino acid) frequency vectors. If the prior probability distribution is flat (i.e., if all possible parameter values have the same prior probability) then the posterior distribution is simply proportional to the likelihood distribution, and the parameter value with the maximum likelihood then also has the maximum posterior probability. www.cbs.dtu.dk /dtucourse/cookbooks/gorm/27615/bayes1.php   (3535 words)

 An Example: The Suspicious Coin For N tosses, the posterior probability distribution is given by The posterior probability distributions are simply saying that a ``good bet'' for the true value of p is n/N although it does not offer any guarantee that it is the best one. First, posterior probability distributions are not symmetric at low S/N. As a matter of fact, many real parameter posterior probability distributions in section www.ucolick.org /~simard/phd/root/node37.html   (537 words)

 Bayesian statistics in a nutshell   (Site not responding. Last check: 2007-10-14) The posteriors that result from conjugate priors can also be seen clearly to the same as the posteriors that would result from a single big experiment that combined the prior and posterior information. For example, the beta distribution is the conjugate prior of the binomial distribution. Let's say the prior on r is normal with mean 0.3 and standard deviation 0.1, and the prior on K is gamma-distributed (a reasonable choice for a parameter that shouldn't be negative, and to me it's easier to think about than the log-normal) with shape parameter a = 10 and scale parameter s = 1e7. www.zoo.ufl.edu /bolker/emd/notes/lect16.html   (1559 words)

 [No title]   (Site not responding. Last check: 2007-10-14) The conditional probability of A given B is defined as  EMBED Equation.3 , (3.2.1) where the left-hand side is read as “Probability of A given B.” For example, suppose there are three different croplands (P1, P2, P3) which have been applied the biological control agents of two species of predators (S1 and S2). P(M) is the probability that M is true before we observe the data, which we call the prior probability. P(MD) is the probability that M is true after we observe the data and is called the posterior probability. www.ent.orst.edu /info2000/Week4/Chapter3entropy.doc   (2689 words)

 [No title]   (Site not responding. Last check: 2007-10-14) Within a Bayesian framework, we define a generative model of image appearance, a robust likelihood function based on image graylevel differences, and a prior probability distribution over pose and joint angles that models how humans move. The posterior probability distribution over model parameters is represented using a discrete set of samples and is propagated over time using particle filtering. The explicit posterior probability distribution represents ambiguities due to image matching, model singularities, and perspective projection. www.cns.nyu.edu /~lcv/meeting/black2_22.html   (167 words)

 Michael J. Black   (Site not responding. Last check: 2007-10-14) Similarly, we learn probability distributions over filter responses for general scenes that define a likelihood of observing the filter responses for arbitrary backgrounds. A prior probability distribution over possible human motions is learned from 3D motion-capture data and is combined with the likelihood for Bayesian tracking using particle filtering. In this approach, a posterior probability distribution over model parameters is represented using a discrete set of samples that is propagated over time. www.cs.rochester.edu /seminars/Sems00-01/Black.html   (255 words)

 Adobe PDF Document - /083883/Text/Meetings/medim98/papers/MCMC/article.dvi   (Site not responding. Last check: 2007-10-14) ABSTRACT The Markov Chain Monte Carlo (MCMC) technique provides a means to generate a random sequence of model realizations that sample the posterior probability distribution of a Bayesian analysis. That sequence may be used to make inferences about the model uncertainties that derive from measurement uncertainties. This paper presents an approach to improving the e ciency of the Metropolis approach to MCMC by incorporating an approximation to the covariance matrix of the posterior distribution. searchpdf.adobe.com /proxies/1/42/2/63.html   (173 words)

 Proc. SPIE (1991) - Abstract   (Site not responding. Last check: 2007-10-14) The optimal solution is provided by a Bayesian approach, which is founded on the posterior probability density distribution. The complete Bayesian procedure requires an integration of the posterior probability over all possible values of the image exterior to the local region being analyzed. In the presented work, the full treatment is approximated by simultaneously estimating the reconstruction outside the local region and the parameters of the model within the local region that maximize the posterior probability. public.lanl.gov /kmh/publications/ipat91.abs.html   (156 words)

 Proof of Proposition 3.1: A Supplement to Common Knowledge that i's posterior probability of event E is q (E) and that j's posterior probability of E is q Since i's posterior probability of event E is common knowledge, it is constant on plato.stanford.edu /entries/common-knowledge/proof3.1.html   (93 words)

 Glossary of research economics This is a one-parameter family of distributions, and the parameter, n, is conventionally labeled the degrees of freedom of the distribution. consistent: An estimator for a parameter is consistent iff the estimator converges in probability to the true value of the parameter; that is, the plim of the estimator, as the sample size goes to infinity, is the parameter itself. A diffuse prior is a distribution of the parameter with equal probability for each possible value, coming as close as possible to representing the notion that the analyst hasn't a clue about the value of the parameter being estimated. econterms.com /econtent.html   (14743 words)

 Posterior Sampling with Improved Efficiency - KM, GS (ResearchIndex)   (Site not responding. Last check: 2007-10-14) This paper presents an approach to improving the efficiency of the Metropolis approach to MCMC by incorporating an approximation to the covariance matrix of the posterior distribution. Markov Chain Monte Carlo Posterior Sampling With The.. Hanson, KM, Cunningham, GS (1998), Posterior sampling with improved efficiency, Medical Imaging: Image Processing, Proc. citeseer.ist.psu.edu /hanson98posterior.html   (604 words)

 Markov Chain Monte Carlo Posterior Sampling With The Hamiltonian Method - KM (ResearchIndex) Abstract: INTRODUCTION A major advantage of Bayesian data analysis is that provides a characterisation of the uncertainty in the model parameters estimated from a given set of measurements in the form of a posterior probability distribution [1]. When the analysis involves a complicated physical phenomenon, the posterior may not be available in analytic form, but only calculable by means of a simulation code. Hanson, KM (2001), Markov Chain Monte Carlo posterior sampling with the Hamiltonian method, Proc. citeseer.ist.psu.edu /km01markov.html   (394 words)

 Using MrBayes   (Site not responding. Last check: 2007-10-14) Bayesian analysis provides a means of estimating the posterior probability distribution for parameters of interest, including the groups of related taxa and substitution model parameters. MrBayes is a flexible and powerful program that uses NEXUS style data formatting for Bayesian phylogenetic analysis of DNA or protein sequences. 4) Pictures of the inferred trees with posterior probability values indicated on the nodes. stripe.colorado.edu /~am/Ass12a.html   (452 words)

 Applications MrBayes is a program for the Bayesian estimation of phylogeny. Bayesian inference of phylogeny is based upon a quantity called the posterior probability distribution of trees, which is the probability of a tree conditioned on the observations. The posterior probability distribution of trees is impossible to calculate analytically; instead, MrBayes uses a simulation technique called Markov chain Monte Carlo (or MCMC) to approximate the posterior probabilities of trees. www.umsl.edu /~beowulf/applications.html   (275 words)

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