Bayes Theorem: A theorem in probability theory named for Thomas Bayes (1702-1761). In epidemiology, it is used to obtain the probability of disease in a group of people with some characteristic on the basis of the overall rate of that disease and of the likelihood of that characteristic in healthy and diseased individuals. The most familiar application is in clinical decision analysis where it is used for estimating the probability of a particular diagnosis given the appearance of some symptoms or test result.Mathematical Concepts: Numeric or quantitative entities, descriptions, properties, relationships, operations, and events.Algorithms: A procedure consisting of a sequence of algebraic formulas and/or logical steps to calculate or determine a given task.Mathematics: The deductive study of shape, quantity, and dependence. (From McGraw-Hill Dictionary of Scientific and Technical Terms, 6th ed)Models, Statistical: Statistical formulations or analyses which, when applied to data and found to fit the data, are then used to verify the assumptions and parameters used in the analysis. Examples of statistical models are the linear model, binomial model, polynomial model, two-parameter model, etc.Information Theory: An interdisciplinary study dealing with the transmission of messages or signals, or the communication of information. Information theory does not directly deal with meaning or content, but with physical representations that have meaning or content. It overlaps considerably with communication theory and CYBERNETICS.Models, Theoretical: Theoretical representations that simulate the behavior or activity of systems, processes, or phenomena. They include the use of mathematical equations, computers, and other electronic equipment.Computer Simulation: Computer-based representation of physical systems and phenomena such as chemical processes.Probability: The study of chance processes or the relative frequency characterizing a chance process.Enzymes: Biological molecules that possess catalytic activity. They may occur naturally or be synthetically created. Enzymes are usually proteins, however CATALYTIC RNA and CATALYTIC DNA molecules have also been identified.Models, Genetic: Theoretical representations that simulate the behavior or activity of genetic processes or phenomena. They include the use of mathematical equations, computers, and other electronic equipment.Achillea: A plant genus of the family ASTERACEAE that has long been used in folk medicine for treating wounds.Models, Biological: Theoretical representations that simulate the behavior or activity of biological processes or diseases. For disease models in living animals, DISEASE MODELS, ANIMAL is available. Biological models include the use of mathematical equations, computers, and other electronic equipment.Likelihood Functions: Functions constructed from a statistical model and a set of observed data which give the probability of that data for various values of the unknown model parameters. Those parameter values that maximize the probability are the maximum likelihood estimates of the parameters.

According to them (p462), it should be decided by application of Bayes theorem, which is an important theorem of probability ... Bayes theorem can never itself give us the probabilities that it needs to get started, in particular the prior probability of ... As an exercise, I have written a judgement for the hypothetical case, which applies Bayes theorem; and set it out in a ... In legal fact-finding, Bayes theorem can alert tribunals to the necessity of taking account of prior probabilities when dealing ...
*  Probability: Bayes Theorem - Probability and Statistics | Coursera
Probability: Bayes Theorem. To view this video please enable JavaScript, and consider upgrading to a web browser that supports ... We then give the definitions of probability and the laws governing it and apply Bayes theorem. We study probability ... And Bayes Theorem states that the probability that an event B will occur, ... So, this is a problem that we'll utilize Bayes Theorem that we've already given. ...
*  Bayes Theorem Limits
... Let ($\displaystyle A_n$) and ($\displaystyle B_n$) be in A with $\displaystyle A_n$--,A and $\ ...
*  What Is Bayes Theorem? | NeuroLogica Blog
I have written a little about Bayes Theorem, mainly on Science-Based Medicine, which is a statistical method for analyzing data ... That is really the basic concept of Bayes Theorem. However, there are some statistical nuances when applying Bayes to specific ... Bayes Theorem is just one of the plethora of tools in the toolbox, but the only tools that apply to the whole toolbox are the ... I think Bayes' theorem is a vital part of understanding the process of Science. Not because people need to be able to do the ...
*  Bayes Theorem-Probabilty Question
From Bayes' Theorem we have:. p(p , c) = p(c , p) * p(p) / p(c) (1). p(ph , c) = p(c , ph) * p(ph) / p(c) (2). p(l , c) = p(c ... From Bayes' Theorem we have:. p(p , c) = p(c , p) * p(c) / p(p) (1). p(ph , c) = p(c , ph) * p(c) / p(ph) (2). p(l , c) = p(c ... Bayes Theorem-Probabilty Question. Hello Hi everybody!. Here is a new thread I faced difficult in solving please give me the ... Bayes' theorem is:. img.top {vertical-align:15%;} which is not what you have here.. CB. ...
*  Bayes Theorem - Who Posted?
Math Help Forum is a free math help forum for Calculus, Algebra, LaTeX, Geometry, Trigonometry, Statistics and Probability, Differential Equations, Discrete Math
*  Wolfram Videos: Parametric Probability Distribution Fitted to Data with Bayes's Theorem
James Rock explains how he's using Bayes's Theorem to fit data to a parametric distribution with Mathematica. Video from the ... Parametric Probability Distribution Fitted to Data with Bayes's Theorem. James Rock. James Rock explains how he's using Bayes's ... Theorem to fit data to a parametric distribution with Mathematica in this talk from the Wolfram Technology Conference. ...
*  Category:Bayes' theorem - Wikimedia Commons
Media in category "Bayes' theorem". The following 61 files are in this category, out of 61 total. ... Retrieved from "https://commons.wikimedia.org/w/index.php?title=Category:Bayes%27_theorem&oldid=83933633" ...
*  Composite Service Recommendation Based on Bayes Theorem: Computer Science & IT Journal Article | IGI Global
Composite Service Recommendation Based on Bayes Theorem: 10.4018/jwsr.2012040104: The number of web services increased ... "Composite Service Recommendation Based on Bayes Theorem," International Journal of Web Services Research (IJWSR) 9 (2012): 2, ... Wu, J., Chen, L., Jian, H., & Wu, Z. (2012). Composite Service Recommendation Based on Bayes Theorem. International Journal of ... "Composite Service Recommendation Based on Bayes Theorem." IJWSR 9.2 (2012): 69-93. Web. 24 Sep. 2018. doi:10.4018/jwsr. ...
*  Bayes' theorem | planetmath.org
Bayes Theorem by apollos ✓. Bayes Theorem by apollos ✓. all events must have nonzero probability (+ other suggestions) by yark ... Bayes' theorem. Let (. A. n. ). subscript. A. n. (A_{n}). be a sequence of mutually exclusive events whose union is the sample ... Bayes' Theorem states. P. (. A. j. ,. E. ). =. P. (. A. j. ). P. (. E. ,. A. j. ). ∑. i. P. (. A. i. ). P. (. E. ,. A. i. ). ...
*  Bayes' Theorem - Probability | Coursera
What is conditional probability and Bayes' theorem? How our plausible reasoning can be interpreted in terms of Bayes' theorem? ... Bayes' Theorem. To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video ... in the probability because of the Bayes' Theorem, doesn't make it very probable. ... Bayes' formula and see the cases when you apply this reasoning in the real life. ...
*  Bayes theorem (disambiguation) - Wikipedia
Bayes theorem may refer to: Bayes' theorem - a theorem which expresses how a subjective degree of belief should rationally ... Bayesian theory in E-discovery - the application of Bayes' theorem in legal evidence diagnostics and E-discovery, where it ... Bayesian theory in marketing - the application of Bayes' theorem in marketing, where it allows for decision making and market ...
*  Bayes' Theorem
... theorem. Shows how to use Bayes' rule to solve conditional probability problems. Includes sample problem with step-by-step ... Bayes Theorem (aka, Bayes Rule). Bayes' theorem (also known as Bayes' rule) is a useful tool for calculating conditional ... Bayes' theorem can be stated as follows:. Bayes' theorem.. Let A1, A2, ... , An be a set of mutually exclusive events that ... When to Apply Bayes' Theorem. Part of the challenge in applying Bayes' theorem involves recognizing the types of problems that ...
*  Evidence under Bayes theorem - Wikipedia
R v Adams - court case about Bayes' Theorem with DNA "Bayes' Theorem in the Court of Appeal , Law Articles", Bernard Robertson ... One area of particular interest and controversy has been Bayes' theorem. Bayes' theorem is an elementary proposition of ... The use of evidence under Bayes' theorem relates to the likelihood of finding evidence in relation to the accused, where Bayes ... If she used Bayes' theorem, she could multiply those prior odds by a "likelihood ratio" in order to update her odds after ...
*  Monkeying with Bayes' theorem
... he describes a translation algorithm based on Bayes' theorem. Pick the English word that has the highest posterior probability ... And I explain why Bayes' Theorem is important in almost every field. Bayes' sets the limit for how much we can learn from ... Monkeying with Bayes' theorem. Posted on 9 March 2012. by John. In Peter Norvig's talk The Unreasonable Effectiveness of Data, ... Bayes theorem is a remarkable thinking tool that has become sort of a revolution. And I think that this tribute is justified. ...
*  Lesson 2.2 Bayes' theorem - Probability and Bayes' Theorem | Coursera
... we review the basics of probability and Bayes' theorem. In Lesson 1, we introduce the different paradigms ... ... Probability and Bayes' Theorem. In this module, we review the basics of probability and Bayes' theorem. In Lesson 1, we ... Lesson 2.2 Bayes' theorem. To view this video please enable JavaScript, and consider upgrading to a web browser that supports ... In Lesson 2, we review the rules of conditional probability and introduce Bayes' theorem. Lesson 3 reviews common probability ...
*  Teaching a short class on Bayes' Theorem?
I think 'Bayes' Theorem' (but perhaps not Bayes's Theorem) is catchier than the latter two suggestions. Also clearer. ... One true story about Bayes' Theorem that grabbed me when I read it:. My most memorable encounter with the Reverend Bayes came ... You will never be able to fit all of the applications of Bayes Theorem in one hour so pick one or two and make it look awesome ... If you want people to sign up for your class, don't call it Bayes Theorem, or anything equally boring (not many people can even ...
*  250 years of Bayes' Theorem | R-bloggers
The Reverend Thomas Bayes died 250 years ago this month. His grave, located near epidemiological centre of excellence St Mary's ... Because since then, Bayes Theorem has been the underpinning of predictive analytics applications from spam detection to medical ... contributions to Statistics with a series of videos on Bayes Theorem, its applications, and the implications for Big Data and ... The Reverend Thomas Bayes died 250 years ago this month. His grave, located near epidemiological centre of excellence St Mary's ...
*  Introduction to Bayesian Probability and Bayes Theorem
This quick refresher about probability shows how to use Bayes' Rule in simple situations and is an introduction to frequentist ... Bayes' theorem is a rule in probability and statistical theory that calculates an event's probability based on related ... That's enough background; now let's derive Bayes' Theorem.. ... Introduction to Bayesian Probability and Bayes Theorem. October ... Here's an explanation of how to use Bayes' Rule in simple situations, and introduce the relationship between Bayesian and ...
*  Using Bayes' theorem and the Neyman-Pearson Lemma to decide - Everything2.com
... theorem -- we don't know P(H , O). This... ... we immediately run into the dilemma described under Bayes' ... In most useful cases, we'll use Bayes' theorem to help us estimate P(H1 , O) and P(H0 , O) (the probabilities that the ... If we wish actually to use the Neyman-Pearson Lemma, we immediately run into the dilemma described under Bayes' theorem -- we ... Bayes' Theorem. Using gzip to do computational linguistics. Shifting the burden of proof. ...
*  Lesson 3.3 Exponential and normal distributions - Probability and Bayes' Theorem | Coursera
... we review the basics of probability and Bayes' theorem. In Lesson 1, we introduce the different paradigms ... ... Probability and Bayes' Theorem. In this module, we review the basics of probability and Bayes' theorem. In Lesson 1, we ... In Lesson 2, we review the rules of conditional probability and introduce Bayes' theorem. Lesson 3 reviews common probability ...
*  Envision the World as a Graph with Bayes' Theorem - dummies
Bayes' theorem can help you deduce how likely something is to happen in a certain context, based on the general probabilities ... The Naïve Bayes algorithm helps you arrange all the evidence you gather and reach a more solid prediction with a higher ... You can further extend Naïve Bayes to represent relationships that are more complex than a series of factors that hint at the ... The following example shows how things work in a Naïve Bayes classification. This is an old, renowned problem, but it ...
*  Bayes's Theorem - British Academy Scholarship
The papers in this book consider the worth and applicability of the theorem. The book sets out the philosophical issues: ... The book ends with the original paper containing the theorem, presented to the Royal Society in 1763. ... and John Earman consider how the theorem can be used in statistical science, in weighing evidence in criminal trials, and in ... theorem is a tool for assessing how probable evidence makes some hypothesis. ...
*  bayes theorem - Notes
bayes theorem posted 24 Feb 2012, 04:50 by David Sherlock [ updated 24 Feb 2012, 04:52 ] Thinking about recursion and self- ... I was panicy over Bayes Theorem but it turns out, its pretty simple. I can thank IBM for a simple explanation. I had made a ... A Bayes filter is an algorithm used in computer science for calculating the probabilities of multiple beliefs to allow a robot ...
*  So You Want to Understand Bayes' Theorem and/or Look at Photos of Cats - Skepchick
You see, Bayes' Theorem is a way of considering our prior knowledge in our calculation of probability. Using Bayes' Theorem we ... Theorem and/or Look at Photos of Cats. Skepticism. So You Want to Understand Bayes' Theorem and/or Look at Photos of Cats. ... Lucky for us Bayes' Theorem has a simple formula: A formula is not very intuitive though, so let's just ignore that for now ... You can apply Bayes' Theorem to any type of "test" where a true positive result is quite rare. For example, the fact that ...

Hyperparameter: In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis.P-adic Hodge theory: In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fieldsIn this article, a local field is complete discrete valuation field whose residue field is perfect. with residual characteristic p (such as Qp).Clonal Selection Algorithm: In artificial immune systems, Clonal selection algorithms are a class of algorithms inspired by the clonal selection theory of acquired immunity that explains how B and T lymphocytes improve their response to antigens over time called affinity maturation. These algorithms focus on the Darwinian attributes of the theory where selection is inspired by the affinity of antigen-antibody interactions, reproduction is inspired by cell division, and variation is inspired by somatic hypermutation.Bill Parry (mathematician)Inverse probability weighting: Inverse probability weighting is a statistical technique for calculating statistics standardized to a population different from that in which the data was collected. Study designs with a disparate sampling population and population of target inference (target population) are common in application.Index of information theory articles: This is a list of information theory topics, by Wikipedia page.Von Neumann regular ring: In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R such that . To avoid the possible confusion with the regular rings and regular local rings of commutative algebra (which are unrelated notions), von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left module is flat.Interval boundary element method: Interval boundary element method is classical boundary element method with the interval parameters.
Negative probability: The probability of the outcome of an experiment is never negative, but quasiprobability distributions can be defined that allow a negative probability for some events. These distributions may apply to unobservable events or conditional probabilities.Enzyme Commission number: The Enzyme Commission number (EC number) is a numerical classification scheme for enzymes, based on the chemical reactions they catalyze.Yarrow oilMatrix model: == Mathematics and physics ==Decoding methods: In coding theory, decoding is the process of translating received messages into codewords of a given code. There have been many common methods of mapping messages to codewords.

(1/6254) Bayesian inference on biopolymer models.

MOTIVATION: Most existing bioinformatics methods are limited to making point estimates of one variable, e.g. the optimal alignment, with fixed input values for all other variables, e.g. gap penalties and scoring matrices. While the requirement to specify parameters remains one of the more vexing issues in bioinformatics, it is a reflection of a larger issue: the need to broaden the view on statistical inference in bioinformatics. RESULTS: The assignment of probabilities for all possible values of all unknown variables in a problem in the form of a posterior distribution is the goal of Bayesian inference. Here we show how this goal can be achieved for most bioinformatics methods that use dynamic programming. Specifically, a tutorial style description of a Bayesian inference procedure for segmentation of a sequence based on the heterogeneity in its composition is given. In addition, full Bayesian inference algorithms for sequence alignment are described. AVAILABILITY: Software and a set of transparencies for a tutorial describing these ideas are available at http://www.wadsworth.org/res&res/bioinfo/  (+info)

(2/6254) Genetic determination of individual birth weight and its association with sow productivity traits using Bayesian analyses.

Genetic association between individual birth weight (IBW) and litter birth weight (LBW) was analyzed on records of 14,950 individual pigs born alive between 1988 and 1994 at the pig breeding farm of the University of Kiel. Dams were from three purebred lines (German Landrace, German Edelschwein, and Large White) and their crosses. Phenotypically, preweaning mortality of pigs decreased substantially from 40% for pigs with < or = 1 kg weight to less than 7% for pigs with > 1.6 kg. For these low to high birth weight categories, preweaning growth (d 21 of age) and early postweaning growth (weaning to 25 kg) increased by more than 28 and 8% per day, respectively. Bayesian analysis was performed based on direct-maternal effects models for IBW and multiple-trait direct effects models for number of pigs born in total (NOBT) and alive (NOBA) and LBW. Bayesian posterior means for direct and maternal heritability and litter proportion of variance in IBW were .09, .26, and .18, respectively. After adjustment for NOBT, these changed to .08, .22, and .09, respectively. Adjustment for NOBT reduced the direct and maternal genetic correlation from -.41 to -.22. For these direct-maternal correlations, the 95% highest posterior density intervals were -.75 to -.07, and -.58 to .17 before and after adjustment for NOBT. Adjustment for NOBT was found to be necessary to obtain unbiased estimates of genetic effects for IBW. The relationship between IBW and NOBT, and thus the adjustment, was linear with a decrease in IBW of 44 g per additionally born pig. For litter traits, direct heritabilities were .10, .08, and .08 for NOBT, NOBA, and LBW, respectively. After adjustment of LBW for NOBA the heritability changed to .43. Expected variance components for LBW derived from estimates of IBW revealed that genetic and environmental covariances between full-sibs and variation in litter size resulted in the large deviation of maternal heritability for IBW and its equivalent estimate for LBW. These covariances among full-sibs could not be estimated if only LBW were recorded. Therefore, selection for increased IBW is recommended, with the opportunity to improve both direct and maternal genetic effects of birth weight of pigs and, thus, their vitality and pre- and postnatal growth.  (+info)

(3/6254) Bayesian mapping of multiple quantitative trait loci from incomplete outbred offspring data.

A general fine-scale Bayesian quantitative trait locus (QTL) mapping method for outcrossing species is presented. It is suitable for an analysis of complete and incomplete data from experimental designs of F2 families or backcrosses. The amount of genotyping of parents and grandparents is optional, as well as the assumption that the QTL alleles in the crossed lines are fixed. Grandparental origin indicators are used, but without forgetting the original genotype or allelic origin information. The method treats the number of QTL in the analyzed chromosome as a random variable and allows some QTL effects from other chromosomes to be taken into account in a composite interval mapping manner. A block-update of ordered genotypes (haplotypes) of the whole family is sampled once in each marker locus during every round of the Markov Chain Monte Carlo algorithm used in the numerical estimation. As a byproduct, the method gives the posterior distributions for linkage phases in the family and therefore it can also be used as a haplotyping algorithm. The Bayesian method is tested and compared with two frequentist methods using simulated data sets, considering two different parental crosses and three different levels of available parental information. The method is implemented as a software package and is freely available under the name Multimapper/outbred at URL http://www.rni.helsinki.fi/mjs/.  (+info)

(4/6254) The validation of interviews for estimating morbidity.

Health interview surveys have been widely used to measure morbidity in developing countries, particularly for infectious diseases. Structured questionnaires using algorithms which derive sign/symptom-based diagnoses seem to be the most reliable but there have been few studies to validate them. The purpose of validation is to evaluate the sensitivity and specificity of brief algorithms (combinations of signs/symptoms) which can then be used for the rapid assessment of community health problems. Validation requires a comparison with an external standard such as physician or serological diagnoses. There are several potential pitfalls in assessing validity, such as selection bias, differences in populations and the pattern of diseases in study populations compared to the community. Validation studies conducted in the community may overcome bias caused by case selection. Health centre derived estimates can be adjusted and applied to the community with caution. Further study is needed to validate algorithms for important diseases in different cultural settings. Community-based studies need to be conducted, and the utility of derived algorithms for tracking disease frequency explored further.  (+info)

(5/6254) Bayesian analysis of birth weight and litter size in Baluchi sheep using Gibbs sampling.

Variance and covariance components for birth weight (BWT), as a lamb trait, and litter size measured on ewes in the first, second, and third parities (LS1 through LS3) were estimated using a Bayesian application of the Gibbs sampler. Data came from Baluchi sheep born between 1966 and 1989 at the Abbasabad sheep breeding station, located northeast of Mashhad, Iran. There were 10,406 records of BWT recorded for all ewe lambs and for ram lambs that later became sires or maternal grandsires. All lambs that later became dams had records of LS1 through LS3. Separate bivariate analyses were done for each combination of BWT and one of the three variables LS1 through LS3. The Gibbs sampler with data augmentation was used to draw samples from the marginal posterior distribution for sire, maternal grandsire, and residual variances and the covariance between the sire and maternal grandsire for BWT, variances for the sire and residual variances for the litter size traits, and the covariances between sire effects for different trait combinations, sire and maternal grandsire effects for different combinations of BWT and LS1 through LS3, and the residual covariations between traits. Although most of the densities of estimates were slightly skewed, they seemed to fit the normal distribution well, because the mean, mode, and median were similar. Direct and maternal heritabilities for BWT were relatively high with marginal posterior modes of .14 and .13, respectively. The average of the three direct-maternal genetic correlation estimates for BWT was low, .10, but had a high standard deviation. Heritability increased from LS1 to LS3 and was relatively high, .29 to .37. Direct genetic correlations between BWT and LS1 and between BWT and LS3 were negative, -.32 and -.43, respectively. Otherwise, the same correlation between BWT and LS2 was positive and low, .06. Genetic correlations between maternal effects for BWT and direct effects for LS1 through LS3 were all highly negative and consistent for all parities, circa -.75. Environmental correlations between BWT and LS1 through LS3 were relatively low and ranged from .18 to .29 and had high standard errors.  (+info)

(6/6254) Thermodynamics and kinetics of a folded-folded' transition at valine-9 of a GCN4-like leucine zipper.

Spin inversion transfer (SIT) NMR experiments are reported probing the thermodynamics and kinetics of interconversion of two folded forms of a GCN4-like leucine zipper near room temperature. The peptide is 13Calpha-labeled at position V9(a) and results are compared with prior findings for position L13(e). The SIT data are interpreted via a Bayesian analysis, yielding local values of T1a, T1b, kab, kba, and Keq as functions of temperature for the transition FaV9 right arrow over left arrow FbV9 between locally folded dimeric forms. Equilibrium constants, determined from relative spin counts at spin equilibrium, agree well with the ratios kab/kba from the dynamic SIT experiments. Thermodynamic and kinetic parameters are similar for V9(a) and L13(e), but not the same, confirming that the molecular conformational population is not two-state. The energetic parameters determined for both sites are examined, yielding conclusions that apply to both and are robust to uncertainties in the preexponential factor (kT/h) of the Eyring equation. These conclusions are 1) the activation free energy is substantial, requiring a sparsely populated transition state; 2) the transition state's enthalpy far exceeds that of either Fa or Fb; 3) the transition state's entropy far exceeds that of Fa, but is comparable to that of Fb; 4) "Arrhenius kinetics" characterize the temperature dependence of both kab and kba, indicating that the temperatures of slow interconversion are not below that of the glass transition. Any postulated free energy surface for these coiled coils must satisfy these constraints.  (+info)

(7/6254) Iterative reconstruction based on median root prior in quantification of myocardial blood flow and oxygen metabolism.

The aim of this study was to compare reproducibility and accuracy of two reconstruction methods in quantification of myocardial blood flow and oxygen metabolism with 15O-labeled tracers and PET. A new iterative Bayesian reconstruction method based on median root prior (MRP) was compared with filtered backprojection (FBP) reconstruction method, which is traditionally used for image reconstruction in PET studies. METHODS: Regional myocardial blood flow (rMBF), oxygen extraction fraction (rOEF) and myocardial metabolic rate of oxygen consumption (rMMRO2) were quantified from images reconstructed in 27 subjects using both MRP and FBP methods. For each subject, regions of interest (ROIs) were drawn on the lateral, anterior and septal regions on four planes. To test reproducibility, the ROI drawing procedure was repeated. By using two sets of ROIs, variability was evaluated from images reconstructed with the MRP and the FBP methods. RESULTS: Correlation coefficients of mean values of rMBF, rOEF and rMMRO2 were significantly higher in the images reconstructed with the MRP reconstruction method compared with the images reconstructed with the FBP method (rMBF: MRP r = 0.896 versus FBP r = 0.737, P < 0.001; rOEF: 0.915 versus 0.855, P < 0.001; rMMRO2: 0.954 versus 0.885, P < 0.001). Coefficient of variation for each parameter was significantly lower in MRP images than in FBP images (rMBF: MRP 23.5% +/- 11.3% versus FBP 30.1% +/- 14.7%, P < 0.001; rOEF: 21.0% +/- 11.1% versus 32.1% +/- 19.8%, P < 0.001; rMMRO2: 23.1% +/- 13.2% versus 30.3% +/- 19.1%, P < 0.001). CONCLUSION: The MRP reconstruction method provides higher reproducibility and lower variability in the quantitative myocardial parameters when compared with the FBP method. This study shows that the new MRP reconstruction method improves accuracy and stability of clinical quantification of myocardial blood flow and oxygen metabolism with 15O and PET.  (+info)

(8/6254) Taking account of between-patient variability when modeling decline in Alzheimer's disease.

The pattern of deterioration in patients with Alzheimer's disease is highly variable within a given population. With recent speculation that the apolipoprotein E allele may influence rate of decline and claims that certain drugs may slow the course of the disease, there is a compelling need for sound statistical methodology to address these questions. Current statistical methods for describing decline do not adequately take into account between-patient variability and possible floor and/or ceiling effects in the scale measuring decline, and they fail to allow for uncertainty in disease onset. In this paper, the authors analyze longitudinal Mini-Mental State Examination scores from two groups of Alzheimer's disease subjects from Palo Alto, California, and Minneapolis, Minnesota, in 1981-1993 and 1986-1988, respectively. A Bayesian hierarchical model is introduced as an elegant means of simultaneously overcoming all of the difficulties referred to above.  (+info)

  • Cox's
  • It would be cool if you found a way to work in the existence of Cox's theorem -- when I encountered it, I had never thought about why the laws of probability were given as they are, or if there could be a different consistent way to represent and calculate probability besides multiplying numbers together. (lesswrong.com)
  • According to the objectivist view, probability is a reasonable expectation that represents the state of knowledge, can be interpreted as an extension of logic, and its rules can be justified by Cox's theorem. (wikipedia.org)
  • posterior
  • 2) While Bayes' theorem describes a way of obtaining the actual posterior probability, maximizing that is only loosely related to any downstream loss function you actually care about, and there are decision-theoretic reasons to add extra parameters (a temperature in this case) to your model to improve a downstream loss. (johndcook.com)
  • and based directly on Bayes theorem, it allows us to make better posterior estimates as more observations become available. (wikipedia.org)
  • Mathematical
  • It says, "Mathematical formulas and theorems are usually not named after their original discoverers" and was named after Carl Boyer, whose book History of Mathematics contains many examples of this law. (wikipedia.org)
  • jury
  • Some observers believe that in recent years (i) the debate about probabilities has become stagnant, (ii) the protagonists in the probabilities debate have been talking past each other, (iii) not much is happening at the high-theory level, and (iv) the most interesting work is in the empirical study of the efficacy of instructions on Bayes' theorem in improving jury accuracy. (wikipedia.org)
  • view
  • According to the subjectivist view, probability quantifies a personal belief, and its rules can be justified by requirements of rationality and coherence following from the Dutch book argument or from the decision theory and de Finetti's theorem. (wikipedia.org)
  • things
  • At the moment, I'm thinking about how to design the class, so I'd appreciate any suggestions as to what content I should cover, the best format, clear ways to explain it, cool things related to Bayes' Theorem, good links, and so forth. (lesswrong.com)
  • case
  • c. 1701 - 7 April 1761) was an English statistician, philosopher and Presbyterian minister who is known for having formulated a specific case of the theorem that bears his name: Bayes' theorem. (wikipedia.org)
  • known
  • Examples include Hubble's law which was derived by Georges Lemaître two years before Edwin Hubble, the Pythagorean theorem although it was known to Babylonian mathematicians before Pythagoras, and Halley's comet which was observed by astronomers since at least 240 BC. (wikipedia.org)
  • better
  • So we experimented some, and we found out that when you raise that first factor [in Bayes' theorem] to the 1.5 power, you get a better result. (johndcook.com)
  • simple
  • Later on, it turned out to have been a simple mistake -- the test was a false positive, and the 999 out of 1,000 figure had been based on a lack of understanding about Bayes' Theorem. (lesswrong.com)
  • List
  • Eponym List of examples of Stigler's law List of misnamed theorems List of persons considered father or mother of a scientific field Matthew effect Matilda effect Obliteration by incorporation Scientific priority Standing on the shoulders of giants Theories and sociology of the history of science Gieryn, T. F., ed. (1980). (wikipedia.org)
  • find
  • No, for the same reason we aren't surprised when we find that logistic regression outperforms naive Bayes. (johndcook.com)