Skip to main content
SpringerLink
Log in
Book cover

Introduction to Bayesian Statistics pp 193–234Cite as

Numerical Methods

  • Chapter

  • 5651 Accesses

Abstract

One distinguishes between generating random numbers, that are values of a random variable which possesses the uniform distribution (2.59) in the interval [0, 1], and generating values of random variables with given distributions. Both mothods give random values, also called random variates.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alenius, S. and U. Ruotsalainen (1997) Bayesian image reconstruction for emission tomography based on median root prior. Eur J Nucl Med, 24:258–265.

    Google Scholar 

  • Alkhatib, H. and W.-D. Schuh (2007) Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for largescale least squares problems. J Geodesy, 81:53–66.

    Article  Google Scholar 

  • Arent, N., G. Hückelheim and K.R. Koch (1992) Method for obtaining geoid undulations from satellite altimetry data by a quasi-geostrophic model of the sea surface topography. Manuscripta geodaetica, 17:174–185.

    Google Scholar 

  • Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis. Springer, Berlin.

    Google Scholar 

  • Bernardo, J.M. and A.F.M. Smith (1994) Bayesian Theory. Wiley, New York.

    Google Scholar 

  • Besag, J.E. (1974) Spatial interaction and the statistical analysis of lattice systems. J Royal Statist Society, B 36:192–236.

    Google Scholar 

  • Besag, J.E. (1986) On the statistical analysis of dirty pictures. J Royal Statist Society, B 48:259–302.

    Google Scholar 

  • Betti, B., M. Crespi and F. Sanso (1993) A geometric illustration of ambiguity resolution in GPS theory and a Bayesian approach. Manuscripta geodaetica, 18:317–330.

    Google Scholar 

  • Bettinardi, V., E. Pagani, M.C. Gilardi, S. Alenius, K. Thielemans, M. Teras and F. Fazio (2002) Implementation and evaluation of a 3D one-step late reconstruction algorithm for 3D positron emission tomography brain studies using median root prior. Eur J Nucl Med, 29:7–18.

    Article  Google Scholar 

  • Bishop, C.M. (2006) Pattern Recognition and Machine Learning. Springer, New York.

    Google Scholar 

  • Blatter, C. (1974) Analysis I, II, III. Springer, Berlin.

    Google Scholar 

  • Box, G.E.P. and M.E. Müller (1958) A note on the generation of random normal deviates. Annals Mathematical Statistics, 29:610–611.

    Google Scholar 

  • Box, G.E.P. and G.C. Tiao (1973) Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading.

    Google Scholar 

  • Chen, M.-H., Q.-M. Shao and J.G. Ibrahim (2000) Monte Carlo Methods in Bayesian Computations. Springer, New York.

    Google Scholar 

  • Cox, R.T. (1946) Probability, frequency and reasonable expectation. American Journal of Physics, 14:1–13.

    Article  Google Scholar 

  • Cressie, N.A.C. (1991) Statistics for Spatial Data. Wiley, New York.

    Google Scholar 

  • Dagpunar, J. (1988) Principles of Random Variate Generation. Clarendon Press, Oxford.

    Google Scholar 

  • Dean, T.L. and M.P. Wellman (1991) Planning and Control. Morgan Kaufmann, San Mateo.

    Google Scholar 

  • DeGroot, M.H. (1970) Optimal Statistical Decisions. McGraw-Hill, New York.

    Google Scholar 

  • Devroye, L. (1986) Non-Uniform Random Variate Generation. Springer, Berlin.

    Google Scholar 

  • Doucet, A., S. Godsill and C. Andrieu (2000) On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing, 10:197–208.

    Article  Google Scholar 

  • Fessler, J.A., H. Erdogan and W.B. Wu (2000) Exact distribution of edge-preserving MAP estimators for linear signal models with Gaussian measurement noise. IEEE Trans Im Proc, 9(6):1049–1056.

    Article  Google Scholar 

  • Förstner, W. (1979) Ein Verfahren zur Schätzung von Varianz-und Kovarianzkomponenten. Allgemeine Vermessungs-Nachrichten, 86:446–453.

    Google Scholar 

  • Gelfand, A.E. and A.F.M. Smith (1990) Sampling-based approaches to calculating marginal densities. J American Statistical Association, 85:398–409.

    Article  Google Scholar 

  • Gelfand, A.E., A.F.M. Smith and T. Lee (1992) Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling. J American Statistical Association, 87:523–532.

    Article  Google Scholar 

  • Gelman, A., J.B. Carlin, H.S. Stern and D.B. Rubin (2004) Bayesian Data Analysis, 2nd Ed. Chapman and Hall, Boca Raton.

    Google Scholar 

  • Geman, D., S. Geman and C. Graffigne (1987) Locating texture and object boundaries. In: Devijver, P.A. and J. Kittler (Eds.), Pattern Recognition Theory and Applications. Springer, Berlin, 165–177.

    Google Scholar 

  • Geman, S. and D. Geman (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Machine Intell, PAMI-6:721–741.

    Article  Google Scholar 

  • Geman, S. and D.E. McClure (1987) Statistical methods for tomographic image reconstruction. Bull Int Statist Inst, 52–21.1:5–21.

    Google Scholar 

  • George, A. and J.W. Liu (1981) Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs.

    Google Scholar 

  • Gilks, W.R. (1996) Full conditional distributions. In: Gilks, W.R., S. Richardson and D. J. Spiegelhalter (Eds.), Markov Chain Monte Carlo in Practice. Chapman and Hall, London, 75–88.

    Google Scholar 

  • Golub, G.H. and U. von Matt (1997) Generalized cross-validation for large-scale problems. J Computational and Graphical Statistics, 6:1–34.

    Article  Google Scholar 

  • Gordon, N. and D. Salmond (1995) Bayesian state estimation for tracking and guidance using the bootstrap filter. J Guidance, Control, and Dynamics, 18:1434–1443.

    Google Scholar 

  • Grafarend, E.W. and B. Schaffrin (1993) Ausgleichungsrechnung in linearen Modellen. B.I. Wissenschaftsverlag, Mannheim.

    Google Scholar 

  • Green, P.J. (1990) Bayesian reconstruction from emission tomography data using a modified EM algorithm. IEEE Trans Med Imaging, 9:84–93.

    Article  Google Scholar 

  • Gui, Q., Y. Gong, G. Li and B. Li (2007) A Bayesian approach to the detection of gross errors based on posterior probability. J Geodesy, DOI 10.1007/s00190-006-0132-y.

    Google Scholar 

  • Gundlich, B. (1998) Konfidenzbereiche für robuste Parameterschätzungen. In: Freeden, W. (Ed.), Progress in Geodetic Science at GW 98. Shaker Verlag, Aachen, 258–265.

    Google Scholar 

  • Gundlich, B. and K.R. Koch (2002) Confidence regions for GPS baselines by Bayesian statistics. J Geodesy, 76:55–62.

    Article  Google Scholar 

  • Gundlich, B., K.R. Koch and J. Kusche (2003) Gibbs sampler for computing and propagating large covariance matrices. J Geodesy, 77:514–528.

    Article  Google Scholar 

  • Gundlich, B., P. Musman, S. Weber, O. Nix and W. Semmler (2006) From 2D PET to 3D PET: Issues of data representation and image reconstruction. Z Med Phys, 16:31–46.

    Google Scholar 

  • Hamilton, A.G. (1988) Logic for Mathematicians. Cambridge University Press, Cambridge.

    Google Scholar 

  • Hampel, F.R., E.M. Ronchetti, P.R. Rousseeuw and W.A. Stahel (1986) Robust Statistics. Wiley, New York.

    Google Scholar 

  • Harville, D.A. (1999) Use of the Gibbs sampler to invert large, possibly sparse, positive definite matrices. Linear Algebra and its Applications, 289:203–224.

    Article  Google Scholar 

  • Heitz, S. (1968) Geoidbestimmung durch Interpolation nach kleinsten Quadraten aufgrund gemessener und interpolierter Lotabweichungen. Reihe C, 124. Deutsche Geodätische Kommission, München.

    Google Scholar 

  • Hörmann, W., J. Leydold and G. Derflinger (2004) Automatic Nonuniform Random Variate Generation. Springer, Berlin.

    Google Scholar 

  • Huber, P.J. (1964) Robust estimation of a location parameter. Annals Mathematical Statistics, 35:73–101.

    Google Scholar 

  • Huber, P.J. (1981) Robust Statistics. Wiley, New York.

    Google Scholar 

  • Hutchinson, M.F. (1990) A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Comun Statistist-Simula, 19:433–450.

    Article  Google Scholar 

  • Jaynes, E.T. (2003) Probability theory. The logic of science. Cambridge University Press, Cambridge.

    Google Scholar 

  • Jazwinski, A.H. (1970) Stochastic Processes and Filtering Theory. Academic Press, New York.

    Google Scholar 

  • Jeffreys, H. (1961) Theory of Probability. Clarendon, Oxford.

    Google Scholar 

  • Jensen, F.V. (1996) An Introduction to Bayesian Networks. UCL Press, London.

    Google Scholar 

  • Johnson, N.L. and S. Kotz (1970) Distributions in Statistics: Continuous Univariate Distributions, Vol. 1, 2. Houghton Mifflin, Boston.

    Google Scholar 

  • Johnson, N.L. and S. Kotz (1972) Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.

    Google Scholar 

  • Junhuan, P. (2005) The asymptotic variance-covariance matrix, Baarda test and the reliability of Li-norm estimates. J Geodesy, 78:668–682.

    Article  Google Scholar 

  • Kirkpatrick, S., C.D. Gelatt and M.P. Vecchi (1983) Optimization by simulated annealing. Science, 220:671–680.

    Article  Google Scholar 

  • Klonowski, J. (1999) Segmentierung und Interpretation digitaler Bilder mit Markoff-Zufallsfeldern. Reihe C, 492. Deutsche Geodätische Kommission, München.

    Google Scholar 

  • Koch, K.R. (1986) Maximum likelihood estimate of variance components; ideas by A.J. Pope. Bulletin Géodésique, 60:329–338.

    Article  Google Scholar 

  • Koch, K.R. (1987) Bayesian inference for variance components. Manuscripta geodaetica, 12:309–313.

    Google Scholar 

  • Koch, K.R. (1990) Bayesian Inference with Geodetic Applications. Springer, Berlin.

    Google Scholar 

  • Koch, K.R. (1994) Bayessche Inferenz für die Prädiktion und Filterung. Z Vermessungswesen, 119:464–470.

    Google Scholar 

  • Koch, K.R. (1995a) Bildinterpretation mit Hilfe eines Bayes-Netzes. Z Vermessungswesen, 120:277–285.

    Google Scholar 

  • Koch, K.R. (1995b) Markov random fields for image interpretation. Z Photogrammetrie und Fernerkundung, 63:84–90, 147.

    Google Scholar 

  • Koch, K.R. (1996) Robuste Parameterschätzung. Allgemeine Vermessungs-Nachrichten, 103:1–18.

    Google Scholar 

  • Koch, K.R. (1999) Parameter Estimation and Hypothesis Testing in Linear Models, 2nd Ed. Springer, Berlin.

    Google Scholar 

  • Koch, K.R. (2000) Numerische Verfahren in der Bayes-Statistik. Z Vermessungswesen, 125:408–414.

    Google Scholar 

  • Koch, K.R. (2002) Monte-Carlo-Simulation für Regularisierungsparameter. ZfV-Z Geodäsie, Geoinformation und Landmanagement, 127:305–309.

    Google Scholar 

  • Koch, K.R. (2005a) Bayesian image restoration by Markov Chain Monte Carlo methods. ZfV-Z Geodäsie, Geoinformation und Landmanagement, 130:318–324.

    Google Scholar 

  • Koch, K.R. (2005b) Determining the maximum degree of harmonic coefficients in geopotential models by Monte Carlo methods. Studia, Geophysica et Geodaetica, 49:259–275.

    Article  Google Scholar 

  • Koch, K.R. (2006) ICM algorithm for the Bayesian reconstruction of tomographic images. Photogrammetrie, Fernerkundung, Geoinformation, 2006(3):229–238.

    Google Scholar 

  • Koch, K.R. (2007) Gibbs sampler by sampling-importance-resampling. J Geodesy, DOI 10.1007/s00190-006-0121-1.

    Google Scholar 

  • Koch, K.R. and J. Kusche (2002) Regularization of geopotential determination from satellite data by variance components. J Geodesy, 76:259–268.

    Article  Google Scholar 

  • Koch, K.R. and J. Kusche (2007) Comments on Xu et al. (2006) Variance component estimation in linear inverse ill-posed models, J Geod 80(1):69–81. J Geodesy, DOI 10.1007/s00190-007-0163-z.

    Google Scholar 

  • Koch, K.R. and H. Papo (2003) The Bayesian approach in two-step modeling of deformations. Allgemeine Vermessungs-Nachrichten, 110,111:365–370,208.

    Google Scholar 

  • Koch, K.R. and M. Schmidt (1994) Deterministische und stochastische Signale. Dümmler, Bonn.

    Google Scholar 

  • Koch, K.R. and Y. Yang (1998a) Konfidenzbereiche und Hypothesentests für robuste Parameterschätzungen. Z Vermessungswesen, 123:20–26.

    Google Scholar 

  • Koch, K.R. and Y. Yang (1998b) Robust Kaiman filter for rank deficient observation models. J Geodesy, 72:436–441.

    Article  Google Scholar 

  • Koch, K.R., H. Fröhlich and G. Bröker (2000) Transformation räumlicher variabler Koordinaten. Allgemeine Vermessungs-Nachrichten, 107:293–295.

    Google Scholar 

  • Koch, K.R., J. Kusche, C. Boxhammer and B. Gundlich (2004) Parallel Gibbs sampling for computing and propagating large covariance matrices. ZfV-Z Geodäsie, Geoinformation und Landmanagement, 129:32–42.

    Google Scholar 

  • Köster, M. (1995) Kontextsensitive Bildinterpretation mit Markoff-Zufallsfeldern. Reihe C, 444. Deutsche Geodätische Kommission, München.

    Google Scholar 

  • Krarup, T. (1969) A contribution to the mathematical foundation of physical geodesy. Geodaetisk Institut, Meddelelse No.44, Kopenhagen.

    Google Scholar 

  • Kulschewski, K. (1999) Modellierung von Unsicherheiten in dynamischen Bayes-Netzen zur qualitativen Gebäudeerkennung. Reihe Geodäsie, Band 3. Shaker Verlag, Aachen.

    Google Scholar 

  • Kusche, J. (2003) A Monte-Carlo technique for weight estimation in satellite geodesy. J Geodesy, 76:641–652.

    Article  Google Scholar 

  • Lange, K. and R. Carson (1984) EM reconstruction algorithms for emission and transmission tomography. J Comput Assist Tomogr, 8:306–316.

    Google Scholar 

  • Lange, K., M. Bahn and R. Little (1987) A theoretical study of some maximum likelihood algorithms for emission and transmission tomography. IEEE Trans Med Imaging, MI-6:106–114.

    Google Scholar 

  • Leahy, R.M. and J. Qi (2000) Statistical approaches in quantitative positron emission tomography. Statistics and Computing, 10:147–165.

    Article  Google Scholar 

  • Leonard, T. and J.S.J. Hsu (1999) Bayesian Methods. Cambridge University Press, Cambridge.

    Google Scholar 

  • Lindley, D.V. (1957) A statistical paradox. Biometrika, 44:187–192.

    Google Scholar 

  • Liu, J.S. (2001) Monte Carlo Strategies in Scientific Computing. Springer, Berlin.

    Google Scholar 

  • Loredo, T. J. (1990) From Laplace to Supernova SN 1987A: Bayesian inference in astrophysics. In: Fougère, P. F. (Ed.), Maximum Entropy and Bayesian Methods. Kluwer Academic Publ., Dordrecht, 81–142.

    Google Scholar 

  • Marsaglia, G. and T.A. Bray (1964) A convenient method for generating normal variables. SIAM Review, 6:260–264.

    Article  Google Scholar 

  • Mayer-Gürr, T., K.H. Ilk, A. Eicker and M. Feuchtinger (2005) ITG-CHAMP01: A CHAMP gravity field model from short kinematical arcs of a one-year observation period. J Geodesy, 78:462–480.

    Article  Google Scholar 

  • Meier, S. and W. Keller (1990) Geostatistik. Springer, Wien.

    Google Scholar 

  • Menz, J. and J. Pilz (1994) Kollokation, Universelles Kriging und Bayesscher Zugang. Markscheidewesen, 101:62–66.

    Google Scholar 

  • Metropolis, N., A.W. Rosenbluth, M.L. Rosenbluth, A.H. Teller and E. Teller (1953) Equation of state calculations by fast computing machines. J Chem Phys, 21:1087–1092.

    Article  Google Scholar 

  • Modestino, J.W. and J. Zhang (1992) A Markov random field modelbased approach to image interpretation. IEEE Trans Pattern Anal Machine Intell, 14:606–615.

    Article  Google Scholar 

  • Moritz, H. (1969) A general theory of gravity processing. Report 122. Department of Geodetic Science, Ohio State University, Columbus, Ohio.

    Google Scholar 

  • Moritz, H. (1973) Least-squares collocation. Reihe A, 75. Deutsche Geodätische Kommission, München.

    Google Scholar 

  • Moritz, H. (1980) Advanced Physical Geodesy. Wichmann, Karlsruhe.

    Google Scholar 

  • Neapolitan, R.E. (1990) Probabilistic Reasoning in Expert Systems. Wiley, New York.

    Google Scholar 

  • Niemann, H. (1990) Pattern Analysis and Understanding. Springer, Berlin.

    Google Scholar 

  • Novikov, P.S. (1973) Grundzüge der mathematischen Logik. Vieweg, Braunschweig.

    Google Scholar 

  • O’Hagan, A. (1994) Bayesian Inference, Kendall’s Advanced Theory of Statistics, Vol. 2B. Wiley, New York.

    Google Scholar 

  • Oliver, R.M. and J.R. Smith (Eds.) (1990) Influence Diagrams, Belief Nets and Decision Analysis. Wiley, New York.

    Google Scholar 

  • O’Sullivan, F. (1986) A statistical perspective on ill-posed inverse problems. Statistical Science, 1:502–527.

    Google Scholar 

  • Ou, Z. (1991) Approximate Bayes estimation for variance components. Manuscripta geodaetica, 16:168–172.

    Google Scholar 

  • Ou, Z. and K.R. Koch (1994) Analytical expressions for Bayes estimates of variance components. Manuscripta geodaetica, 19:284–293.

    Google Scholar 

  • Pearl, J. (1986) Fusion, propagation, and structuring in belief networks. Artificial Intelligence, 29:241–288.

    Article  Google Scholar 

  • Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Mateo.

    Google Scholar 

  • Pilz, J. (1983) Bayesian Estimation and Experimental Design in Linear Regression Models. Teubner, Leipzig.

    Google Scholar 

  • Pilz, J. and V. Weber (1998) Bayessches Kriging zur Erhöhung der Prognosegenauigkeit im Zusammenhang mit der UVP für den Bergbau. Markscheidewesen, 105:213–221.

    Google Scholar 

  • Press, S.J. (1989) Bayesian Statistics: Principles, Models, and Applications. Wiley, New York.

    Google Scholar 

  • Qi, J., R.M. Leahy, S.R. Cherry, A. Chatziioannou and T.H. Farquhar (1998) High-resolution 3D Bayesian image reconstruction using the microPET small-animal scanner. Phys Med Biol, 43:1001–1013.

    Article  Google Scholar 

  • Raiffa, H. and R. Schlaifer (1961) Applied Statistical Decision Theory. Graduate School of Business Administration, Harvard University, Boston.

    Google Scholar 

  • Reigber, Ch., H. Jochmann, J. Wünsch, S. Petrovic, P. Schwintzer, F. Barthelmes, K.-H. Neumayer, R. König, Ch. Förste, G. Balmino, R. Biancale, J.-M. Lemoine, S. Loyer and F. Perosanz (2005) Earth gravity field and seasonal variability from CHAMP. In: Reigber, Ch., H. Lühr, P. Schwintzer and J. Wickert (Eds.), Earth Observation with CHAMP-Results from Three Years in Orbit. Springer, Berlin, 25–30.

    Chapter  Google Scholar 

  • Riesmeier, K. (1984) Test von Ungleichungshypothesen in linearen Modellen mit Bayes-Verfahren. Reihe C, 292. Deutsche Geodätische Kommission, München.

    Google Scholar 

  • Ripley, B.D. (1987) Stochastic Simulation. Wiley, New York.

    Google Scholar 

  • Ripley, B.D. (1996) Pattern Recognition and Neural Networks. University Press, Cambridge.

    Google Scholar 

  • Robert, C.P. (1994) The Bayesian Choice. Springer, Berlin.

    Google Scholar 

  • Roberts, G.O and A.F.M. Smith (1994) Simple conditions for the convergence of the Gibbs sampler and Metropolis-Hastings algorithms. Stochastic Processes and their Applications, 49:207–216.

    Article  Google Scholar 

  • Rousseeuw, P.J. (1984) Least median of squares regression. J American Statistical Association, 79:871–880.

    Article  Google Scholar 

  • Rousseeuw, P.J. and A.M. Leroy (1987) Robust Regression and Outlier Detection. Wiley, New York.

    Google Scholar 

  • Rubin, D.B. (1988) Using the SIR algorithm to simulate posterior distributions. In: Bernardo, J.M., M.H. DeGroot, D.V. Lindley and A.F.M. Smith (Eds.), Bayesian Statistics 3. Oxford University Press, Oxford, 395–402.

    Google Scholar 

  • Rubinstein, R.Y. (1981) Simulation and the Monte Carlo Method. Wiley, New York.

    Google Scholar 

  • Shepp, L.A. and Y. Vardi (1982) Maximum likelihood reconstruction for emission tomography. IEEE Trans Med Imaging, MI-1:113–122.

    Article  Google Scholar 

  • Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.

    Google Scholar 

  • Sivia, D.S. (1996) Data Analysis, a Bayesian Tutorial. Clarendon Press, Oxford.

    Google Scholar 

  • Skare, O., E. Bolviken and L. Holden (2003) Improved samplingimportance resampling and reduced bias importance sampling. Scandinavian Journal of Statistics, 30:719–737.

    Article  Google Scholar 

  • Smith, A.F.M. and A.E. Gelfand (1992) Bayesian statistics without tears: a sampling-resampling perspective. American Statistician, 46:84–88.

    Article  Google Scholar 

  • Smith, A.F.M. and G.O. Roberts (1993) Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo methods. J Royal Statist Society, B 55:3–23.

    Google Scholar 

  • Späth, H. (1987) Mathematische Software zur linearen Regression. Oldenbourg, München.

    Google Scholar 

  • Stassopoulou, A., M. Petrou and J. Kittler (1998) Application of a Bayesian network in a GIS based decision making system. Int J Geographical Information Science, 12:23–45.

    Article  Google Scholar 

  • Tikhonov, A.N. and V.Y. Arsenin (1977) Solutions of Ill-Posed Problems. Wiley, New York.

    Google Scholar 

  • Vardi, Y., L.A. Shepp and L. Kaufman (1985) A statistical model for positron emission tomography. J American Statist Ass, 80:8–37.

    Article  Google Scholar 

  • Vinod, H.D. and A. Ullah (1981) Recent Advances in Regression Methods. Dekker, New York.

    Google Scholar 

  • Wang, W. and G. Gindi (1997) Noise analysis of MAP-EM algorithms for emission tomography. Phys Med Biol, 42:2215–2232.

    Article  Google Scholar 

  • West, M. and J. Harrison (1989) Bayesian Forecasting and Dynamic Models. Springer, Berlin.

    Google Scholar 

  • Whitesitt, J.E. (1969) Boolesche Algebra und ihre Anwendungen. Vieweg, Braunschweig.

    Google Scholar 

  • Wiener, N. (1949) Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications. Wiley, New York.

    Google Scholar 

  • Wolf, H. (1968) Ausgleichungsrechnung nach der Methode der kleinsten Quadrate. Dümmler, Bonn.

    Google Scholar 

  • Wolf, H. (1975) Ausgleichungsrechnung, Formeln zur praktischen Anwendung. Dümmler, Bonn.

    Google Scholar 

  • Wolf, H. (1979) Ausgleichungsrechnung II, Aufgaben und Beispiele zur praktischen Anwendung. Dümmler, Bonn.

    Google Scholar 

  • Xu, P. (2005) Sign-constrained robust least squares, subjective breakdown point and the effect of weights of observations on robustness. J Geodesy, 79:146–159.

    Article  Google Scholar 

  • Xu, P., Y. Shen, Y. Fukuda and Y. Liu (2006) Variance component estimation in linear inverse ill-posed models. J Geodesy, 80:69–81.

    Article  Google Scholar 

  • Yang, Y. and W. Gao (2006) An optimal adaptive Kaiman filter. J Geodesy, 80:177–183.

    Article  Google Scholar 

  • Yang, Y., L. Song and T. Xu (2002) Robust estimator for correlated observations based on bifactor equivalent weights. J Geodesy, 76:353–358.

    Article  Google Scholar 

  • Zellner, A. (1971) An Introduction to Bayesian Inference in Econometrics. Wiley, New York.

    Google Scholar 

Download references

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2007). Numerical Methods. In: Introduction to Bayesian Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72726-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation