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1

Lindsay, Bruce G. Mixture Models. Haywood CA and Alexandria VA: Institute of Mathematical Statistics and American Statistical Association, 1995. http://dx.doi.org/10.1214/cbms/1462106013.

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2

McLachlan, Geoffrey, and David Peel. Finite Mixture Models. Hoboken, NJ, USA: John Wiley & Sons, Inc., 2000. http://dx.doi.org/10.1002/0471721182.

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3

1952-, Basford Kaye E., ed. Mixture models: Inference and applications to clustering. New York, N.Y: M. Dekker, 1988.

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4

Bouguila, Nizar, and Wentao Fan, eds. Mixture Models and Applications. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-23876-6.

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5

Chen, Jiahua. Statistical Inference Under Mixture Models. Singapore: Springer Nature Singapore, 2023. http://dx.doi.org/10.1007/978-981-99-6141-2.

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6

von Davier, Matthias. Multivariate and Mixture Distribution Rasch Models. Edited by Claus H. Carstensen. New York, NY: Springer New York, 2007. http://dx.doi.org/10.1007/978-0-387-49839-3.

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7

R, Hancock Gregory, and Samuelsen Karen M, eds. Advances in latent variable mixture models. Charlotte, NC: Information Age Pub., 2008.

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8

Mixture models: Theory, geometry, and applications. Hayward, Calif: Institute of Mathematical Statistics, 1995.

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9

service), SpringerLink (Online, ed. Medical Applications of Finite Mixture Models. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2009.

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10

Jepson, Allan D. Mixture models for optical flow computation. Toronto: University of Toronto, Dept. of Computer Science, 1993.

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11

Experiments with mixtures: Designs, models, and the analysis of mixture data. 3rd ed. New York: Wiley, 2002.

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12

A, Cornell John. Experiments with mixtures: Designs, models, and the analysis of mixture data. 2nd ed. New York: Wiley, 1990.

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13

Chew, Soo Hong. Mixture symmetric utility theory. Toronto: Dept. of Economics, Institute for Policy Analysis, University of Toronto, 1988.

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14

Eva, Gregorová, ed. Phase mixture models for the properties of nanoceramics. Hauppauge, N.Y: Nova Science Publishers, 2010.

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15

Sims, Christopher A. MCMC method for Markov mixture simultaneous-equation models: A note. [Atlanta]: Federal Reserve Bank of Atlanta, 2004.

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16

H, Stokes Eric, and NASA Glenn Research Center, eds. Porous media and mixture models for hygrothermal behavior of phenolic composites. [Cleveland, Ohio]: National Aeronautics and Space Administration, Glenn Research Center, 1999.

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17

Gebhardt, Christoph. Variational reconstruction of Quaternary temperature fields using mixture models as botanical-climatological transfer functions. St. Augustin [Germany]: Asgard Verlag, 2003.

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18

Gellert, Charles Lawrence. Misaligned (shuffled) data analysis with application to gene regulation. [Albany, N.Y.?]: University at Albany, School of Public Health, Dept. of Biometry and Statistics, 1990.

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19

L, Dryer F., and United States. National Aeronautics and Space Administration., eds. Transient numerical modeling of the combustion of bi-component liquid droplets: Methanol/water mixture. [Washington, DC: National Aeronautics and Space Administration, 1994.

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20

Lin, Li, He Guoqi, and United States. National Aeronautics and Space Administration., eds. Nonlinear spectral mixture modeling of lunar multispectral: Implications for lateral transport. [Washington, DC: National Aeronautics and Space Administration, 1997.

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21

Gerke, Udo. Numerical analysis of mixture formation and combustion in a hydrogen direct-injection internal combustion engine. Göttingen: Cuvillier, 2007.

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22

G, Collins F., Aumalis A. E, and United States. National Aeronautics and Space Administration. Scientific and Technical Information Branch., eds. An exact solution for the solidification of a liquid slab of binary mixture. [Washington, D.C.]: National Aeronautics and Space Administration, Scientific and Technical Information Branch, 1986.

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23

1939-, Hettmansperger Thomas P., Pennsylvania State University. Department of Statistics, ebrary Inc, and Nonparametric Statistics and Mixture Models : Past, Present, and Future (2008 : Pennsylvania State University), eds. Nonparametric statistics and mixture models: A festschrift in honor of Thomas P. Hettmansperger, the Pennsylvania State University, USA, 23-24 May, 2008. Hackensack, N.J: World Scientific, 2011.

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24

Pella, J. Search algorithms for computing stock composition of a mixture from traits of individuals by maximum likelihood / by J. Pella, M. Masuda, and S. Nelson. Seattle, WA: U.S. Dept. of Commerce, National Oceanic and Atmospheric Administration, National Marine Fisheries Service, Alaska Fisheries Science Center, 1996.

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25

McNicholas, Paul D. Mixture Model-Based Classification. Boca Raton : Taylor & Francis, 2016.: Chapman and Hall/CRC, 2016. http://dx.doi.org/10.1201/9781315373577.

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26

1940-, Yang Raymond Shih-hsien, Elias Robert W, and National Institute of Environmental Health Sciences, eds. Chemical mixtures: Lead model validation. Research Triangle Park, NC: U.S. Dept. of Health and Human Services, Public Health Service, National Institutes of Health, National Institute of Environmental Health Sciences, 1998.

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27

Theory of structured multiphase mixtures. Berlin: Springer, 1991.

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28

Dobran, F. Theory of structured multiphase mixtures. Berlin: Springer-Verlag, 1991.

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29

Srinivasan, Seshasai. Thermodiffusion in Multicomponent Mixtures: Thermodynamic, Algebraic, and Neuro-Computing Models. New York, NY: Springer New York, 2013.

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30

Peel, David, and Geoffrey J. McLachlan. Finite Mixture Models. Wiley & Sons, Incorporated, John, 2005.

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31

Cheng, Russell. Finite Mixture Models. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198505044.003.0017.

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Fitting a finite mixture model when the number of components, k, is unknown can be carried out using the maximum likelihood (ML) method though it is non-standard. Two well-known Bayesian Markov chain Monte Carlo (MCMC) methods are reviewed and compared with ML: the reversible jump method and one using an approximating Dirichlet process. Another Bayesian method, to be called MAPIS, is examined that first obtains point estimates for the component parameters by the maximum a posteriori method for different k and then estimates posterior distributions, including that for k, using importance sampling. MAPIS is compared with ML and the MCMC methods. The MCMC methods produce multimodal posterior parameter distributions in overfitted models. This results in the posterior distribution of k being biased towards high k. It is shown that MAPIS does not suffer from this problem. A simple numerical example is discussed.
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32

Peel, David, and Geoffrey J. McLachlan. Finite Mixture Models. Wiley & Sons, Incorporated, John, 2007.

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33

Peel, David, and Geoffrey J. McLachlan. Finite Mixture Models. Wiley & Sons, Incorporated, John, 2004.

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34

Bouguila, Nizar, and Wentao Fan. Mixture Models and Applications. Springer, 2019.

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35

Bouguila, Nizar, and Wentao Fan. Mixture Models and Applications. Springer International Publishing AG, 2020.

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36

Frühwirth-Schnatter, Sylvia. Finite Mixture and Markov Switching Models. Springer New York, 2010.

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37

Medical Applications of Finite Mixture Models. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-68651-4.

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38

Finite Mixture and Markov Switching Models. Springer New York, 2006. http://dx.doi.org/10.1007/978-0-387-35768-3.

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39

Finite Mixture and Markov Switching Models. Springer, 2006.

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40

Tatarinova, Tatiana V., and Alan Schumitzky. Nonlinear Mixture Models: A Bayesian Approach. Imperial College Press, 2015.

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41

Schlattmann, Peter. Medical Applications of Finite Mixture Models. Springer, 2010.

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42

Finite Mixture and Markov Switching Models. Springer London, Limited, 2006.

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43

Hancock, Gregory R., and Karen M. Samuelsen. Advances in Latent Variable Mixture Models. Information Age Publishing, Incorporated, 2007.

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44

Cornell, John A. Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data. Wiley & Sons, Incorporated, John, 2011.

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45

Cornell, John A. Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data. Wiley & Sons, Limited, John, 2011.

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46

Visser, Ingmar, and Maarten Speekenbrink. Mixture and Hidden Markov Models with R. Springer International Publishing AG, 2022.

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47

Jain, Sonia. Split-merge techniques for bayesian mixture models. 2002.

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48

Rodriguez, Abel, and Athanasios Kottas. Bayesian Nonparametric Mixture Models: Methods and Applications. Taylor & Francis Group, 2023.

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49

Jain, Sonia. Split-merge techniques for Bayesian mixture models. 2002, 2002.

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50

(Editor), Gregory R. Hancock, and Karen M. Samuelsen (Editor), eds. Advances in Latent Variable Mixture Models (HC). Information Age Publishing, 2007.

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