Abstract
Recently, statistical distributions have been explored to provide estimates of the mineralogical diversity of Earth, and Earth-like planets. In this paper, a Bayesian approach is introduced to estimate Earth’s undiscovered mineralogical diversity. Samples are generated from a posterior distribution of the model parameters using Markov chain Monte Carlo simulations such that estimates and inference are directly obtained. It was previously shown that the mineral species frequency distribution conforms to a generalized inverse Gauss–Poisson (GIGP) large number of rare events model. Even though the model fit was good, the population size estimate obtained by using this model was found to be unreasonably low by mineralogists. In this paper, several zero-truncated, mixed Poisson distributions are fitted and compared, where the Poisson-lognormal distribution is found to provide the best fit. Subsequently, the population size estimates obtained by Bayesian methods are compared to the empirical Bayes estimates. Species accumulation curves are constructed and employed to estimate the population size as a function of sampling size. Finally, the relative abundances, and hence the occurrence probabilities of species in a random sample, are calculated numerically for all mineral species in Earth’s crust using the Poisson-lognormal distribution. These calculations are connected and compared to the calculations obtained in a previous paper using the GIGP model for which mineralogical criteria of an Earth-like planet were given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baayen RH (2001) Word frequency distributions, text, speech and language technology, vol 18. Kluwer Academic Publishers, Dordrecht
Barger K, Bunge J (2008) Bayesian estimation of the number of species using noninformative priors. Biom J 50(6):1064–1076
Barger K, Bunge J (2010) Objective bayesian estimation for the number of species. Bayesian Anal 5(4):765–785
Baroni M, Evert S (2007) Words and echoes: assessing and mitigating the non-randomness problem in word frequency distribution modeling. In: Proceedings of the 45th annual meeting of the association for computational linguistics, Prague, Czech Republic, pp 904–911
Bernardo JM (1979) Reference posterior distributions for bayesian inference. J R Stat Soc B 41:113–147
Bernardo JM, Ramón JM (1998) An introduction to bayesian reference analysis: inference on the ratio of multinomial parameters. J R Stat Soc D 47:101–135
Bunge J, Barger K (2008) Parametric models for estimating the number of classes. Biom J 50(6):971–982
Bunge J, Fitzpatrick M (1993) Estimating the number of species: a review. J Am Stat Assoc 88(421):364–373
Carroll JB (1967) On sampling from a lognormal model of word frequency distribution. In: Kučera H, Francis WN (eds) Computational analysis of present-day American English. Brown University Press, Providence, pp 406–424
Chao A, Bunge J (2002) Estimating the number of species in a stochastic abundance model. Biometrics 58(3):531–539
Chib S, Greenberg E (1995) Understanding the metropolis-hastings algorithm. Am Stat 49(4):327–335
Fisher RA, Corbet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J Anim Ecol 12(1):42–58
Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2013) Bayesian data analysis, 3rd edn. CRC Press, Boca Raton
Golden J, McMillan M, Downs RT, Hystad G, Goldstein I, Stein HJ, Zimmerman A, Sverjensky DA, Armstrong JT, Hazen RM (2013) Rhenium variations in molybdenite \((MoS_{2})\): evidence for progressive subsurface oxidation. Earth Planet Sci Lett 366:1–5
Grew ES, Krivovichev SV, Hazen RM, Hystad G (2016) Evolution of structural complexity in boron minerals. Can Mineral 54:125–143
Grøtan V, Engen S (2015) Poisson lognormal and bivariate Poisson lognormal distribution, package poilog. https://cran.r-project.org/web/packages/poilog/poilog.pdf. Accessed 20 Feb 2015
Gutiérrez-Peña E, Rueda R (2003) Reference priors for exponential families. J Stat Plan Inference 110:35–54
Hazen RM (2017) Chance, necessity, and the origins of life: a physical sciences perspective. Philos Trans R Soc A 375:20160353
Hazen RM, Grew ES, Downs RT, Golden J, Hystad G (2015a) Mineral ecology: chance and necessity in the mineral diversity of terrestrial planets. Can Mineral 53:295–324
Hazen RM, Hystad G, Downs RT, Golden JJ, Pires AJ, Grew ES (2015b) Earth’s ’missing’ minerals. Am Mineral 100:2344–2347
Hazen RM, Hummer DR, Hystad G, Downs RT, Golden JJ (2016) Carbon mineral ecology: predicting the undiscovered minerals of carbon. Am Mineral 101:889–906
Hazen RM, Hystad G, Golden JJ, Hummer DR, Liu C, Downs RT, Morrison SM, Ralph J, Grew ES (2017) Cobalt mineral ecology. Am Mineral 102:108–116
Hystad G, Downs RT, Grew ES, Hazen RM (2015a) Statistical analysis of mineral diversity and distribution: earth’s mineralogy is unique. Earth Planet Sci Lett 426:154–157
Hystad G, Downs RT, Hazen RM (2015b) Mineral species fequency distribution conforms to a large number of rare events model: prediction of Earth’s missing minerals. Math Geosci 47:647–661
Hystad G, Downs RT, Hazen RM, Golden JJ (2017) Relative abundances of mineral species: a statistical measure to characterize earth-like planets based on earth’s mineralogy. Math Geosci 49:179–194
Jeffreys H (1946) An invariant form for the prior probability in estimation problems. Proc R Soc Ser A 186:453–461
Jørgensen B (1982) Statistical properties of the generalized inverse Gaussian distribution, 1st edn. Springer, New York
Lindsay BG, Roeder K (1987) A unified treatment of integer parameter models. J Am Stat Assoc 82:758–764
McGill BJ et al (2007) Species abundance distributions: moving beyond single prediction theories to integration within an ecological framework. Ecol Lett 10:995–1015
Plummer M, Best N, Cowles K, Vines K, Sarkar D, Bates D, Almond R, Magnusson A (2018) Output analysis and diagnostics for MCMC, package ’coda’. https://cran.r-project.org/web/packages/coda/coda.pdf. Accessed 8 Oct 2018
Quince C, Curtis TP, Sloan WT (2008) The rational exploration of microbial diversity. Int Soc Microb Ecol J 2:997–1006
Rodrigues J, Milan LA, Leite JG (2001) Hierarchical Bayesian estimation for the number of species. Biom J 43(6):737–746
Sichel HS (1971) On a family of discrete distributions particularly suited to represent long-tailed frequency data. In: Proceedings of the third symposium on mathematical statistics, Pretoria, South Africa, pp 51–97
Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc B 64:583–639
Stan Development Team (2017) RStan: the R interface to stan. http://mc-stan.org/, R package version 2.17.2
Stasinopoulos M, Rigby B (2018) Generalised additive models for location scale and shape, package GAMLSS. https://cran.r-project.org/web/packages/gamlss/gamlss.pdf. Accessed 6 Oct 2018
Stein GZ, Zucchini W, Juritz JM (1987) Parameter estimation for the sichel distribution and its multivariate extension. J Am Stat Assoc 82:938–944
Acknowledgements
We would like to thank the reviewers for their valuable comments to improve the paper. We gratefully acknowledge support from NASA Mars Science Laboratory Mission NNX11AP82A, as well as support from the Alfred P. Sloan Foundation (Grant No. 2013-10-01), the W.M. Keck Foundation (Grant No. 140002372), the John Templeton Foundation (Grant No. 60645), the Deep Carbon Observatory, the Carnegie Institution for Science, and an anonymous private foundation.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Hystad, G., Eleish, A., Hazen, R.M. et al. Bayesian Estimation of Earth’s Undiscovered Mineralogical Diversity Using Noninformative Priors. Math Geosci 51, 401–417 (2019). https://doi.org/10.1007/s11004-019-09795-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-019-09795-8