# zbMATH — the first resource for mathematics

Approximate Bayesian computation in controlled branching processes: the role of summary statistics. (English) Zbl 1437.60054
Summary: A controlled branching process is a stochastic growth population model in which the number of individuals with reproductive capacity in each generation is given by a random control function. The purpose of the present work was to examine the Approximate Bayesian Computation sequential Monte Carlo method, and to propose appropriate summary statistics for these processes. The method’s success is shown to rely on this latter issue, and its accuracy is illustrated and compared with a “likelihood free” Markov chain Monte Carlo technique by means of a simulated example. How to extend the method to a controlled multitype branching process is also illustrated, and an application is made to model real data from the cell kinetics field. Both illustrations are developed using the R statistical software environment.
##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 62F15 Bayesian inference
##### Software:
CODA; MASS (R); mvtnorm; R
Full Text:
##### References:
 [1] Beaumont, MA; Cornuet, JM; Marin, JM; Robert, CP, Adaptive approximate bayesian computation, Biometrika, 96, 4, 983-990 (2009) · Zbl 1437.62393 [2] Beaumont, MA; Zhang, W.; Balding, DJ, Approximate bayesian computation in population genetics, Genetic, 162, 4, 2025-2035 (2002) [3] Becker, N., On parametric estimation for mortal branching processes, Biometrika, 61, 3, 393-399 (1974) · Zbl 0296.62080 [4] Corral, A.; García-Millán, R.; Font-Clos, F., Exact derivation of a finite-size scaling law and corrections to scaling in the geometric Galton-Watson process, PLoS One, 11, 9, e0161586 (2016) [5] Drovandi, CC; Pettitt, AN; McCutchan, RA, Exact and approximate Bayesian inference for low integer-valued time series models with intractable likelihoods, Bayesian Anal., 11, 2, 325-352 (2016) · Zbl 1359.62365 [6] Filippi, S.; Barnes, CP; Cornebise, J.; Stumpf, MPH, On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo, Stat. Appl. Genet. Mol. Biol., 12, 1, 87-107 (2013) [7] Frazier, D.T., Robert, C.P., Rousseau, J.: Model misspecification in approximate Bayesian computation: consequences and diagnostics. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 10.1111/rssb.12356,(2020) [8] Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., Scheipl, F., Hothorn, T.: mvtnorm: Multivariate Normal and t Distributions (2020) R package version 1.1-0. https://CRAN.R-project.org/package=mvtnorm [9] González, M., Gutiérrez, C., Martínez, R., Minuesa, C., del Puerto, I.: Bayesian analysis for controlled branching processes. In: I. del Puerto, M. González, C. Gutiérrez, R. Martínez, C. Minuesa, M. Molina, M. Mota, A. Ramos (eds.) Branching Processes and Their Applications, Lecture Notes in Statistics, vol. 219, pp. 185-205. Springer, Switzerland (2016) · Zbl 1354.60092 [10] González, M.; Gutiérrez, C.; Martínez, R.; del Puerto, I., Bayesian inference for controlled branching processes through MCMC and ABC methodologies, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 107, 2, 459-473 (2013) · Zbl 1277.62083 [11] González, M.; Minuesa, C.; del Puerto, I., Maximum likelihood estimation and Expectation-Maximization algorithm for controlled branching processes, Comput. Stat. Data Anal., 93, 209-227 (2016) · Zbl 06918698 [12] González, M.; Minuesa, C.; del Puerto, I., Minimum disparity estimation in controlled branching processes, Electron. J. Stat., 11, 1, 295-325 (2017) · Zbl 1356.60130 [13] González, M., Minuesa, C., del Puerto, I., Vidyashankar, A.N.: Robust estimation in controlled branching processes: Bayesian estimators via disparities, 1-62 (2018) arXiv:1802.05917 [14] González, M.; del Puerto, I.; Yanev, GP, Controlled branching processes (2018), Hoboken: ISTE Ltd, London, Wiley,, Hoboken [15] Guttorp, P., Statistical inference for branching processes (1991), New York: Wiley, New York · Zbl 0778.62077 [16] Guttorp, P.; Perlman, MD, Predicting extinction or explosion in a Galton-Watson branching process with power series offspring distribution, J. Stat. Plan. Infer., 165, 193-215 (2015) · Zbl 1326.62176 [17] Holgate, P.; Lakhani, KH, Effect of offspring distribution on population survival, Bull. Math. Biophys., 29, 4, 831-839 (1967) [18] Hyrien, O.; Ambeskovic, I.; Mayer-Proschel, M.; Noble, M.; Yakovlev, A., Stochastic modeling of oligodendrocyte generation in cell culture: model validation with time-lapse data, Theor. Biol. Med. Model., 3, 1, 21 (2006) [19] Lintusaari, J.; Gutmann, MU; Dutta, R.; Kaski, S.; Corander, J., Fundamentals and recent developments in approximate Bayesian Computation, Syst. Biol., 66, 1, e66-e68 (2017) [20] Martínez, R.; Mota, M.; del Puerto, I., On asymptotic posterior normality for controlled branching processes, Statistics, 43, 4, 367-378 (2009) · Zbl 1278.60133 [21] McKinley, T.; Cook, AR; Deardon, R., Inference in epidemic models without likelihoods, Int. J. Biostat., 5, 1, 24 (2009) [22] Owen, J.; Wilkinson, DJ; Gillespie, CS, Likelihood free inference for Markov processes: a comparison, Stat. Appl. Genet. Mol. Biol., 14, 2, 189-209 (2015) · Zbl 1311.60079 [23] Plummer, M.; Best, N.; Cowles, K.; Vines, K., CODA: convergence diagnosis and output analysis for MCMC, R News, 6, 1, 7-11 (2006) [24] Prangle, D., Adapting the ABC distance function, Bayesian Anal., 12, 1, 289-309 (2017) · Zbl 1384.62098 [25] R Core Team: R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2020). http://www.R-project.org/ [26] Robert, C.P.: Approximate Bayesian Computation: A survey on recent results. In: R. Cools, D. Nuyens (eds.) Monte Carlo and Quasi-Monte Carlo Methods, Springer Proceedings in Mathematics & Statistics, vol. 163, pp. 185-205. Springer International Publishing (2016) · Zbl 1356.65031 [27] Sevast’yanov, BA; Zubkov, AM, Controlled branching processes, Theory Prob. Appl., 19, 1, 15-25 (1974) [28] Venables, WN; Ripley, BD, Modern applied statistics with S. Statistics and computing (2002), New York: Springer, New York [29] Weiß, CH, Fully observed INAR(1) processes, J. Appl. Stat., 39, 3, 581-598 (2012) [30] Yakovlev, AY; Stoimenova, VK; Yanev, NM, Branching processes as models of progenitor cell populations and estimation of the offspring distributions, J. Am. Stat. Assoc., 103, 484, 1357-1366 (2008) · Zbl 1286.62073 [31] Yanev, NM, Conditions for degeneracy of $$\phi$$-branching processes with random $$\phi$$, Theory Prob. Appl., 20, 421-428 (1975)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.