Bayesian inference for a partially observed birth-death process using data on proportions

Stochastic kinetic models are often used to describe complex biological processes. Typically these models are analytically intractable and have unknown parameters which need to be estimated from observed data. Ideally we would have measurements on all interacting chemical species in the process, observed continuously in time. However, in practice, measurements are taken only at a relatively few time-points. In some situations, only very limited observation of the process is available, such as when experimenters can only observe noisy observations on the proportion of cells that are alive. This makes the inference task even more problematic. We consider a range of data-poor scenarios and investigate the performance of various computationally intensive Bayesian algorithms in determining the posterior distribution using data on proportions from a simple birth-death process.

[1]  A. Doucet,et al.  Particle Markov chain Monte Carlo methods , 2010 .

[2]  Darren J Wilkinson,et al.  Likelihood free inference for Markov processes: a comparison , 2014, Statistical applications in genetics and molecular biology.

[3]  William Feller.  Die Grundlagen der Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung , 1939 .

[4]  Andrew Gordon Wilson,et al.  Student-t Processes as Alternatives to Gaussian Processes , 2014, AISTATS.

[5]  D. Wilkinson,et al.  Bayesian Inference for Stochastic Kinetic Models Using a Diffusion Approximation , 2005, Biometrics.

[6]  D A Henderson,et al.  Bayesian Calibration of a Stochastic Kinetic Computer Model Using Multiple Data Sources , 2010, Biometrics.

[7]  Anthony O'Hagan,et al.  Diagnostics for Gaussian Process Emulators , 2009, Technometrics.

[8]  A. Raftery,et al.  Probabilistic forecasts, calibration and sharpness , 2007 .

[9]  Marc A Suchard,et al.  Estimation for General Birth-Death Processes , 2011, Journal of the American Statistical Association.

[10]  Derek Bingham,et al.  Efficient emulators of computer experiments using compactly supported correlation functions, with an application to cosmology , 2011, 1107.0749.

[11]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[12]  Darren J. Wilkinson,et al.  Bayesian inference for a discretely observed stochastic kinetic model , 2008, Stat. Comput..

[13]  Forrest W. Crawford,et al.  Transition probabilities for general birth–death processes with applications in ecology, genetics, and evolution , 2011, Journal of Mathematical Biology.

[14]  A. O'Hagan,et al.  Bayesian calibration of computer models , 2001 .

[15]  Gavin J. Gibson,et al.  Estimating parameters in stochastic compartmental models using Markov chain methods , 1998 .

[16]  Gavin J. Gibson,et al.  Inference for immigration-death processes with single and paired immigrants , 2001 .

[17]  Jianfeng Yao,et al.  ON LIKELIHOOD ESTIMATION FOR DISCRETELY OBSERVED MARKOV JUMP PROCESSES , 2007 .

[18]  G. Yule,et al.  A Mathematical Theory of Evolution Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[19]  Richard J. Boys,et al.  Inference for population dynamics in the Neolithic period , 2013, 1301.1525.

[20]  A. O'Hagan,et al.  Predicting the output from a complex computer code when fast approximations are available , 2000 .

[21]  E. Renshaw,et al.  The evolution of a single-paired immigration death process , 2008 .

[22]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[23]  M. Bladt,et al.  Statistical inference for discretely observed Markov jump processes , 2005 .

[24]  Darren J. Wilkinson,et al.  Bayesian Emulation and Calibration of a Stochastic Computer Model of Mitochondrial DNA Deletions in Substantia Nigra Neurons , 2009 .

[25]  M. Burrows,et al.  Modelling biological populations in space and time: Cambridge studies in mathematical biology: 11 , 1993 .

[26]  D. Kendall.  On the Generalized "Birth-and-Death" Process , 1948 .