Extending the latent multinomial model with complex error processes and dynamic Markov bases

The latent multinomial model (LMM) of Link et al. [Biometrics 66 (2010) 178-185] provides a framework for modelling mark-recapture data with potential identification errors. Key is a Markov chain Monte Carlo (MCMC) scheme for sampling configurations of the latent counts of the true capture histories that could have generated the observed data. Assuming a linear map between the observed and latent counts, the MCMC algorithm uses vectors from a basis of the kernel to move between configurations of the latent data. Schofield and Bonner [Biometrics 71 (2015) 1070-1080] shows that this is sufficient for some models within the framework but that a larger set called a Markov basis is required when errors are more complex. We address two further challenges: (1) that models with complex error mechanisms may not fit within the LMM framework and (2) that Markov bases can be difficult to compute for studies of even moderate size. We extend the framework to model the capture/demographic and error processes separately and develop a new MCMC algorithm using dynamic Markov bases. Our work is motivated by a study of queen snakes (Regina septemvittata) and we use simulation to compare estimates of survival rates when snakes are marked with PIT tags which have perfect identification versus brands which are prone to error.