# On the Bayesian Analysis of Population Size

### Ruth King and Stephen P. Brooks

### University of Bristol and University of Cambridge, England

## Summary

In this paper we consider the problem of estimating the total size of a population from
a series of incomplete census data.
We observe that inference is typically highly sensitive to the choice
of model and we demonstrate how Bayesian model averaging techniques
easily overcome this problem. We combine and extend the work of
Madigan and York (1997) and Dellaportas and Forster (1999) using reversible jump MCMC simulation to
calculate posterior model probabilities which can then be used to estimate
model averaged statistics of interest. We provide a detailed description
of the simulation procedures involved and consider a wide variety of modelling
issues, such as the range of models considered, their parameterisation,
both prior choice and sensitivity, and computational efficiency.
We consider a detailed example concerning adolescent injuries in
Pennsylvania on the basis of medical, school and survey data.
In the context of this example, we discuss the relationship between
posterior model probabilities and the associated
information criteria values for model selection. We also discuss cost-efficiency
issues with particular reference to inclusion and exclusion of sources on the
grounds of cost. We consider a decision theoretic approach, which balances
the cost and accuracy of different combinations of data sources to guide future decisions on data
collection.

### Keywords:

Contingency table; Unobserved data;
Log-linear models; Markov chain Monte Carlo; Reversible jump MCMC; Posterior
model probabilities; Decision theory;
Cost-effectiveness.

Appeared as King, R. and Brooks, S.P. (2001) "On the Bayesian Analysis of Population Size".
*Biometrika* **88** pp317--336.