# Classical Model Selection via Simulated Annealing

### Stephen P. Brooks, Nial Friel and Ruth King

### University of Cambridge

## Summary

The classical approach to statistical analysis is usually based upon finding values
for model parameters that maximise the likelihood function. Model choice
in this context is often also based upon the likelihood function, but with the addition of
a penalty term for the number of parameters.
Though models may be
compared pairwise using likelihood ratio tests for example, various
criteria such as the AIC have been proposed as alternatives when multiple
models need to be compared. In practical terms, the classical approach
to model selection usually involves maximising the likelihood function associated
with each competing model and then calculating the corresponding criteria
value(s). However, when large numbers of models are possible, this quickly
becomes infeasible unless a method that simultaneously
maximises over both parameter and model space is available.
In this paper we propose an extension to the traditional simulated
annealing algorithm that allows for moves that not only
change parameter values but that also move between competing models.
This trans-dimensional simulated annealing algorithm can therefore be
used to locate models and parameters that minimise criteria
such as the AIC, but within a single algorithm, removing the
need for large numbers of simulations to be run.
We discuss the implementation of the trans-dimensional
simulated annealing algorithm and use simulation studies to
examine their performance in realistically complex modelling
situations.
We illustrate our ideas
with a pedagogic example based upon the analysis of an autoregressive
time series and two more detailed examples: one on variable selection for
logistic regression and the other on model selection for the analysis
of integrated recapture/recovery data.

### Keywords:

Autoregressive time series; Capture-recapture;
Classical statistics; Information criteria; Logistic Regression;
Markov chain Monte Carlo; Optimisation; Reversible jump MCMC;
Variable selection.

Appeared as Brooks, S. P., Friel, N. and King, R. (2003) "Classical Model Selection
via Simulated Annealing". * Royal Statistical Society, Series B * **65** pp 503-520.